Wavelet Transforms and Localization Operators

Operator Theory: Advances and Applications Vol. 136

Editor: I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel

Editorial Board: J . Arazy (Haifa) A. Atzmon (Tel Aviv) J . A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J . A. Helton (La Jolla) M.A. Kaashoek (Amsterdam) H. G. Kaper (Argonne) ST . Kuroda (Tokyo)

P. Lancaster (Calgary) L.E. Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) J . D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J . Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Upmeier (Marburg) S. M. Verduyn Lunel (Amsterdam) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes)

Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Haimos (Santa Clara) T. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)

Wavelet Transforms and Localization Operators

M. W. Wong

Springer Basel AG

Author:

M.W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J IP3 Canada e-mail: [email protected]

2000 Mathematics Subject Classification 47-02, 47G10, 47G30; 22A10, 42C40, 81S40, 94A12

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data

Wong, Man-Wan: Wavelet transforms and localization operators / M . W. Wong. - Basel; Boston ; Berlin : Birkhäuser, 2002

(Operator theory ; Vol. 136) ISBN 978-3-0348-9478-4 ISBN 978-3-0348-8217-0 (eBook) DOI 10.1007/978-3-0348-8217-0

This work is subject to copyright. A l l rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfiln or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

©2002 Springer Basel A G Originally published by Birkhäuser Verlag in 2002 Softcover reprint of the hardcover 1st edition 2002 Printed on acid-free paper produced from chlorine-free pulp. TCF ° o Cover design: Heinz Hiltbrunner, Basel

ISBN 978-3-0348-9478-4

Contents

Preface vii

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Schatten-von Neumann Classes 11

3 Topological Groups 21

4 Haar Measures and Modular Functions 25

5 Unitary Representations 34

6 Square-Integrable Representations 39

7 VVavelet Transforros 48

8 Jl SaInpling TheoreD1 51

9 VVavelet Constants 53

10 Jldjoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57

11 COD1pact Groups 60

12 Localization Operators 63

13 Sp NorD1 Inequalities, 1 :::; p :::; 00 67

14 Trace Class NorD1 Inequalities 71

15 Hilbert-SchD1idt Localization Operators 79

16 Two-VVavelet Theory 84

17 The VVeyl-Heisenberg Group 90

18 The Jlffine Group..................................................... 98

19 VVavelet Multipliers 107

vi Contents

20 The Landau-Pollak-Slepian Operator 113

21 Products of Wavelet Multipliers 117

22 Products of Daubechies Operators 124

23 Gaussians 129

24 Group Actions and Homogeneous Spaces 141

25 A Unification 143

26 The Affine Group Action on IR 147

References 149

Index 155

Preface

This book is based on lectures given at the Global Analysis Research Center(GARC) of Seoul National University in 1999 and at Peking University in 1999 and2000. Preliminary versions of the book have been used for various topics coursesin analysis for graduate students at York University.

We study in this book wavelet transforms and localization operators in thecontext of infinite-dimensional and square-integrable representations of locallycompact and Hausdorff groups. The wavelet transforms studied in this book, whichinclude the ones that come from the Weyl-Heisenberg group and the well-knownaffine group, are the building blocks of localization operators. The theme thatdominates the book is the spectral theory of wavelet transforms and localizationoperators in the form of Schatten-von Neumann norm inequalities. Several chapters are also devoted to the product formulas for concrete localization operatorssuch as Daubechies operators and wavelet multipliers.

This book is a natural sequel to the book on pseudo-differential operators[103] and the book on Weyl transforms [102] by the author. Indeed, localizationoperators on the Weyl-Heisenberg group are Weyl transforms, which are in factpseudo-differential operators. Details on the perspective and the organization ofthe book are laid out in the first chapter.

This is a book on mathematics and is written for anyone who has taken basicgraduate courses in measure theory and functional analysis. Some knowledge ofgroup theory and general topology at the undergraduate level is also assumed.As such, the book is suitable for graduate students and mathematicians who areinterested in operator theory and harmonic analysis.

and

1 Introduction

The study of wavelet transforms and localization operators undertaken in this bookcan best be motivated by the study of a class of pseudo-differential operators, whichwe now recall.

Let x = (XI, X2, ... , xn) and Y = (Y1, Y2,.'" Yn) be any two points in JRn.The inner product x . Y of x and Y is defined by

n

x· Y = LXjYjj=1

and the norm Ixi of x is defined by

Ixi = (t x;)!J=1

For j = 1,2, ... , we denote 8~ by OJ and we define the partial differential operatorJ

D j byD j = -ioj,

where i is the complex number such that i 2 = -1.

Let a be a multi-index, i.e., a = (al' a2, .. . , an), where at, a2, ... , an arenonnegative integers. Then we define lal, 0C> and DC> by

n

lal = Laj,j=1

respectively. We call lal the length of the multi-index a. For all x in JRn, we alsodefine xC> by

For mE JR, we let sm be the set of all complex-valued functions (J in COO(JRn xJRn) such that for all multi-indices a and {3, there exists a positive constant Cc>,f3for which

I(D~Df(J)(X,~)1 ::; Cc>,f3(l + IWm- f3 , x,~ E JRn.

Let (J E sm. Then we call (J a symbol of order m and we define a linearoperator Ta on the Schwartz space 5 by

(Ta<p)(x) = (21f)-~ r eix'~(J(x,~)~(~)d~, <p E 5,JRnM. W. Wong, Wavelet Transforms and Localization Operators© Springer Basel AG 2002

2 1. Introduction

sup Ix'"Y(Doep)(x)1 < 00xElRn

where the Schwartz space S is defined to be the set of all complex-valued functionsep in Coo (IRn ) such that

for all multi-indices, and 8, and rjJ is the Fourier transform of ep defined by

rjJ(~) = (27r)-~ r e-ix'~ep(x)dx, ~ E IRn .JlRn

We call Tu the pseudo-differential operator associated to the symbol 0". The mostfundamental properties of pseudo-differential operators which are useful in thestudy of partial differential equations are listed in Theorems 1.1-1.4.

Theorem 1.1 If 0" E 8 m1 and T E 8 m2 , then TuTr = T).., where A E 8 m1 +m2 and

'"' (-i) IJLI JL JLA rv L..J -1-(&~O")(&xT).

JL J.L.

The asymptotic expansion A rv I:JL (-~i"l (&rO")(&t:T) in Theorem 1.1 meansthat

( .) IJLIA - L -=;-(&rO")(&!:T) E 8ml+m2-N

IJLI<N J.L.

for all positive integers N.

Theorem 1.2 If 0" E 8 m , then T: = Tr , where T E 8 m and

(-i)IJLIT rv L --,-&!:&ra.

JL J.L.

Here, T: is the formal adjoint ofTu.

In Theorem 1.2, the formal adjoint T: of Tu is defined by

(Tuep, 'l/J)p(lRn) = (ep, T:'l/J)p(lRn)

for all ep and 'l/J in S, where (, )p(lRn) is the inner product in L2 (lRn ). The asymp

totic expansion T rv I:JL (_~tl &t:&ra means that

for all positive integers N.

Before we state Theorem 1.3, let us recall that a symbol 0" in 8 m is said tobe elliptic if there exist positive constants C and R such that

Of course, a pseudo-differential operator Tu is said to be elliptic if 0" is elliptic.

1. Introduction 3

Theorem 1.3 Let (J E sm. Then there exists a symbol T in s-m such that TrTO' =1+ Rand TO'Tr = 1+ S, where Rand S are pseudo-differential operators withsymbols in nkEIR Sk if and only if (J is elliptic.

Pseudo-differential operators with symbols in n kEIR Sk are considered to benegligible in the regularity theory of partial differential equations. Thus, Theorem 1.3 says that elliptic pseudo-differential operators have approximate inverses.Approximate inverses are also known as parametrices.

Theorem 1.4 If (J E So, then TO', initially defined on S, can be uniquely extendedto a bounded linear operator from L 2 (lRn ) into L 2 (lRn ).

Theorem 1.4 allows us to study pseudo-differential operators as boundedlinear operators on L 2 (lRn ).

The theory of pseudo-differential operators that we have described can befound in Hormander [47], Kumano-go [58], Saint Raymond [75], Shubin [81], Stein[87], Taylor [93], Treves [96], Wong [103] and others.

The class of pseudo-differential operators that we have introduced arise naturally in quantum physics. To wit, let us begin with a very simple sketch of thefoundations of physics. It is well known that among the fundamental objects ofstudy are phase spaces and observables. In classical mechanics, the phase space isIRn x IRn and the observables are given by real-valued functions on IRn x IRn . Inquantum mechanics, the phase space is L 2 (lRn ) and the observables are self-adjointoperators on L 2 (lRn ). The mechanism that allows us to pass from classical mechanics to quantum mechanics is known as quantization. The rules of quantization dueto von Neumann [97] consist of replacing the position Xj in the jth coordinate bythe multiplication operator by Xj, and substituting the momentum E;,j in the jth

coordinate by the partial differential operator D j . We make this more transparent by means of an example in relativistic quantum mechanics. The book [1] byAitchison and the book [6] by Bjorken and Drell are good references in relativisticquantum mechanics.

Example 1.5 Consider a relativistic particle of mass m moving on IR under theinfluence of a potential V(x). Then the momentum E;, and the kinetic energy E k

are given, respectively, by

~= mv

Jl-~and

where v is the velocity of the particle and c is the velocity of light. Following thestandard practice of mathematicians, we let m = c = 1. Then

4

So,

1. Introduction

E~ = 1 +e ::::} Ek =~,and hence by von Neumann's rules of quantization, the total energy E of therelativistic particle is given by

E= { ~+V(x),V1- ~ + V(x),

If we let a be the function on JR x JR defined by

Classical,

Quantum.

a(x,~) = VI +~2 + V(x), x,~ E JR,

then the total energy E of the relativistic particle in quantum mechanics is givenby

E = a(x,D),

where D = -id~.

What is the linear operator a(x, D) that arises in quantizing the relativisticenergy in classical mechanics in accordance with von Neumann's rules of quantization? A plausible answer is provided by the class of pseudo-differential operatorsthat we have just introduced. We can try to define a(x, D) as Ta . Is a ---. Ta aquantization? To answer this question, we need to recall that a requirement for acorrect quantization is that the linear operator Ta corresponding to a real-valuedsymbol a should be self-adjoint and hence at least formally self-adjoint. However,by Theorem 1.2, we see that for a real-valued symbol a, T: i:- Ta in general. So,a ---. Ta is not a quantization.

In order to arrive at a correct quantization, let us recall that for all x in JRn ,

(Ta<p)(x) (27f)-~ r eix'~a(x,~)<p(~)~Jan

(27f)-n r r ei(X-Y)'~a(x, ~)<p(y)dy~,Jan Jan

where the last integral is to be understood as an iterated integral in which theintegration with respect to y has to be performed first. In fact, we can associateto a another linear operator Wa on S defined by

for all <p in S. This formula can be traced back to the work [100] by Weyl and hencewe call Wa the Weyl transform associated to the symbol a. In fact, we have thefollowing connection between Weyl transforms and pseudo-differential operators.

1. Introduction 5

Theorem 1.6 Let (j E sm. Then there exists a symbol T in sm such that T" = W'Tand there exists a symbol K, in sm such that WeT = T".

Thus, there is a one-to-one correspondence between the set of pseudo-differential operators and the set of Weyl transforms.

One of the most important properties of Weyl transforms is the followingtheorem.

Theorem 1.7 Let (j E sm. Then WI = Wlf , where WI is the formal adjointofW".

An immediate consequence of Theorem 1.7 is the following fact.

Corollary 1.8 Let (j E sm be real-valued. Then WI = W".

In view of Corollary 1.8, we can conclude that the Weyl transform WeT associated to a real-valued symbol is formally self-adjoint. Hence (j ---t WeT is a goodcandidate for quantization. That this is indeed the correct quantization is explainedin the book [102] by Wong.

Now, let (j E SO be real-valued. Then, by Theorems 1.4 and 1.6, WeT canbe extended uniquely to a bounded linear operator, again denoted by WeT, fromL2(JRn) into L2(JRn). By Corollary 1.8, it can be proved that WeT : L2(JRn) ---t

L 2 (JRn) is self-adjoint and hence a bona fide observable in quantum mechanics. Itis a doctrine in quantum mechanics that the numerical values of the measurementsof the observable WeT : L2 (JRn) ---t L 2 (JRn) are precisely provided by the spectrum~(W,,). Thus, the spectral analysis of the linear operator WeT : L2 (JRn) ---t L2 (JRn)or, equivalently, the study of the spectrum ~(W,,), is of paramount importance inmathematics. Unfortunately, this is often a very difficult problem and very littleis known about the spectrum ~(W,,) of a Weyl transform associated to a symbolin So. Results on the spectra ~(W,,) of Weyl transforms associated to symbols inL 1 (JRn x JRn) U L 2 (JRn x JRn) or modulation spaces satisfying appropriate conditionsdo exist and can be found in, e.g., the papers [16, 17] by Du and Wong, [42] byHeil, Ramanathan and Topiwala, and [70] by Ramanathan and Topiwala. Theconnection between modulation spaces and pseudo-differential operators can alsobe found in [33] by Grochenig, [34] by Grochenig and Heil, and [92] by Tachizawa.

We can now look at some recent developments in wavelet analysis that shedsome light on the spectral analysis of pseudo-differential operators and Weyl transforms.

Let U be the upper half plane given by

U = {(b, a) : bE JR, a > O}.

Then we define the binary operation . on U by

(b1,a1)' (b2,a2) = (b1+a1b2,a1a2)

6 1. Introduction

for all (b1 , al) and (b2, a2) in U. With respect to the binary operation ., U is anon-abelian group in which (0,1) is the identity element and the inverse elementof (b, a) is (- ~, ~) for all (b, a) in U. In fact, U is a locally compact and Hausdorffgroup on which the left and right Haar measures are given by

and dv = dbdaa

respectively. The group U is called the affine group and is non-unimodular.

Let Hi(JR) be the subspace of L 2 (JR) defined by

H~(JR) = {f E L 2 (JR) : supp(j) ~ [O,oon,

where supp(}) is the support of j. The function j is the Fourier transform of fdefined by

~ 1 100

. ~f((,) = lim tn= e- tX XR(x)f(x)dx,R-+oo v 27r -00

where XR is the characteristic function on the interval [-R, R]. We can also defineH:' (JR) to be the subspace of L2 (JR) by

H:' (JR) = {f E L2 (JR) : supp(}) ~ (-00, OJ}.

Hi(JR) and H:'(JR) are known as the Hardy space and the conjugate Hardy spacerespectively. They can be shown to be closed subspaces of L2 (JR).

Let U(Hi(JR)) be the group of all unitary operators on Hi(JR) and let 7r :U ---. U(Hi(JR)) be the mapping defined by

(7r(b, a)f)(x) = )af (x: b), x E JR. (1.2)

Then it can be proved that 7r : U ---. U(Hi(JR)) is an irreducible and unitaryrepresentation of U on Hi(JR). In fact, it can be proved that 7r : U ---. U(H~(JR))is a square-integrable representation of U on H~(JR) in the sense that there existsa function cp in H~(JR) such that IIcpll£2(IR) = 1 and

Such a function cp is known as an admissible wavelet for the square-integrablerepresentation 7r : U ---. U(Hi(JR)) of U on Hi (JR). It can be proved that theadmissibility condition holds if and only if

roo I<P((,W~ < 00.10 (,

1. Introduction 7

The above-mentioned facts on the affine group and the square-integrablerepresentation 1r : U --+ U(H~(l~)) can be found in the paper [43] by Heil andWalnut and the book [104] by Wong.

Let cP be an admissible wavelet for the square-integrable representation 1r :

U --+ U(H~(IR)) of U on H~(IR). Then the wavelet transform A<p : H~(IR) --+ L2 (U)associated to the admissible wavelet cP is defined by

1(A<pJ)(b, a) = -(I, 1r(b, a)cp)L2(R)' (b, a) E U,

VC;

where

C<p = Ll(cp,1r(b,a)cp)L2(R)1 2dj.£(b,a).

The wavelet transform just introduced is the one that has been studied mostextensively in mathematics, statistics, science and engineering. See, for instance,Blatter [7], Chan [10], Chui [11]' Daubechies [13], Gasquet and Witomsky [29],Grochenig [33], Hernandez and Weiss [44]' Holschneider [46], Kaiser [49], Krantz[57], Meyer [63], Pinsky [67], Rao and Bopardikar [71]' Strang and Nguyen [89],Strichartz [90], Walker [98,99], Wojtaszczyk [101] and Young [107] in this connection. However, we prefer to look at the wavelet transform in the context of squareintegrable representations of locally compact and Hausdorff groups on infinitedimensional, separable and complex Hilbert spaces. The group representation thatunderscores this most widely used and most popular wavelet transform is the representation 1r : U --+ U(H~(IR)) of the affine group U on the Hardy space H~(IR)

defined by (1.2). Looking at wavelet transforms in this perspective is a fairly recent venture and has been studied in Ali, Antoine and Gazeau [2], Ali, Antoine,Gazeau and Mueller [3], Du and Wong [19], Du, Wong and Zhang [23], J. He [37],J. He and Liu [38], Z. He [39], Jiang and Peng [48], Kawazoe [51, 52, 53], Liu[61], Liu and Peng [62], Stark [86] and Wong [104, 105]. Of particular interest inthe study of pseudo-differential operators and Weyl transforms is the case whenthe group representation is the Schrodinger representation of the Weyl-Heisenberggroup (WH)n on L 2 (IRn), which we now describe.

Let IRn x IRn = {(q,p) : q, p E IRn} and let Z be the set of all integers. Let(WH)n = IRn x IRn x 1R/21rZ. Then we define the binary operation· on (WH)n by

for all points (ql,Pl, h) and (q2,P2, t2) in (WH)n, where tl, t2 and tl + t2 +ql .P2are cosets in the quotient group 1R/21rZ in which the group law is addition modulo21r. With respect to the binary operation ., (WH)n is a non-abelian group in which(0, 0, 0) is the identity element and the inverse element of (q, P, t) is (-q, -P, -t +q. p) for all (q,p,t) in (WH)n. If we identify 1R/21rZ with the interval [0,21r],then (W H)n can be identified with IRn x IRn x [0, 21r]. It is a locally compact and

8 1. Introduction

Hausdorff group on which the left and right Haar measure is the Lebesgue measuredqdpdt. (WH)n is known as the Weyl-Heisenberg group and it is unimodular.

Let U(L2(I~n)) be the group of all unitary operators on L2(I~n) and let 1r :

(WH)n -+ U(L2(lRn)) be the mapping defined by

(1r(q,p,t)f)(x) = ei(p.x-q.p+t) f(x - q), x E lRn,

for all (q,p,t) in (WH)n and all f in L2(lRn). Then 1r : (WH)n -+ U(L2(lRn))is an irreducible and unitary representation of (W H)n on L2(lRn). In fact, it isa square-integrable representation of (WH)n on L2(lRn) in the sense that everyfunction cP in L2 (lRn ) with Ilcpll£2(JRn) = 1 satisfies the admissibility condition that

r I(cp, 1r(q,p, t)cp)£2(JRn) 12dq dpdt < 00.

J(WH)n

Thus, every function cp in L2 (lRn ) with IlcpIlL2(JRn) = 1 is an admissible waveletfor the square-integrable representation 1r : (WH)n -+ U(L2(lRn)) of (WH)non L2(lRn). The representation 1r : (WH)n -+ U(L2(lRn)) is often called theSchr6dinger representation of (WH)n on L2(lRn). The paper [43] and the book[104] contain the basic facts on the Weyl-Heisenberg group and the Schr6dingerrepresentation.

Let cp E L2 (lRn) be such that IIcpll£2(JRn) = 1. Then the wavelet transformAep : L2 (lRn) -+ L2((WH)n) associated to the admissible wavelet cp is defined by

1(Aepf)(q,p,t) = ~(j,1r(q,p,t)cp)L2(JRn), (q,p,t) E (WH)n,

vCep

where

cep =1 !(cp, 1r(q,p, t)cp)£2(JRn) 12dq dpdt.

(WH)n

In fact, cep = (21r)n+l.

In the paper [12] by Daubechies, a class of bounded linear operatorsDF,ep : L2(lRn) -+ L2(lRn) associated to F in L1 (lRn x lRn) and cp in L2(lRn) withIIcpll£2(JRn) = 1 is studied'in the context of signal analysis. In fact,

(DF,epu,V)L2(JRn) = (21r)-n r r F(q,p)(u,CPq,p)£2(JRn)(cpq,p,V)£2(JRn)dqdpJJRn JJRn

for all u and v in L2 (lRn ), where

CPq,p(x) = eip,xcp(x - q), x E lRn,

for all q and p in lRn . We can prove that

(DF,epU, V)£2 (JRn)

_1_ r F(q,p)(u, 1r(q,p, t)cp)£2(JRn) (1r(q,p, t)cp, V)£2(JRn)dqdpdt..;c; J(WH)n

1. Introduction 9

for all u and v in L2(lRn ). Thus, the linear operator DF,'P : L2(lRn ) --+ L2(lRn ),

which is called the Daubechies operator in [20, 21], is the same as the localizationoperator LF,'P : L2(lRn ) --+ L2(lRn ) associated to the symbol F and the admissiblewavelet cp for the Schr6dinger representation of the Weyl-Heisenberg group (WH)non L 2 (lRn ). A Daubechies operator associated to an admissible wavelet is like awindowed Fourier transform used by Gabor [28] in time-frequency analysis. Theadmissible wavelet plays the same role in the Daubechies operator that the windowplays in the windowed Fourier transform. A symbolic calculus for the product oftwo such localization operators is given in [18].

Daubechies operators are also the same as pseudo-differential operators withanti-Wick symbols studied by Boggiatto, Buzano and Rodino [8], and Shubin [81],among others, in the context of quantization.

It is a result, i.e., Theorem 17.1 in the book [102] by Wong, that if cp is thefunction on IRn defined by

then the Daubechies operator DF,'P : L2(lRn ) --+ L2(lRn ) is the same as the Weyltransform WF*A : L2(lRn

) --+ L2(lRn), where

A(x,~) = 7r-ne-(lxI2+1~1\ x,~ E IRn ,

and F * A is the convolution of F and A given by

Motivated by Daubechies operators, which are localization operators on theWeyl-Heisenberg group (WH)n equipped with the Schr6dinger representation of(WH)n on L 2 (IRn ), we give a systematic study of wavelet transforms and localization operators on locally compact and Hausdorff groups G in this book.These wavelet transforms and localization operators are based on coherent statesparametrized by elements in the group G and admissible wavelets defined in termsof the coherent states. Daubechies operators turn out to be just localization operators on the Weyl-Heisenberg group defined in terms of the coherent states firstenvisaged in the 1926 paper [78] by Schr6dinger. The books [56] by Klauder andSkagerstam and [66] by Perelomov are definitive accounts on coherent states.

In another direction, guided by the Landau-Pollak-Slepian operator in signalanalysis, a theory of wavelet multipliers has been initiated in the paper [41] byHe and Wong, developed in the paper [22] by Du and Wong, and detailed in thebook [104] by Wong. Wavelet multipliers are localization operators on the additivegroup IRn defined in terms of coherent states parametrized by points in IRn .

In the case of localization operators on a locally compact and Hausdorffgroup G endowed with a left Haar measure, the coherent states originate from

10 1. Introduction

a unitary representation of G on an infinite-dimensional, separable and complexHilbert space. For wavelet multipliers, the coherent states stem from the unitaryrepresentation of IRn on L2 (IRn ) given by modulation. Suggested by the book [2] ofAli, Antoine and Gazeau, and the paper [3] of Ali, Antoine, Gazeau and Mueller,the Daubechies operators, localization operators on locally compact and Hausdorffgroups and wavelet multipliers can be looked at as localization operators on homogeneous spaces. The book [2] contains an extensive list of references on coherentstates parametrized by points in a homogeneous space.

The aim of this book is to give a compact account of the Schatten-von Neumann property of localization operators and wavelet multipliers, which include thevery important class of Weyl transforms dubbed as Daubechies operators. Basic tothe construction of localization operators and wavelet multipliers are the wavelettransforms on locally compact and Hausdorff groups, which are also studied inthis book as objects of interest in their own right. We also give a symbolic calculus for Daubechies operators and two symbolic calculi for wavelet multipliers.These symbolic calculi are rudimentary in the sense that only the product of twoDaubechies operators and the product of two wavelet multipliers are considered.

The genesis of the book is as follows. Chapters 2-5 are devoted to background materials which are fundamental in an understanding of the contents inthis book. This book really begins with Chapter 6 in which an exposition of thetheory of square-integrable representations is given. The exposition is based onthe excellent paper [36] by Grossmann, Morlet and Paul. Earlier contributions tothis subject include the works of Carey [9], Dixmier [14], and Duflo and Moore[24]. Admissible wavelets and wavelet transforms, which inevitably arise in thestudy of square-integrable representations, are studied in Chapters 7-11 as objects of interest in their own right. Localization operators associated to admissiblewavelets for square-integrable representations of locally compact and Hausdorffgroups on infinite-dimensional, separable and complex Hilbert spaces are introduced in Chapter 12. The Schatten-von Neumann property in general, the traceclass property and the Hilbert-Schmidt theory in particular, for these localizationoperators are developed in Chapters 13-16. We specialize to the Weyl-Heisenberggroup in Chapter 17, the affine group in Chapter 18, wavelet multipliers in Chapter19 and the Landau-Pollak-Slepian operator in Chapter 20. These four canonicalexamples of localization operators are the main impetus for the abstract theorydeveloped in the book and their importance can hardly be over-emphasized. Product formulas for wavelet multipliers are given in Chapter 21. The more importantformula for the product of two localization operators is derived in Chapter 22 usinga product formula of Grossmann, Loupias and Stein [35] for Weyl transforms. Tothis end, a new twisted convolution for functions in L2 (lRn x IRn ) is introduced,and in Chapter 23 we tackle the problem of finding subspaces M of L2 (lRn x IRn )

such that the new twisted convolution is a binary operation on M. In the lastthree chapters, we introduce localization operators on homogeneous spaces andshow that they can be used to unify the various localization operators studied inthe book into a single theory.

2 Schatten-von Neumann Classes

We devote this chapter to a brief and fairly self-contained study on the Schattenvon Neumann classes. The topics are selected and organized in such a way that wecan use them easily later in this book. Some proofs are omitted whenever they areeasily available in the literature. More comprehensive accounts on the Schatten-vonNeumann classes can be found in Dunford and Schwartz [25], Gohberg, Goldbergand Krupnik [31], Reed and Simon [72], Simon [82] and Zhu [108]. Recommendedreferences for basic functional analysis that we use in this book are Douglas [15],Gohberg and Goldberg [30], Reed and Simon [72], Schechter [76] and Young [106].

Let X be a separable and complex Hilbert space in which the inner productand the norm are denoted by (, ) and 1111 respectively. Let A : X ---. X be abounded linear operator. We define IAI : X ---. X by IAI = (A* A)!, where A*is the adjoint of A. We call IAI : X ---. X the absolute value of A : X ---. X. Abounded linear operator V : X ---. X is said to be a partial isometry if V is anisometry when restricted to the orthogonal complement N(V).l of the null spaceN(V) of V. The starting point is the following well-known theorem, which givesthe polar form of a bounded linear operator on a separable and complex Hilbertspace. A proof can be found on pages 88 and 89 of the book [15] by Douglas or onpages 197 and 198 of the book [72] by Reed and Simon.

Theorem 2.1 Let A : X ---. X be a bounded linear operator. Then there exists apartial isometry V : X ---. X such that A = VIAl. The partial isometry V : X ---. Xis uniquely determined by the condition N(V) = N(A). FUrthermore, the range ofV : X ---. X is equal to the closure of the range of A : X ---. X.

Let A : X ---. X be a compact operator. Then the linear operator IAI : X ---. Xis positive and compact. Let {cpk : k = 1,2, ...} be an orthonormal basis for Xconsisting of eigenvectors of IAI : X ---. X, and let sk(A) be the eigenvalue ofIAI :X ---. X corresponding to the eigenvector cpk, k = 1,2, .... We call sk(A), k =1,2, ... , the singular values of A : X ---. X. A basic result that we need is thefollowing canonical form for compact operators.

Theorem 2.2 Let A : X ---. X be a compact operator. Then we can find an orthonormal basis {Uk: k = 1,2, ...} for N(A).l consisting of eigenvectors ofIAI : X ---. X and an orthonormal set {Vk : k = 1,2, ...} in X such that

00

A = L sk(A)(·, Uk)Vk,k=l

M. W. Wong, Wavelet Transforms and Localization Operators© Springer Basel AG 2002

12 2. Schatten-von Neumann Classes

where sk(A), k = 1,2, ... , are the positive singular values of A : X ---+ X and theseries converges to A strongly.

Proof. By Theorem 2.1, we can write A = VIAl, where V : X ---+ X is thepartial isometry uniquely determined by the condition N(V) = N(A). We writeX = N(V).L EB N(V). Moreover,

x E N(V) = N(A) ~ VIAlx = Ax = 0 ~ IAlx E N(V)

andx E N(V).L,y E N(V) ~ (IAlx,y) = (x, IAly) = O.

Thus, IAlx E N(V).L. Therefore N(V).L and N(V) are invariant subspaces of Xwith respect to IAI : X ---+ X. Thus, we can pick an orthonormal basis {Uk :k = 1,2, ...} for N(V).L and an orthonormal basis {Wk : k = 1,2, ...} for N(V)consisting of eigenvectors of IAI : X ---+ X. Putting the orthonormal bases forN(V).L and for N(V) together, we get an orthonormal basis for X. For k = 1,2, ... ,let sk(A) be the eigenvalue of IAI : X ---+ X corresponding to Uk and let tk be theeigenvalue of IAI :X ---+ X corresponding to Wk. Then

IAlwk = tkwk ~ 0 = A*AWk = IAI2wk = t~Wk ~ tk = 0

for k = 1,2, .... Now, the spectral theorem gives

00

IAI = L sk(A)(·, Uk)Uk,k=l

where the series converges to A strongly. Hence

00

A = LSk(A)("Uk)VUk,k=l

where the series converges to A strongly. For k = 1,2, ... , let Vk = VUk. SinceUk E N(V).L, k = 1,2, ... , and V : X ---+ X is a partial isometry, it follows that{Vk : k = 1,2, ...} is an orthonormal set in X, and the proof is complete. 0

A compact operator A : X ---+ X is said to be in the Schatten-von Neumannclass Sp, 1 ::; p < 00, if

00

2)Sk(A))P < 00.

k=l

It can be shown that Sp, 1 ::; p < 00, is a complex Banach space in which thenorm I1l1 s is given by

p

1

IIAll sp = {~(Sk(A))P } P, A ESp.

2. Schatten-von Neumann Classes 13

We let Soo be the C*-algebra B(X) of all bounded linear operators on X. Thus,IllIsoo = 1111*, where 1111* denotes the norm in B(X). It is obvious that Sp ~ Sq, 1 ~p ~ q ~ 00.

It is customary to call Sl the trace class and S2 the Hilbert-Schmidt class.

Proposition 2.3 Let A : X ---+ X be a positive operator. If there exists an orthonormal basis {CPk : k = 1,2, ...} for X such that

00

~)ACPk'CPk) < 00,

k=l

then A : X ---+ X is compact.

Proof. Let B = VA and let {CPk : k = 1,2, ...} be an orthonormal basis for X forwhich

00

L(Acpk, CPk) < 00.

k=lThen

00 00

L IIBcpkll 2= L(Acpk, CPk) < 00.

k=l k=lNow, there exist complex numbers bjk, j, k = 1,2, ... , such that

00

BCPk = L bjkCPj,j=l

where the convergence of the series is in X, and the Parseval's identity gives

00

IIBcpkl12

= L Ibj kl 2

j=l

for k = 1,2, . ... Thus, by (2.1) and (2.2),

00 00

L Ibjk l2

= L IIBcpkl12 < 00.

j,k=l k=l

For any positive integer N, let

(2.1)

(2.2)

(2.3)

1 ~ j,k ~ N,

otherwise.(2.4)

Let BN : X ---+ X be the bounded linear operator given by

~f=1 bffccpj,

0,

1 ~ k ~ N,

k>N.(2.5)


Then BN : X --+ X is a finite rank operator and for all x in X, we get, by Schwarz'inequality and Parseval's identity,

00

< L I(X,CPk)III(B - BN)CPkllk=1

By (2.2), (2.4) and (2.5),

(B - BN)'" ~ {

So, by (2.7) and Parseval's identity,

'£';N bjkCPj,

'£';1 bjkCPj,

1 ::; k ::; N,

k>N.(2.7)

1 ::; k ::; N,

k>N.(2.8)

00

L II(B - B N)CPkI1 2

k=1

N 00 00 00

L L Ibjk l2 + L L Ibj kl 2

k=1 j=N k=N j=1ex:> CX) 00 00

< L L Ibj kl2 + L L Ibjk l

2--+ 0 (2.9)

j=N k=1 k=N j=1

as N --+ 00. Thus, by (2.6) and (2.9), BN --+ B in the norm of B(X). Hence B iscompact. Since A = B 2

, it follows that A : X --+ X is compact. 0

We have the following criterion for a positive operator to be in the traceclass 81.

Proposition 2.4 Let A : X --+ X be a positive operator such that00

for all orthonormal bases {CPk : k = 1,2, ...} for X. Then A : X --+ X is in thetrace class 81,


Proof. By Proposition 2.3, A : X --. X is compact. Let {¢k : k = 1,2, ...} be anorthonormal basis for X consisting of eigenvectors of A : X --. X. Let s k (A) bethe eigenvalue of A : X --. X corresponding to the eigenvector ¢k, k = 1,2, ....Then

00 00

L sk(A) = L(A¢k, ¢k) < 00,

k=l k=l

and the proof is complete. 0

We have the following criterion for a compact operator A : X --. X to be inthe trace class 8 1 . It is worth noting that positivity is no longer required.

Proposition 2.5 Let A : X --. X be a compact operator such that

00

L I(Acpk,¢k)1 < 00

k=l

for all orthonormal sets {CPk : k = 1,2, ...} and {¢k : k = 1,2, ...} in X. ThenA: X --. X is in 8 1 ,

Proof. Using the canonical form for compact operators given by Theorem 2.2, weget

00

A = L sk(A)(-, Uk)Vk,k=l

where {Uk: k = 1,2, ...} is an orthonormal basis for N(A)-L consisting of eigenvectors of IAI : X --. X, Vk = VUk, k = 1,2, ... , and the series converges to Astrongly. Thus,

00 00

LI(Auj,vj)1 = LSj(A) < 00,

j=l j=l

and the proof is complete. D

The following proposition is some sort of a converse of Proposition 2.4.

Proposition 2.6 Let A : X --. X be a bounded linear operator in the trace class 81 .

Then 2:%:1 (Acpk' CPk) is absolutely convergent for all orthonormal bases {CPk : k =1,2, ...} for X. Moreover, the sum of the series is independent of the choice of theorthonormal basis.

Proof. Let {CPk : k = 1,2, ...} be an orthonormal basis for X and let {¢k : k =1,2, ...} be an orthonormal basis for X consisting of eigenvectors of IAI : X --. X.Then, by Parseval's identity,

00

L(Acpj, CPj)j=l

00

= L(cpj, A*cpj)j=l


ex> ex>

L L('Pj, 7/Jk)(7/Jk, A*'Pj)j=lk=l

ex> ex>

L L('Pj, 7/Jk)(A7/Jk, 'Pj)k=l j=l

ex>

provided that the interchange of the order of summation is justified. But, by Theorem 2.1, we get A = VIAl, where V : X --+ X is a partial isometry. If we letsk(A) be the eigenvalue of IAI : X --+ X corresponding to 7/Jk : k = 1,2, , then,by Schwarz' inequality, Parseval's identity, lI'Pkll = II7/Jkll = 1, k = 1,2, , andIIVII* = 1, we get

ex> ex>

L L 1('Pj, 7/Jk)(A7/Jk, 'Pj)1k=lj=l

1 1

< ~ {t.IC'I';'",,)I'r{t.I(A,h, '1';)1'rex> 00

L IIA7/Jkll = L IIVIAI7/Jkllk=l k=l

ex>

= L sk(A) IIV7/Jk II :s; IIAllsl < 00,

k=l

and this completes the proof. D

In view of Proposition 2.6, we can define the trace tr(A) of any boundedlinear operator A : X --+ X in the trace class 8 1 by

ex>

tr(A) = L(A'Pk' 'Pk),k=l

where {'Pk : k = 1,2, ...} is any orthonormal basis for X.

Proposition 2.7 Let A : X --+ X be a positive operator in the trace class 8 1 , Then

IIAllsl = tr(A).

Proof. Using the definition of tr(A),

ex>

tr(A) = L(A7/Jk' 7/Jk),k=l

(2.10)


where {'l/Jk : k = 1,2, ...} is an orthonormal basis for X consisting of eigenvectorsof A : X --+ X. If sk(A) is the eigenvalue of A : X --+ X corresponding to theeigenvector 'l/Jk, k = 1,2, ... , then, by (2.10),

00

tr(A) = L sk(A) = IIAlls!,k=l


The following theorem gives a criterion for a bounded linear operator A :X --+ X to be in the Hilbert-Schmidt class S2 and a formula for the norm IIAlls2of A : X --+ X in S2.

Proposition 2.8 Let A : X --+ X be a bounded linear operator such that

00

L II A'PkIl2 < 00

k=l

for all orthonormal bases {'Pk : k = 1,2, ...} for X. Then A : X --+ X is in theHilbert-Schmidt class S2 and

00

IIAII~2 = L II A'PkI1 2,

k=l

where {'Pk : k = 1,2, ...} is any orthonormal basis for X.

We need the following lemma in the proof of Proposition 2.8.

Lemma 2.9 Let {'Pk : k = 1,2, ...} be an orthonormal basis for X consisting ofeigenvectors of a bounded linear operator A : X --+ X. For k = 1,2, ... , let Ak be theeigenvalue of A : X --+ X corresponding to 'Pk. Then the spectrum of A : X --+ Xis the closure in C of the set {Ak : k = 1,2, ...}.

Proof. Let A be any complex number which is not in the closure of the set{Ak : k = 1, 2, ...}. Then there exists a positive number 15 such that

IAk - AI ~ 15, k = 1,2, ....

Let y EX. Then let x E X be defined by

(2.11)

where the convergence of the series is understood to be in X. By (2.11) andParseval's identity,

(A - AI)x = f ~' ~kl (A - AI)'Pk = f(y, 'Pk)'Pk = y,k=l k=l


where I is the identity operator on X. Therefore the linear operator A - AI : X --+

X is onto. Moreover, for any x in X, we get, by Parseval's identity,

1

II(A - AI)xll {~IAk _ AI 2 1(x, cpk)12} '2

1

> 0 {~I(X' cpk)12} '2 = ollxll.

Therefore A belongs to the resolvent set of A : X --+ X, and the proof is complete.D

Proof of Proposition 2.8. Let B = A* A. Then B : X --+ X is a positive operator.Since

00 00 00

~)Bcpk'cpk) = ~)A*cpk' cpk) = L II Acpkl1 2 < 00

k=1 k=1 k=1for all orthonormal bases {cpk : k = 1,2, ...}, it follows from Proposition 2.4 thatB : X --+ X is in the trace class 81 . Since IAI = VB, it follows that IAI : X --+ Xis compact. By Theorem 2.1, we can write A = VIAl, where V : X --+ X is apartial isometry. Therefore A : X --+ X is compact. Let {'l/Jk : k = 1,2, ...} bean orthonormal basis for X consisting of eigenvectors of IAI : X --+ X and fork = 1,2, ... , let sk(A) be the eigenvalue of IAI : X --+ X corresponding to 'l/Jk'Then

B'l/Jk = IAI 2'l/Jk = (sk(A))2'l/Jk' k = 1,2, ....

So, by Lemma 2.9, the spectrum of B : X --+ X is the closure in C of the set{(sk(A))2 : k = 1,2, ...}. Thus,

00 00

L(Sk(A))2 = L sk(B) < 00,

k=1 k=1

and hence A : X --+ X is in the Hilbert-Schmidt class 82 • By Propositions 2.6 and2.7, we get

00 00

II Alb L(Sk(A))2 = L sk(B)k=1 k=1

00

IIBllsl = tr(B) = L(Bcpk, cpk)k=1

00 00

L(A* ACPk,CPk) = L IIAcpk11 2,k=1 k=1

where {cpk : k = 1,2, ...} is any orthonormal basis for X. This completes the proof.D


In order to obtain some information on the Schatten-von Neumann classSp, 1 :s P :s 00, we need interpolation theory, which we now recall. Good referencesfor interpolation theory include the book [5] by Bergh and Lofstrom, the book [77]by Schechter and the book [108] by Zhu.

Let Bo and B l be complex Banach spaces in which the norms are denotedby IllIBo and 1IIIB1 respectively. We say that Bo and B l are compatible if thereis a complex vector space V such that Bo ~ V and B l ~ V. If this is the case,then the subspaces Bo n B l and Bo + B 1 of V are complex Banach spaces whenequipped with the norms 1111 BonB, and 1111 Bo+B, given by

for all v in Bo n B l , and

for all v in Bo + B l , respectively.

Let Bo and B 1 be compatible Banach spaces. A complex Banach space B iscalled an intermediate space between Bo and B 1 if

where the inclusions are continuous. An intermediate space B between Bo andB l is said to be an interpolation space between Bo and B l if any bounded linearoperator on Bo+B l , which is bounded from Bk into Bk, k = 0, 1, is also boundedfrom B into B.

Let S = {z E C : °< Re z < I} and let B be any complex Banach space. Afunction 1 : S -+ B is said to be analytic on S if for any bounded linear functionalb' on B, the complex-valued function b' 0 I: S -+ C is analytic on S.

Let Bo and B 1 be compatible Banach spaces. Then we define F(Bo, Bd tobe the set of all bounded and continuous functions 1 from the closure S of S intoBo+ B l such that 1 is analytic on S and the mappings

IR 3 Y ~ I(k + iy) E B k , k = 0,1,

are continuous from IR into Bk' k = 0,1. Then it can be shown that F(Bo, Bd isa complex Banach space with respect to the norm IIIIF given by

II/I1F = max sup 11/(k + iy)IIBk' 1 E F(Bo, B l ).k=O,l yEIR

For any number () in [0,1]' we let Be be the subspace of Bo + B l consistingof all elements bin Bo+ B l such that b = I(()) for some 1 in F(Bo, B l ). Then we


can show that Be is a complex Banach space with respect to the norm 11110 givenby

IIbll B8 = inf IlfllF, bE Bo,b=f(O)

and Bo is an interpolation space between Bo and B 1 • We denote Bo by [Bo, B 1]0.

We have the following result on the boundedness of linear operators from aninterpolation space between a pair of compatible Banach spaces into the corresponding interpolation space between another pair of compatible Banach spaces.

Theorem 2.10 Let Bo, B 1 and Bo, B1 be two pairs of compatible Banach spaces.Let A be a bounded linear operator from Bo + B 1 into Bo + B1 such that A is abounded linear operator from B k into Bk with norm ::; Mk, k = 0,1. Then for anynumber () in (0,1), A is a bounded linear operator from [Bo, B 1]e into [Bo,B1]0

with norm ::; MJ-o Mf.

The Lebesgue space LP(M, J.L), where (M, J.L) is a measure space, and theSchatten-von Neumann class Sp, 1 ::; p ::; 00, are standard examples of interpolation spaces. These facts are made precise by the following theorem.

Theorem 2.11 For 1 ::; P ::; 00,

and[SI, Soo]..1- = Sp,

1"

where (M, J.L) is a measure space and p' is the conjugate index of p.

3 Topological Groups

This chapter contains the basic information on topological groups. Analysis ontopological groups requires a study of Haar measures and modular functions, whichwe give in the next chapter. Basic references include Folland [27] and Pontryagin[68]. The book [45] is a good reference for the basic group theory used in thisbook. As for general topology, the books [54] and [64] by Kelley and Munkresrespectively are standard references.

Let G be a group on which the binary operation is denoted by '. Suppose thatG is also a topological space such that the mappings G x G 3 (g, h) f-+ 9 . h E Gand G 3 9 f-+ g-l E G are continuous, where g-l is the inverse of g. Then we callG a topological group.

Some remarks on the definition of a topological group are in order.

Remark 3.1 The continuity of the mapping G x G 3 (g, h) f-+ g. h E G means thatfor all 9 and h in G and any neighborhood W of 9 . h, we can find a neighborhoodU of 9 and a neighborhood V of h such that u·v E W for all u in U and all v in V.

Remark 3.2 The continuity of the mapping G 3 9 f-+ g-l E G means that for all9 in G and any neighborhood V of g-l, there exists a neighborhood U of 9 suchthat

u E U =} u- 1 E V.

Remark 3.3 For all g and h in G, g. h is also denoted by gh.

Theorem 3.4 Let G be a topological group. Then G is a To-space {:} G is a T1-space{:} G is a T2 -space {:}

n V = {e},VEr,eEV

where T is the topology in G and e is the identity element in G.

(3.1)

Let us recall that G is a To-space if for any two distinct elements in G, thereexists an open set that contains exactly one of them. G is a T1-space if for anytwo distinct elements 9 and h in G, there exist two open sets U and V such that9 E U, h E V, 9 f/. V and h f/. U. G is a T2-space means that for any two distinctelements 9 and h in G, we can find open sets U and V for which 9 E U, h E Vand U n V = cP. A T2-space is also known as a Hausdorff space.


22 3. Topological Groups

Before the proof of Theorem 3.4, we introduce some notation. Let U ~ G.Then we denote the set {u- 1 : u E U} by U-1. For all elements 9 and h in G, welet gU and Uh be the sets defined by

gU = {gu : u E U} and Uh = {uh: u E U}.

If U ~ G and V ~ G, then we define U . V by

U . V = {uv : u E U, v E V}.

Proof of Theorem 3.4. Suppose that G is a To-space and let 9 and h be twodistinct elements in G. Then we can find an open set U that contains either 9 orh, but not both. We assume that 9 E U and h rt. U. Let

V = hU- 1g = {hu- 1g : u E U}.

Then it is easy to see that V is an open set and hE V. Now, 9 rt. V. Indeed, if 9 E V,then there exists an element u in U such that 9 = hu-1g. Thus, h = u E U andthis is a contradiction. Hence G is a T1-space. Next, suppose that G is a T1-spaceand let 9 and h be two distinct elements in G. Let H = G - {h- 1g}, i.e., H isthe complement of the set {h- 1g} in G. Then e E H. Let u E H. Then u =I- h- 1g.Since G is a T 1-space, it follows that there exist open sets 0 1 and 02 such thatu E 01> h-1g E O2 , U rt. 02 and h-1g rt. 0 1 , Thus, 0 1 is an open neighborhoodof u, which is contained in H. Therefore every point in H is an interior point.Hence H is an open set. Using the continuity of the multiplication and the factthat ee = e, there exist open neighborhoods U and V of e such that

u E U, v E V => uv E H.

LetW = Un u- 1 nvnv-1 .

Then W is an open set and e E W. Furthermore, W = W- 1 and

W·W- 1 ~ H.

If there exist elements 8 and t in W such that g8 = ht, then 9 = ht8-1 E hH =G - {g}, and this is impossible. Therefore gW n hW = <p. In other words, G is aT2-space. Now, suppose that G is a T2-space. Then for all 9 in G such that 9 =I- e,there exist open sets U and V for which 9 E U, e E V and Un V = <p. Hence(3.1) is proved. Finally, suppose that (3.1) is valid and let 9 and h be two distinctelements in G. Then g-lh =I- e. So, there exists an open set U such that e E U andg-lh rt. U. Therefore 9 E gU and h rt. gUo This proves that G is a To-space, andthe proof of Theorem 3.4 is complete. 0

We assume that all topological groups in this book are T2-topological groups,which we simply call Hausdorff groups. In view of Theorem 3.4, this is not a severerestriction.

3. Topological Groups 23

Let G be a Hausdorff group. Let f be a complex-valued function on G.Suppose that for all positive numbers c, there exists a neighborhood U of e suchthat

g-lh E U =? If(g) - f(h)1 < c.

Then we say that f is left uniformly continuous on G. If we replace g-lh E U byhg- 1 E U in the definition, then we have the notion of right uniform continuityon G.

Remark 3.5 Let U be any neighborhood of the identity element e in G. Then wecan always find a neighborhood V of e such that V ~ U and V is symmetric, i.e.,V = V- 1 . Indeed, we can pick V to be the neighborhood un U- 1 . Thus,

and

Therefore the roles of 9 and h in the definitions of left and right uniform continuityare symmetric.

Let f be a continuous and complex-valued function on G. Then the supportsupp(J) of the function f is defined to be the closure ofthe set {g E G : f (g) =I- O}.We let Co(G) be the set of all continuous and complex-valued functions f on Gsuch that supp(J) is compact.

Proposition 3.6 Let G be a Hausdorff group. Let f E Co(G). Then f is left andright uniformly continuous on G.

Proof. Let K = supp(J). Then for all positive numbers c and for all 9 in G, thereexists an open set Ug such that 9 E Ug and

hE Ug =} If(g) - f(h)1 < c.

Let W g = g-lUg . Then Wg is an open set and e E W g . By Remark 3.5, we canfind a symmetric and open set Vg of e such that Vg ~ Wg and ~. Vg ~ Wg . Now,{gVg : 9 E K} is an open cover of K. Since K is compact, we can find elementsg1, g2,···, gN in K such that

N

K c U(gjVgj ).

j=l

LetN

V = nVgj '

j=l

Then V is a symmetric and open neighborhood of e. Now, let 9 and h be elementsin G. If 9 E K and g-lh E V, then there exists an integer j, j = 1,2, ... , N, suchthat 9 E gj Vgj ' So, on one hand, 9 E gjWgj = Ugj ' and hence

(3.2)

24 3. Topological Groups

If(h) - f(gj)1 < E.

Thus, by (3.2) and (3.3), we get

If(g) - f(h)1 < 2E.

(3.3)

If hE K and g-lh E V, then, by Remark 3.5, h-1g E V, and by what we havejust proved, we also get If(g) - f(h)1 < 2E. If g-lh E V, 9 f/. K and h f/. K,then it is obvious that f(g) = f(h) = O. Hence we have proved that f is leftuniformly continuous on G. The proof for right uniform continuity is similar andhence omitted. 0

A complex-valued function on G that is left and right uniformly continuouson G is said to be uniformly continuous on G.

4 Haar Measures and Modular Functions

We assume a basic knowledge of measure theory on locally compact and Hausdorfftopological spaces, which can be found in [73] by Royden and [74] by Rudin, amongothers.

Let p, be a nonzero Radon measure on a locally compact and Hausdorff groupG. This means that p, is a nonzero Borel measure on G such that p,(K) < 00 forall compact subsets K of G,

p,(B) = inf p,(U)UE'T,B~U

for all Borel subsets B of G and

p,(U) = sup p,(U)K~U

(4.1)

(4.2)

for all U in the topology T, where the supremum in (4.2) is taken over all compactsets K contained in U. If

p,(gB) = p,(B) (4.3)

for all 9 in G and all Borel subsets B of G, then we call p, a left Haar measure onG'If

(4.4)

for all 9 in G and all Borel subsets B of G, then we call p, a right Haar measureon G. A Radon measure that is both a left and right Haar measure on G is saidto be a Haar measure on G.

Remark 4.1 Conditions (4.1) and (4.2) are, respectively, known as the outer regularity and the inner regularity of the measure p,. Conditions (4.3) and (4.4) give,respectively, the left invariance and the right invariance of the measure p,.

The following theorem is a fundamental result in measure theory on locallycompact and Hausdorff groups. The proof is not necessary for an understandingof the contents in this book. We are content with the statement of the theoremand the proof is omitted. For a proof, see Section 2.2 of the book [27] by Folland.

Theorem 4.2 There exists a left Haar measure on a locally compact and Hausdorffgroup.


26 4. Haar Measures and Modular Functions

Let J.1- be a Radon measure on a locally compact and Hausdorff group G. Iffor all Borel subsets B of G, we define jJ,(B) by

jJ,(B) = J.1-(B- 1), (4.5)

(4.6)

then it is easy to see that ji is a Radon measure on G and we have the followingsimple proposition.

Proposition 4.3 J.1- is a left Haar measure on G if and only if ji is a right Haarmeasure on G.

Proof. Suppose that J.1- is a left Haar measure on G. Then for all 9 in G and allBorel sets B, we can use (4.5) and the left invariance of J.1- to get

Therefore ji is a right Haar measure on G. Conversely, if ji is a right Haar measureon G, then for all 9 in G and all Borel sets B, we get, by (4.5),

J.1-(gB) = jJ,((gB)-l) = ji(B- 1g- 1) = ji(B-1) = J.1-(B).

Therefore J.1- is a left Haar measure on G. 0

We can now give a corollary of Theorem 4.2 and Proposition 4.3.

Corollary 4.4 There exists a right Haar measure on a locally compact and Hausdorff group.

In order to investigate the "uniqueness" of the left Haar measure on a locallycompact and Hausdorff group, we need Propositions 4.5 and 4.6.

Proposition 4.5 Let J.1- be a left Haar measure on a locally compact and Hausdorffgroup G. Then J.1-(U) > 0 for all nonempty and open subsets U of G, and

fa f(g)dJ.1-(g) > 0

for all nonzero and nonnegative real-valued functions f in Co (G).

Proof. There exists a compact subset K of G such that J.1-(K) > O. Otherwise,J.1-(K) = 0 for all compact sets K, and then, using the inner regularity of J.1-,

J.1-(G) = sup J.1-(K) = 0,Kt;;.a

where the supremum is taken over all compact subsets K of G. Thus, J.1- is the zeromeasure. Let U be any nonempty and open subset of G. Then {gU: 9 E K} is anopen cover of K. Hence there exist gl, g2, ... , 9N in G for which

N

K c U(gjU).j=l

4. Haar Measures and Modular Functions

Therefore, by (4.6) and the left invariance of JL, we get

N

o< JL(K) $ L JL(gjU) = N JL(U).j=l

27

Next, let f be a nonzero and nonnegative real-valued function in Co(G). Thenf(go) > 0 for some go E G. By continuity, there exists an open neighborhood U ofgo such that

9 E U => If(g) - f(go)1 < f(go) => f(g) > f(go)22'

So,

fc f(g)dJL(g) ~Lf(g)dJL(g) > /(;0) LdJL(g) = f(;o) JL(U) > 0,

and the proof is complete. o

Proposition 4.6 Let JL be a left Haar measure on a locally compact and Hausdorffgroup G. Let cp E Co(G). Then the function

G 3 9 ~ fc cp(kg)dJL(k) E C

is continuous on G.

Proof. Let go E G and let c be any positive number. Then, by Proposition3.6, cp is left uniformly continuous on G. So, there exists a symmetric and openneighborhood V of e such that

(4.7)

Since G is locally compact, it follows that the identity element e has a compactneighborhood N. Without loss of generality, we can assume that V c N. Then forall k in G and all gin goV, we get, by (4.7),

and hence

where K = supp(cp), and the proof is complete. o

Theorem 4.7 Let J.L and 1/ be left Haar measures on a locally compact and Hausdorffgroup G. Then there exists a positive number a such that JL = al/.


Proof. Let <p be a nonzero and nonnegative real-valued function in Co(G). Thenfor all f in Co (G), let F be the function on G x G defined by

f(g)<p(hg)F(g, h) = fa <p(kg)dv(k) ' g, hE G. (4.8)

Then, by Propositions 4.5 and 4.6, the denominator in (4.8) is a positive andcontinuous function on G and hence F is a continuous function on G x G withcompact support. Using the left invariance of f.l and Fubini's theorem,

So, by (4.8), (4.9) and Fubini's theorem,

Let

1 f(h-l)AU, <Pi v) = a fa <p(kh-1 )dv(k) dv(h).

Then, by (4.10) and (4.11), we get

fa f(g)df.l(g) = AU, <Pi v) fa <p(g)df.l(g).

Similarly,

fa f(g)dv(g) = AU, <Pi v) fa <p(g)dv(g).

If we let a be the number defined by

fa <p(g)df.l (g)a = fa <p(g)dv(g) '

(4.11)

(4.12)

(4.13)

(4.14)


then, by (4.12), (4.13) and (4.14),

l f(g)d/.L(g) = a).,,(j, cp; v) l cp(g)dv(g) = al f(g)dv(g).

29

(4.15)

Obviously, a > 0 and is independent of f. Since f is an arbitrary function inCo(G), we conclude from (4.15) that /.L = av. 0

Remark 4.8 For the validity of Fubini's theorem used in the proof of Theorem 4.7,we need the Haar measures /.L and v on the locally compact and Hausdorff groupG to be a-finite. The a-finiteness is guaranteed if we assume that G is a-compact,which means that G is a countable union of compact sets. Thus, we assume that allthe locally compact and Hausdorff groups encountered in this book are a-compact.

We can now develop the properties of Haar measures on locally compact andHausdorff groups.

Proposition 4.9 Let /.L be a left Haar measure on a locally compact and Hausdorffgroup G. Then /.L is a finite measure if and only if G is compact.

Proof. Suppose that /.L(G) < 00. By the inner regularity of /.L, we can find acompact subset K of G such that /.L(K) > O. Consider the family {gK : 9 E G}of compact subsets of G. Let gl E G. If (gK) n (glK) =f. ¢ for all 9 in G, thenwe stop. If there exists an element 9 in G such that (gK) n (glK) = ¢, then wepick such an element and call it g2. If (gK) n U~=l (gjK) =f. ¢ for all 9 in G, thenwe stop. Otherwise, we pick an element g3 in G such that glK, g2K and g3K arepairwise disjoint. Repeating this argument, we can get elements gl, g2,"" gN inG for which the sets glK, g2K, ... ,gNK are pairwise disjoint and

N

(gK) n U(gjK) =f. ¢, 9 E G.j=l

Otherwise, we can get an infinite sequence {gjK}~l of pairwise disjoint compactsubsets of G, and hence, using the left invariance of /.L, we get

00 00

/.L(G) ~ L/.L(gjK) = L/.L(K) = 00,

j=l j=l

and consequently a contradiction. Thus,

N

G = U(gjK) . K-l,j=l

and the proof is complete. o


Let j.L be a left Haar measure on a locally compact and Hausdorff group G.Let 9 E G. Then the mapping G 3 h.-. hg EGis a homeomorphism of G onto G.If for all Borel subsets B of G, we define j.Lg(B) by

j.Lg(B) = j.L(Bg), (4.16)

then it can be checked easily that j.Lg is also a left Haar measure on G. Therefore,by Theorem 4.7, we can find a positive number 6.(g) such that

(4.17)

If v is another left Haar measure on G, then, by Theorem 4.7 again, there is apositive number a for which v = aj.L. Hence, by (4.16) and (4.17),

vg = aj.Lg = a6.(g)j.L = 6.(g)v.

This observation tells us that the positive function 6. on G is independent of thechoice of the left Haar measure j.L. It is determined by the group G and is calledthe modular function on G.

Let f be a complex-valued function on G. For all elements gin G, we definethe right translation Rgf of f by 9 by

(Rgf)(h) = f(hg-1), h E G.

Now, let B be a Borel subset of G. Then for all gin G,

(4.18)

(4.19)

hE Bg,= XBg(h).

h rt Bg,

where XB and XBg are the characteristic functions on Band Bg respectively.Indeed, for all h in G, we get, by (4.18),

1 {I, hg-1 EB,(RgXB)(h) = XB(hg- ) =

0, hg-1 rt B,

Consequently, by (4.16), (4.17) and (4.19),

[(RgXB)(h)dj.L(h) = j.L(Bg) = 6.(g)j.L(B) = 6.(g) [XB(h)dj.L(h). (4.20)

Therefore for all functions f in L1(G,j.L), we get from (4.20)

[(Rgf)(h)dj.L(h) = 6.(g) [f(h)dj.L(h)

or, by (4.18),

[f(h9-1)dj.L(h) = 6.(g) [f(h)dj.L(h). (4.21 )

4. Haar Measures and Modular Functions 31

Proposition 4.10 The modular function !:i. on a locally compact and Hausdorffgroup G is a continuous function on G such that

!:i.(gh) = !:i.(g)!:i.(h), g, hE G.

Proposition 4.10 tells us that the modular function !:i. : G ----> IR x is a grouphomomorphism, where IR x is the group of all positive numbers with respect tomultiplication.

Proof of Proposition 4·10. Let J.L be a left Haar measure on G. Let f E Co(G) besuch that

fa f(h)dJ.L(h) = 1.

Then, by (4.21) and Proposition 4.6, !:i.: G ----> (0,00) is continuous. Now, for anyBorel subset B of G, we get, by (4.16) and (4.17),

!:i.(gh)J.L(B) = J.L(Bgh) = !:i.(h)J.L(Bg) = !:i.(h)!:i.(g)J.L(B)

for all 9 and h in G. Thus, by (4.22),

!:i.(gh) = !:i.(g)!:i.(h), g, hE G,

(4.22)

and the proof of Proposition 4.10 is complete. 0

A locally compact and Hausdorff group G is said to be unimodular if

!:i.(g) = 1, 9 E G.

Proposition 4.11 Let J.L be a left Haar measure on a locally compact and Hausdorffgroup G. Then G is unimodular if and only if ji, = J.L, where ji, is defined by (4.5).

We need the following lemma for the proof of Proposition 4.11.

Lemma 4.12 Let J.L be a left Haar measure and let!:i. be the modular function on alocally compact and Hausdorff group G. Then for all Borel subsets B of G,

where jl is defined by (4.5).

Proof. Let f E Co(G) be such that

fa f(h)dJ.L(h) = 1.

Then, by (4.21) and (4.23),

!:i.(g-l) = faf(h9)dJ.L(h),

(4.23)


and hence for all Borel subsets B of G, we can use Fubini's theorem, the leftinvariance of J..L and (4.5) to get

h6.(g-1)dJ..L(9)

fa XB (g)6.(g-1)dJ..L(9)

fa (faXB(9)f(h9)dJ..L(h) ) dJ..L(g)

fa (faXB(9)f(h9)dJ..L(9) ) dJ..L(h)

fa (fa XB (h- 1g)f(9)dJ..L(9)) dJ..L(h)

fa (fa XB (h- 1g)dJ..L(h)) f(g)dJ..L(g)

fa (fa XB (h- 1)dJ..L(h)) f(g)dJ..L(g)

faXB-1(h)dJ..L(h) = J..L(B- 1) = jj(B).

oProof of Proposition 4.11. Suppose that G is unimodular. Then 6.(g) = 1 for allgin G. Let B be a Borel subset of G. Then, by Lemma 4.12,

ji(B) = L6.(g-1)dJ..L(g) = LdJ..L(g) = J..L(B).

Thus, ji = J..L. Conversely, suppose that ji = J..L. If 6.(go) =I- 1 for some go in G. Then,by continuity, we can find an open neighborhood U of go such that 6.(g-1) =I- 1for all 9 in U. To be specific, we assume that 6.(g-1) > 1 for all 9 in U. Therefore,by Lemma 4.12,

and this is a contradiction. Therefore 6.(g) = 1 for all gin G, i.e., G is unimodular.o

Proposition 4.13 Every abelian, locally compact and Hausdorff group is unimodular.

Proof. Let 9 E G and let B be any Borel subset of G. Then, using (4.16), theassumption that G is abelian and the left invariance of J..L, we get

J..Lg(B) = J..L(Bg) = J..L(gB) = J..L(B). (4.24)


On the other hand, by (4.17),

By (4.24) and (4.25), we get

and hence .6.(g) = 1 for all g in G. Therefore G is unimodular.

Proposition 4.14 Every compact and Hausdorff group is unimodular.

33

(4.25)

D

Proof. Let.6. be the modular function on G. Then, using the fact that.6. : G -> IR x

is a group homomorphism, we get

(4.26)

for all positive integers n. If there exists an element go in G for which .6.(go) =I- 1,then .6.(go) > lor .6.(gol) > 1. Let us assume that .6.(go) > 1. Then, by (4.26),.6.(go) -> 00 as n -> 00. This contradicts the fact that .6. is a continuous functionon the compact group G. Therefore .6.(g) = 1 for all g in G, i.e., G is unimodular.

Di

The Weyl-Heisenberg group and the Heisenberg group, to be studied in Chap-ter 17, are concrete unimodular groups, which are neither abelian nor compact.The affine group, to be studied in Chapter 18, is a non-unimodular group.

We assume throughout the book that a locally compact and Hausdorff groupG is always equipped with a left Haar measure, which we denote by f.J,.

5 Unitary Representations

This chapter is a brief account on unitary representations of locally compact andHausdorff groups on separable and complex Hilbert spaces. Only the most basic topics are touched on in this chapter. The more advanced theory of squareintegrable representations is given in the next chapter. A good reference for thischapter is Chapter 3 of the book [27] by Folland. A more comprehensive treatiseis the book [55] by Kirillov.

Let G be a locally compact and Hausdorff group. Let X be a separable andcomplex Hilbert space in which the inner product and the norm are denoted by( , ) and 1111 respectively. The group of all unitary operators on X with respect tothe usual composition of mappings is denoted by U(X). A group homomorphism11" : G ---+ U(X) is said to be a unitary representation of G on X if it is stronglycontinuous, i.e., G :3 9 f--+ 1I"(g)x E X is a continuous mapping for all x in X.The Hilbert space X is called the representation space of 11" : G ---+ U(X) and thedimension of X is known as the dimension or the degree of 11" : G ---+ U(X).

Remark 5.1 We can replace the requirement that 11" : G ---+ U(X) be stronglycontinuous by the weaker condition of weak continuity, i.e., the condition that thefunction G :3 9 f--+ (1I"(g )x, Y) E C be continuous for all x and y in X. Indeed, let{gj}jEJ be a net in G such that gj ---+ 9 for some gin G, then

(5.1)

for all x and y in X. Since 1I"(g) and 1I"(gj), j E J, are unitary operators on X, itfollows from (5.1) that

111I"(gj)x - 1I"(g)xI1 2 = 111I"(gj)xI1 2+ 1I11"(g)xIl 2- 2Re(1I"(gj)x, 1I"(g)x)

= 211xl12 - 2Re(1I"(gj)x, 1I"(g)x)

---+ 211xl1 2 - 21111"(g)xIl 2 = 211xll2 - 211xll 2 = o.Thus, 11" : G ---+ U(X) is strongly continuous.

A closed subspace M of X is said to be invariant with respect to the unitaryrepresentation 11" : G ---+ U(X) of G on X if 1I"(g)M ~ M for all gin G. {a} and Xare the trivial invariant subspaces. It is important to emphasize that all invariantsubspaces are closed by definition. A unitary representation 11" : G ---+ U(X) of Gon X is said to be irreducible if it has only the trivial invariant subspaces.

A fundamental result in representation theory is the following theorem, whichis usually referred to as Schur's lemma.


5. Unitary Representations 35

Theorem 5.2 A unitary representation 1f : G --t U(X) of a locally compact andHausdorff group G on a separable and complex Hilbert space X is irreducible ifand only if the only bounded linear operators on X that commute with 1f(g) for allg in G are scalar multiples of the identity operator on X.

We need the following lemma to prove Theorem 5.2.

Lemma 5.3 Let M be an invariant subspace of X with respect to the unitary representation 1f : G --t U(X) of G on X. Then the same is true for the orthogonalcomplement M 1. of M in X.

Proof. Let x E M 1.. Then for all g in G and all y in M, we can use the fact that1f : G --t U(X) is a unitary representation to obtain

(1f(g)x,y) = (x, (1f(g))*y) = (x, (1f(g))-l y) = (x, 1f(g-l)y) = 0,

where (1f(g))* is the adjoint of 1f(g). Therefore 1f(g)x E M1. for all gin G. 0

Another ingredient in the proof of Schur's lemma is the spectral theorem forself-adjoint operators on separable and complex Hilbert spaces. References for thespectral theorem abound in the literature. A good one is the book [72] by Reedand Simon.

Proof of Theorem 5.2. Suppose that 1f : G --t U(X) is not irreducible. Let M bean invariant subspace of X with respect to 1f : G --t U(X) such that M =I- {O} andM =I- X. Let P be the orthogonal projection of X onto M. Then P is a boundedlinear operator on X. Moreover, for all gin G, we get

1f(g)Px = 1f(g)x = P1f(g)x, x E M,

and1f(g)Px = °= P1f(g)x, x E M1..

Thus, P is a bounded linear operator on X that commutes with 1f(g) for all gin G,and P is not a scalar multiple of the identity operator on X. Conversely, supposethat A is a bounded linear operator on X such that A is not a scalar multiple ofthe identity operator on X and A commutes with 1f(g) for all 9 in G. Then thebounded linear operators Sand T on X, defined by

S=~(A+A*) and T = -~i(A - A*),2

where A* is the adjoint of A, are self-adjoint. Since 1f : G --t U(X) is a unitaryrepresentation of G on X for all 9 in G and A commutes with 1f(g) for all 9 in G,it follows that

(A*1f(g)x, y) (1f(g)x, Ay) = (x, (1f(g))* Ay)

(x, (1f(g))-l Ay) = (x,1f(g-l)Ay)

(x,A1f(g-l)y) = (A*X,1f(g-l)y) = (1f(g)A*x,y)

36 5. Unitary Representations

for all x and y in X. Thus, A* commutes with 1r(g) for all 9 in G. Therefore bothSand T commute with 1r(g) for all 9 in G. At least one of them is not a scalarmultiple of the identity operator on X. To be specific, let us suppose that S isnot a scalar multiple of the identity operator on X. Let {E(..\) : ..\ E JR.} be thespectral family of the self-adjoint operator S. Then, by the spectral theorem, theprojection E(..\) commutes with 1r(g) for all ..\ in JR. and all 9 in G. Let P be onesuch nonzero projection that is not the identity operator on X and let M be itsrange. Then M is a nontrivial closed subspace of X. Furthermore, for all 9 in Gand all x in M, we get

1r(g)x = 1r(g)Px = P1r(g)x E M.

Thus, 1r : G -t U(X) is not irreducible. This completes the proof.

We can now give two consequences of Schur's lemma.

D

Theorem 5.4 Let G be an abelian, locally compact and Hausdorff group. Then everyirreducible and unitary representation of G on a separable and complex Hilbertspace is one-dimensional.

Proof. Let 1r : G -t U(X) be an irreducible and unitary representation of G on aseparable and complex Hilbert space X and let 9 E G. Then, using the fact thatG is abelian, 1r(g) commutes with 1r(h) for all h in G. By Schur's lemma, thereexists a complex number cg such that 1r(g) = cgI, where I is the identity operatoron X. Now, suppose that dim(X) > 1. Let M be a closed subspace of X such thatM # {O} and M # X. Then for all 9 in G and all x in M,

1r(g)x = cgx E M.

Therefore M is an invariant subspace of X and this contradicts the irreducibilityof 1r : G -t U(X). D

Theorem 5.5 Let G be a compact and Hausdorff group. Then every irreducibleand unitary representation of G on a separable and complex Hilbert space is finitedimensional.

Proof. Let 1r : G -t U(X) be an irreducible and unitary representation of G onX. Let cp E X be such that Ilcpli = 1. We define the linear operator Tcp : X -t Xby

(Tcpx, y) = L(x, 1r(g)cp)(1r(g)cp, y)dJ.L(g) , x, Y E X. (5.2)

Then Tcp : X -t X is a bounded linear operator. Indeed, for all x and y in X, weget, by Schwarz' inequality, the compactness of G, Ilcpll = 1 and the fact that 1r(g)is a unitary operator for all 9 in G,

I(Tcpx, y)1 ~LI(x, 1r(g)cp)II(1r(g)cp, y)ldJ.L(g) ~ J.L(G)llxllllyll·

5. Unitary Representations

Moreover, Tcp : X --+ X is a positive operator. Indeed, let x E X. Then

37

Now, let {<pj : j = 1,2, ...} be an orthonormal basis for X. Then, by Fubini'stheorem, Parseval's identity, II<p11 = 1 and the fact that 7f(g) is a unitary operatorfor all 9 in G, we get

00

L (Tcp<pj, <pj)j=1

= f jl(<pj, 7f(g)<pWdJl(g)j=1 G

= L t, I(<pj ,7f(g)<pWdJl(g)

= L 117f(g)<p112dJl(g) = Jl(G) < 00.

Thus, by Proposition 2.4, Tcp : X --+ X is in 8 1 and hence compact. Furthermore,the function G :1 9 1--+ I(u, n(g)<p)12 E IR is continuous and is equal to 1 when 9 = e.It follows that it is strictly positive in a neighborhood of e. Thus,

In other words, the compact operator Tcp : X --+ X is nonzero. Now, for all hin G, using the fact that 7f : G --+ U(X) is a unitary representation and the leftinvariance of the Haar measure 1-£, we get

(Tcp7f(h)x, y) L (7f(h)x, 7f(g)<p) (7f(g)<p, y)dJl(g)

= L (x, 7f(h- 1g)<p)(7f(g)<p, y)dJl(g)

L(x,7f(g)<P)(7f(h9)<P,y)dJl(9)

= L (x, 7f(g)<p)(7f(g)<p, n(h-1 )y)dJl(g)

= (Tcpx,7f(h- 1 )y) = (7f(h)Tcpx,y)

for all x and y in X. Thus, Tcp commutes with 7f(g) for all 9 in G. So, usingthe irreducibility of 7f : G --+ U(X), we can find a nonzero constant c such thatTcp = el, where I is the identity operator on X. Thus, the identity operator I onX is compact. So, X has to be finite-dimensional. 0

38 5. Unitary Representations

Remark 5.6 Theorem 5.5 is a well-known result in the representation theory ofcompact groups. See, for instance, Chapter 5 of the book [27] by Folland. It isremarkable that in the proof of Theorem 5.5, the function

in (5.2) is a wavelet transform, and the linear operator Tcp : X ---+ X defined by(5.2) is a localization operator associated to the admissible wavelet <p and thesymbol F on G given by

F(g) = 1, 9 E G.

See Chapters 7 and 12 in this connection. Thus, we have already come acrosswavelet transforms and localization operators in the context of the representationtheory of compact groups.

The unitary representations of locally compact and Hausdorff groups thatare of most interest to us in this book are infinite-dimensional. We assume fromnow on that all Hilbert spaces are infinite-dimensional, separable and complex.

We give an example of a unitary representation of a locally compact groupon a Hilbert space.

Example 5.7 Let G be a locally compact and Hausdorff group endowed with a leftHaar measure J-L. Let L : G ---+ U(L2(G)) be the mapping defined by

(L(g)f)(h) = f(g-lh), g, hE G,

for all fin L2(G). Then we leave it as an exercise to prove that L : G ---+ U(L2(G))is a unitary representation of G on L2(G). We call it the left regular representationofG.

6 Square-Integrable Representations

We are now well-equipped to study the main contents in this book. We begin with astudy of square-integrable representations of locally compact and Hausdorff groupson Hilbert spaces. This chapter can be considered as a continuation of the studyof unitary representations begun in Chapter 5.

Let G be a locally compact and Hausdorff group. Let X be a Hilbert space. Asbefore, we denote the inner product and the norm in X by ( , ) and II II respectively.A unitary representation 7r : G ---. U(X) of G on X is said to be square-integrableif there exists a nonzero element cp in X such that

(6.1)

The condition (6.1) is known as the admissibility condition for the square-integrable representation of G on X. We call any element cp in X for which II cpll = 1 andthe admissibility condition is valid an admissible wavelet for the square-integrablerepresentation 7r : G ---. U(X), and we define the constant crp by

(6.2)

We call Crp the wavelet constant associated to the admissible wavelet cp.

Theorem 6.1 Let 7r : G ---. U(X) be an irreducible and square-integrable representation of G on X. If cp is an admissible wavelet for 7r : G ---. U(X), then

(x, y) = ~1(x, 7r(g)cp) (7r(g)cp, y)dp,(g)Crp G

(6.3)

for all x and y in X.

Remark 6.2 In order to understand the formula (6.3) better, let us note that ittells us, informally, that

I = ~1(-,7r(g)Cp)7r(g)cpdp,(g),Crp G

where I is the identity operator on X. In other words, the identity operator Ion X can be resolved into a sum of rank-one operators -1.(.,7r(g)Cp)7r(g)cp, which

Cop

are parametrized by elements 9 in G. Thus, the formula (6.3) is known as theresolution of the identity formula.


40 6. Square-Integrable Representations

To prove Theorem 6.1, we need a lemma.

Lemma 6.3 The subspace M of X defined by

M = {x EX: fc l(x,1f(g)<pWdj.£(g) < oo} (6.4)

is a closed subspace of X.

The role of Lemma 6.3 in the proof of Theorem 6.1 is to show that thesubspace M is in fact the entire Hilbert space X and hence a closed linear operatordefined on it is a bounded linear operator by the closed graph theorem.

Proof of Theorem 6.1. Using the fact that 1f : G ~ U(X) is a representation, theleft invariance of j.£ and (6.4), we get

fc I(x, 1f(h-1g)<p)12dj.£(g)

fc I(x, 1f(g)<p) [2dj.£(g) < 00

for all h in G and all x in M. Hence, by (6.4), 1f(h)x E M for all x in M and all hin G. Therefore, using the fact that <P E M, M is a nonzero subspace of X, whichis invariant with respect to the irreducible representation 1f : G ~ U(X). Since Mis a closed subspace of X by Lemma 6.3, it follows that M = X. Now, we definethe linear operator Acp : X ~ L2(G) by

(Acpx)(g) = (x,1f(g)<p), x E X, 9 E G. (6.5)

Then for x E X and g, h E G, by (6.5) and the fact that 1f : G ~ U(X) is arepresentation, we have

(Acp1f(h)x)(g) = (1f(h)x,1f(g)<p) = (x,1f(h-1g)<p) = (Acpx)(h-1g),

and so,Acp1f(h) = L(h)Acp, (6.6)

where L : G ~ U(L2 (G)) is the left regular representation of G in Example 5.7,i.e.,

(L(h)f)(g) = f(h-1g), g, hE G, (6.7)

for all f in L2(G). Let {Xk}k"=l be a sequence of elements in X such that Xk ~ x inX and AcpXk ~ f in L2(G) as k ~ 00. Then there is a subsequence of {AcpXk}k::l'again denoted by {Acpxdk::l' such that

(6.8)

a.e. on G as k ~ 00. Since

(Xk, 1f(g)<p) ~ (x, 1f(g)<p) , 9 E G,

6. Square-Integrable Representations

as k ---+ 00, it follows from (6.5) that

41

(6.9)

Thus, by (6.8) and (6.9), A<px = f. Hence A<p : X ---+ L 2 (G) is a closed linearoperator, and, by the closed graph theorem, A<p : X ---+ L 2 (G) is a bounded linearoperator. Finally, for all x and y in X, we get, by (6.5)-(6.7) and the left invarianceof f.J"

(A~L(g)A<px, y) (L(g)A<px, A<pY)£2(G)

= fa (A<pX)(g-lh) (A<py)(h)df.J,(h)

fa (A<px)(h)(A<py)(gh)df.J,(h)

(A<px, L(g-l )A<pY)£2(G)

= (A<px,A<p7r(g-1)y)L2(G)

(7r(g)A~A<px, y)

for all 9 in G, where A~ is the adjoint of A<p, and hence

(6.10)

Moreover, by (6.6),A~L(g)A<p = A~A<p7r(g), 9 E G. (6.11)

Thus, by (6.10), (6.11) and the fact that 7r : G ---+ U(X) is irreducible, we can useSchur's lemma to conclude that there exists a constant c such that

(6.12)

where I is the identity operator on X. Thus, for all x and y in X, we get, by (6.12),

c(x,y) (A~A<px,y) = (A<px,A<pY)£2(G)

fa(x,7r(g)~)(y,7r(9)~)df.J,(9)

= fa (x, 7r(g)~)(7r(g)~, y)df.J,(g).

So, by (6.2) and (6.13),

c = c(~,~) = fa 1(~,7r(g)~)12df.J,(g)= c<p'

(6.13)

(6.14)

Hence, by (6.13) and (6.14), the proof is complete provided that we can proveLemma 6.3. 0


To prove Lemma 6.3, we use the extended Schur's lemma used by Grossmann,Morlet and Paul in the paper [36]. We formulate it as Theorem 6.4 and we omitthe proof.

Theorem 6.4 Let G be a locally compact and Hausdorff group. Let Xl and X 2 beHilbert spaces in which the norms are denoted by 1IIIxl and II11x2 respectively. Let71"1 : G ~ U(X1) be an irreducible and unitary representation of G on Xl and let71"2 : G ~ U(X2) be a unitary representation of G on X 2. Let A be a closed linearoperator from Xl into X 2 such that the domain V(A) of A is dense in Xl and

A7I"1(g)X = 7I"2(g)Ax, x E V(A).

Then A is a scalar multiple of an isometry from V(A) into X 2, where V(A) is theHilbert space equipped with the graph norm IIIIA of A defined by

IIxll~ = IIxl13cl + IIAx 113c2 , x E V(A).

Proof of Lemma 6.3. As has been shown in the proof of Theorem 6.1, the linearoperator Alp from X into X with domain M is a closed linear operator. It hasbeen shown in the proof of Theorem 6.1 that M is an invariant subspace of Xwith respect to 71" : G ~ U(X). So, for all x in the closure M of M in X, we get asequence {Xd~l in M such that

(6.15)

in X as k ~ 00, and hence, by (6.15),

7I"(g )Xk ~ 7I"(g)x

in X as k ~ 00 for all 9 in G. Thus, 7I"(g)x E M for all 9 in G. Therefore Mis an invariant subspace of X with respect to 71" : G ~ U(X). Since r.p E M and71" : G ~ U(X) is irreducible, we can use Schur's lemma to conclude that M = X.Therefore M is dense in X. By (6.6),

Acp7l"(g)x = L(g)Acpx, x E M,

where L : G ~ U(L2(G)) is the left regular representation of G defined by

(L(g)f)(h) = f(g-lh), hE G,

for all 9 in G and all f in L 2 (G). Now, M becomes a Hilbert space, denoted byMcp, if we equip it with the graph norm 1111'1' of Acp given by

(6.16)

Thus, by the extended Schur's lemma, we can conclude that Alp is a scalar multipleof an isometry from Mcp into L2(G). So, by (6.16), there exists a positive constantA such that

(6.17)

6. Square-Integrable Representations

Hence, by (6.17), A < 1 and

2 A 2IIAcp x ll£2(G) = 1 _ Allxll , x E M.

43

(6.18)

Using (6.18) and the density of M in X, we can extend Acp : M --+ L2 (G) to abounded linear operator from X into L2 (G), which we denote by Acp : X --+ L2 (G).So, if {Xk}~l is a sequence of elements in M such that Xk --+ x in X as k --+ 00,

then AcpXk --+ Acpx in L2(G) as k --+ 00. Since Acp is a closed linear operator fromX into L2 (G) with domain M, it follows that x EM. Therefore M is a closedsubspace of X. 0

Remark 6.5 Theorem 6.1 is a simplified version of Theorem 3.1 in the paper [36]by Grossmann, Morlet and Paul, where the original contributions due to Dufio andMoore [24] are acknowledged. Chapter 14 of the book [14] by Dixmier is devotedto a study of square-integrable representations. See also the paper [9] by Carey inthis connection.

We can give some information on the set AW (7r) of admissible wavelets associated to an irreducible and unitary representation 7r : G --+ U(X) for unimodulargroups G.

Theorem 6.6 Let G be a unimodular group and let 7r : G --+ U(X) be an irreducibleand unitary representation of G on X. Then AW (7r) = ¢ or AW(7r) = {x EX:Ilxll = I}.

For a proof of Theorem 6.6, we use the theory of quadratic forms, which werecall without proofs. Details can be found in Section 6 of Chapter 8 of the book[72] by Reed and Simon.

Let M be a dense subspace of X. A mapping q : M x M --+ C is said tobe a quadratic form on X with form domain M if q(., y) is linear and q(x, .) isconjugate linear for all x and y in M. A quadratic form q : M x M --+ C on Xwith form domain M is said to be symmetric if

q(x,y)=q(y,x), x,yEM,

and is said to be positive if

q(x,x) ~ 0, x E M.

A positive quadratic form q : M x M --+ C on X with form domain M is said tobe closed if M is complete with respect to the norm IIl1q given by

IIxll~ = IIxll 2 + q(x, x), x E M. (6.19)

We need the following result, which is known as the second representationtheorem given on page 331 of the book [50] by Kato.


(6.20)

Theorem 6.7 Let q : M x M --t C be a symmetric, positive and closed quadraticform on X with form domain M. Then there exists a unique positive and selfadjoint operator A from X into X with domain M such that

(Ax,Ay) = q(x,y), x,y E M.

Proof of Theorem 6.6. Let D be the subspace of X defined by

D = {x EX: fa I(x, 1r(g)x)12dJ.l(g) < oo} .

Suppose that D =I- {O}. Let xED and h E G. Then, using the fact that 1r : G --t

U(X) is a representation, the unimodularity of G and (6.20),

fa 1(1r(h)x, 1r(g)1r(h)xWdJ.l(g) fa I(x, 1r(h-1gh)xWdJ.l(g)

fa I(x, 1r(g)x)12dJ.l(g) < 00,

and hence, by (6.20), 1r(h)x E D. Therefore D is an invariant subspace of X withrespect to 1r : G --t U(X). Thus, D = X. Let <p E D and let Acp : X --t L2(G)be the linear operator defined by (6.5). Then, as has been shown in the proof ofTheorem 6.1,

Acp1r(h) = L(h)Acp, hE G, (6.21)

where L : G --t U(L2(G)) is the left regular representation of G. Thus, for all <pand'ljJ in D, we get, by (6.21),

(6.22)

where A:;j, is the adjoint of A1/>' Using the argument in the derivation of (6.10), weget

A:;j,L(h)Acp = 1r(h)A~Acp, hE G.

So, by (6.22) and (6.23),

A~Acp1r(h) = 1r(h)A~Acp, hE G,

and hence, by Schur's lemma, there exists a constant cCP,1/> such that

A~Acp = ccp,1/>I,

where I is the identity operator on X. Thus, by (6.5) and (6.24),

(6.23)

(6.24)

CCP,1/> = 1\:1\2(A~AcpX,x) = 1\:11 2 (Acpx,A1/>x)£2(G)

1\:1\2 fa (x, 1r(g)<p) (1r(g)'ljJ, x)dJ.l(g) (6.25)

6. Square-Integrable Representations 45

for all nonzero elements x in X. Let q : V x V -+ C be the quadratic form on Xwith form domain V defined by

q('P,'Ij;) = cep,"', 'P, 'Ij; E V, (6.26)

where cep,'" is given by (6.25). Then, obviously, q : V x V -+ C is symmetric andpositive. Let {'Pk}~l be a Cauchy sequence in V with respect to the norm II Ilqgiven by (6.19). Then {'Pk}~l is a Cauchy sequence in X. Hence

(6.27)

(6.28)

for some 'P in X as k -+ 00. Now, for any x in X, and k = 1,2, ... , we define thefunction fk on G by

fk(g) = (x, n(g)'Pk), 9 E G.

If x i 0, then by (6.19) and (6.28),

Ilfj - fk'lli2(G) 11:11 2 fc I(x, n(g)('Pj - 'Pk))12d/L(g)

q('Pj - 'Pk, 'Pj - 'Pk)

< II'Pj - 'Pk II~ -+ 0

as j, k -+ 00. Thus, {fk}k'=1 is a Cauchy sequence in L2 (G) for all x in X. So,

(6.29)

for some f in L2 (G) as k -+ 00. Also, by (6.27), (6.28), Schwarz' inequality, andthe fact that n(g) is unitary for all 9 in G,

fk'(g) = (x,n(g)'Pk) -+ (x,n(g)'P), 9 E G,

as k -+ 00. Thus, by (6.29) and (6.30), (x, n(·)'P) = f a.e. and hence

fc I(x, n(g)'P)12 d/L(g) < 00

for all x in X. Therefore, by (6.31),

(6.30)

(6.31)

i.e., 'P E V. Now, for any nonzero element x in X, we get, by (6.27), (6.29) and(6.30),

II'Pk - 'PII~ = lI'Pk - 'P11 2 + q('Pk - 'P, 'Pk - 'P)

= II'Pk - 'P11 2 + 11:11 2 fc I(x, n(g)('Pk - 'P)Wd/L(g) -+ 0


as k ---t 00. Hence q : V x V ---t C is closed. By Theorem 6.7, i.e., the secondrepresentation theorem, there exists a positive and self-adjoint operator A from Xinto X with domain V such that

(6.32)

Now, using (6.25), (6.26) and the unimodularity of G, the fact that V is an invariantsubspace of X with respect to rr : G ---t U(X) and the fact that rr : G ---t U(X) isa representation, we get, for any nonzero element x in X,

q(rr(h)<p, rr(h)'ljJ) 11:112[(x, rr(gh)<p) (rr(gh)'ljJ, x)df.l(g)

11:112[(x, rr(g)<p)(rr(g)'ljJ, x)df.l(g)

q(<p,'ljJ) (6.33)

for all <p and 'ljJ in V, and all h in G. Thus, by (6.32), (6.33), and the fact that therepresentation rr : G ---t U(X) is unitary,

(A<p,A'ljJ) = (Arr(h)<p,Arr(h)'ljJ)= (rr(h)rr(h- 1 )Arr(h)<p, Arr(h)'ljJ)= (rr(h -1 )Arr(h)<p, rr(h -1 )Arr(h)'ljJ)

for all <p and 'ljJ in V, and all h in G. Hence, on V,

rr(h- 1)Arr(h) = A, hE G,

or equivalently,Arr(h) = rr(h)A, h E G.

Let V be the Hilbert space equipped with the inner product of which the inducednorm is II Ilq given by (6.19). Then, by Theorem 6.4, i.e., the extended Schur'slemma, A is a scalar multiple of an isometry from V into L 2 (G). So, we can finda positive number ,.\ such that

IIA<p112 = "\11<p11~ = ,.\1I<p1l2 + ,.\IIA<p112, <p E V.

Hence, by (6.34), ,.\ < 1 and

IIA<P112 = 1 ~,.\ 11<p112, <p E V.

(6.34)

(6.35)

Using (6.35) and the density of V in X, we can extend A : V ---t X to a boundedlinear operator from X into X, which we denote by A: X ---t X. So, if {<Pkl~l isa sequence of elements in V such that <Pk ---t <p in X as k ---t 00, then A<Pk ---t A<p inL 2(G) as k ---t 00. Since A is a closed linear operator from X into X with domainV, it follows that <p E V. Therefore V is a closed subspace of X. Thus, usingthe irreducibility of rr : G ---t U(X), we conclude that V = X and the proof iscomplete. 0

6. Square-Integrable Representations 47

Remark 6.8 We give in Chapter 17 a unimodular group G, and an irreducible andunitary representation 7r : G ~ U(X) of G on X for which AW(7r) = {x EX:Ilxll = I}. A different unimodular group G', and a new irreducible and unitaryrepresentation 7r' : G' ~ U(X) for which AW(7r) = ¢ are also given. It is worthemphasizing the fact that Theorem 6.6 is false, in general, for non-unimodulargroups, and Chapter 18 is devoted to a study of a non-unimodular group forwhich the conclusion of Theorem 6.6 is not true.

7 Wavelet Transforms

The linear operator Aep : X ---+ L2(G) used in the proof of the resolution of theidentity formula (6.3) in the previous chapter plays a pivotal role in this book. Itis in fact the wavelet transform associated to the admissible wavelet <p for the irreducible and square-integrable representation 7f : G ---+ U(X) of a locally compactand Hausdorff group G on a Hilbert space. To be more precise, we introduce thefollowing definition.

Definition 7.1 Let <p be an admissible wavelet for a square-integrable representation 7f : G ---+ U(X) of a locally compact and Hausdorff group G on a Hilbert spaceX. Then the wavelet transform associated to the admissible wavelet <p is the linearoperator Aep : X ---+ C(G) defined by

1(Aepx)(g) = ;r-(x,7f(g)<p), 9 E G, (7.1)

vCep

where C(G) is the set of all continuous and complex-valued functions on G.

In the case when the representation 7f : G ---+ U(X) in Definition 7.1 is alsoirreducible, the wavelet transform associated to <p is in fact equal to );;Aep, whereAep is the linear operator used extensively in Chapter 6. To emphasize the fact thatthe wavelet transform is something we have come across before, we prefer to denotethe wavelet transform associated to <p again by Aep. There should be no dangerof confusion. Another point to note is that the wavelet transform is defined forall square-integrable representations. The irreducibility condition is invoked onlywhen it is necessary.

From the proof of Theorem 6.1, we see that if <p is an admissible wavelet foran irreducible and square-integrable representation 7f : G ---+ U(X) of G on X, thenAep : X ---+ L 2 (G) is a bounded linear operator. In this chapter and the followingfour chapters, we give a more detailed study of wavelet transforms.

We begin with the following reformulation of Theorem 6.1 in terms of thewavelet transform Aep : X ---+ L 2 (G).

Theorem 7.2 Let <p be an admissible wavelet for an irreducible and square-integrable representation 7f : G ---+ U(X) of a locally compact and Hausdorff group Gon a Hilbert space X. Then

(x, y) = fa (Aepx)(g) (Aepy)(g)df.L(g)

for all x and yin X.


7. Wavelet Transforms 49

Remark 7.3 Theorem 7.2 can be considered as Plancherel's theorem for the wavelettransform Aep : X --7 L2 (G).

Remark 7.4 That the wavelet transform Aep : X --7 L2(G) is an isometry is animmediate consequence of Theorem 7.2. Thus, the wavelet transform Aep : X --7

L2 (G) is an isometry of X onto its range R(Aep). This is why the range R(Aep) ofthe wavelet transform Aep : X --7 L2(G) should be studied in some detail.

Proposition 7.5 Let cp be an admissible wavelet for an irreducible and squareintegrable representation of a locally compact and Hausdorff group G on a Hilbertspace X. Then the range R(Aep) of Aep : X --7 L2 (G) is a closed subspace of L2 (G).

Proof. Let {Fj }~1 be a sequence of functions in R(Aep) such that

(7.2)

for some F in L 2 (G) as j --7 00. For j = 1,2, ..., let Xj be such that Aepxj = Fj .Now, using Plancherel's theorem for Aep : X --7 L2 (G),

as j,k --700. Therefore {Xj}~l is a Cauchy sequence in X. So, Xj --7 x for somex in X as j --7 00. Thus,

(7.3)

in L2 (G) as j --7 00. By (7.2) and (7.3), F = Aepx. Hence F E R(Aep) and theproof is complete. 0

It follows from Proposition 7.5 that the range R(Aep) ofthe wavelet transformAep : X --7 L2 (G) is a Hilbert space. In fact, it is much more than a Hilbert space.

Theorem 7.6 Let cp be an admissible wavelet for an irreducible and square-integrable representation of a locally compact and Hausdorff group G on a Hilbertspace X. Then the range R(Aep) of the wavelet transform Aep : X --7 L2 (G) is areproducing kernel Hilbert space with reproducing kernel pep given by

(7.4)

In other words, if FE R(Aep), then

Proof. Let FE R(Aep). Then there exists an element x in X such that

(7.5)

50 7. Wavelet Transforms

Using the resolution of the identity formula in Chapter 6, (7.1), (7.4), (7.5) andthe fact that 1r : G --+ U(X) is a unitary representation, we get

fa F(h)pcp(g-lh)dJL(h)

!-1 (Acpx)(h)(1r(g-lh)cp,cp)dJL(h)Ccp G

31/21 (x, 1r(h)cp) (1r(h)cp, 1r(g)cp)dJL(h)

Ccp G

1-(x,1r(g)cp) = (Acpx)(g) = F(g)...;c;

for all 9 in G. D

8 A Sampling Theorem

As an application of the fact that the range of the wavelet transform associated toan admissible wavelet for an irreducible and square-integrable representation is areproducing kernel Hilbert space, we give in this chapter a sampling theorem on alocally compact and Hausdorff group. This is an analogue of Shannon's samplingtheorem given in Section 2.4 of the book [7] by Blatter and Section 2.1 of thebook [13] by Daubechies among others. The origin of the theorem is rooted in thepapers [79, 80] by Shannon.

Let 'P be an admissible wavelet for an irreducible and square-integrable representation 7f : G ~ U(X) of a locally compact and Hausdorff group G on a Hilbertspace X. Let {gj : j = 1,2, ...} be a countable collection of elements in G suchthat {7f(gj)'P : j = 1,2, ...} is an orthonormal basis for X. As has been indicatedin Remark 7.3, A<p : X ~ L2(G) is an isometry. Thus, {A<p7f(gj)'P : j = 1,2, }is an orthonormal basis for the range R(A<p) of A<p : X ~ L2 (G). For j = 1,2, ,let k~ be the function on G defined by

(8.1)

Theorem 8.1 Let F E R(A<p). Then

00

F = L(F,k~)£2(G)k~,j=l

where the series is convergent in L2 (G) and is absolutely convergent on G.

Proof. The convergence of the series in L2(G) follows from the fact that {k~

j = 1,2, ...} is an orthonormal basis for R(A<p). So, we only need to prove absoluteconvergence on G. Indeed, using Schwarz' inequality and Parseval's identity, weget

00

L I(F, k~)£2(G)llk~ (g)1 <j=l

=

1 1

{t, I(F, k~ )£,(G)I' }' {t, Ik~ (g)I' } ,

1

IIFIIL'(G) {t, Ik~ (g )1'r (8.2)


52 8. A Sampling Theorem

for all 9 in G. Since k~ E R(A<p), j = 1,2, ... , and R(A<p) is a reproducing kernelHilbert space, it follows that

for j = 1,2, ... , where P~ is the function on G given by

(8.4)

So, by (8.2)-(8.4) and Schwarz' inequality,

00

L I(F, k~ )L2(G)IJk~j (g)J 5 11F11L2(G)"~IIL2(G)j=l


The absolute convergence on G of the series in Theorem 8.1 says that

F(g) = t,(F, k~)L2(G)k~(g) = ~ {fa F(h)k~ (h)dj1-(h)} k~ (g) (8.5)

for all 9 in G. By (7.1), (7.4), (8.1) and the fact that 7f : G ---+ U(X) is a unitaryrepresentation, we get for all h in G,

k~(h)

(8.6)

for j = 1,2, .... So, by (8.5) and (8.6),

F(g) = VC;t, {fa F(h)P<p(gjl h)dj1-(h)} k~(g) = VC;t,F(9j)k~(9)

for all 9 in G. Thus,00

F = VC;LF(gj)k~,j=l

i.e., each signal F processed by means of the wavelet transform A<p : X ---+ L2 (G)can be reconstructed in terms of its sampled values {F(gj) : j = 1,2, ...} on G.Therefore Theorem 8.1 is a sampling theorem.

9 Wavelet Constants

We begin with the following theorem, which is an extension of the resolution of theidentity formula given in Theorem 6.1. This extension allows us to obtain someinteresting results on the wavelet constants for unimodular groups.

Theorem 9.1 Let <p and'IjJ be two admissible wavelets for an irreducible and squareintegrable representation 7f : G -+ U(X) of a locally compact and Hausdorff groupG on a Hilbert space X. Then for all x and y in X,

where

fa (x, 7f(g)<p)(7f(g)'IjJ, y)dj1.(g) = ccp,'" (x, y), (9.1)

(9.2)

Proof. Using the definition of the wavelet transform and the fact that 7f : G -+

U(X) is a unitary representation, we get

for all x in X, and all 9 and h in G. So,

(9.4)

where L : G -+ U(L2 (G)) is the left regular representation of G . Now, for allx and y in X, we get, by (9.3), (9.4), the left invariance of j1. and the fact that7f : G -+ U(X) is a unitary representation,

(A~L(g)Acpx,y) (LgAcpx, A",Yh2(G)

fa (AcpX)(g-lh)(A",y)(h)dj1.(h)

fa (Acpx)(h) (A",y) (gh)dj1.(h)

(Acpx, Lg-l A",y)L2(G)

(Acpx, A",7f(g-l )Y)£2(G)

= (7f(g)A~Acpx, Y)£2(G), 9 E G, (9.5)


54 9. Wavelet Constants

where A~ : L2 (G) ---+ X is the adjoint of A,p : X ---+ L 2 (G). Thus, by (9.5), we get

A~L(g)Acp = 7r(g)A~Acp, g E G.

Hence, by (9.4) and (9.6), we get

(9.6)

(9.7)

From (9.7) and the fact that 7r : G ---+ U(X) is an irreducible representation, wecan use Schur's lemma to conclude that there exists a constant acp,,p such that

(9.8)

where I is the identity operator on X. So, by (9.8), we get

(9.9)

for all x and y in X. If we let x = y = ep in (9.9) and use (9.2), then we get

and (9.1) follows. D

We call the number ccp,,p in Theorem 9.1 the two-wavelet constant associatedto the admissible wavelets ep and "l/J. It is obvious that ccp,cp is the same as thewavelet constant Ccp associated to the admissible wavelet ep defined by (6.2). As anapplication of two-wavelet constants, we give the following immediate consequenceof Theorem 9.1.

Corollary 9.2 Let cp and"l/J be two admissible wavelets for an irreducible and squareintegrable representation of a locally compact and Hausdorff group G on a Hilbertspace X. If Ccp,,p = 0, then R(Acp) and R(A,p) are orthogonal.

For unimodular groups, the wavelet constants are particularly illuminating.The following theorem gives us very important information on two-wavelet constants.

Theorem 9.3 Let G be a unimodular group, and let ep and "l/J be two admissiblewavelets for an irreducible and square-integrable representation 7r : G ---+ U(X) ofG on a Hilbert space X. Then

(9.10)

(9.11)

9. Wavelet Constants

Proof. By (9.2) and the fact that 7r : G ---+ U(X) is a unitary representation,

fa (<p, 7r(g)<p)(7r(g )'l/J, <p)dp,(g)

fa ('l/J, 7r(g-1 )<p)(7r(g-1 )<p, <p)dp,(g)

fa ('l/J, 7r(g)<p) (7r(g)<p, <p)diJ,(g),

55

where ji, is defined by (4.5). So, using the unimodularity of G, Theorem 6.1 and(9.11), we get (9.10). 0

The following consequence of (9.2) and Theorem 9.3 holds for unimodulargroups, and is an interesting result in its own right.

Theorem 9.4 Let G be a unimodular group, and let <p and 'l/J be two admissiblewavelets for an irreducible and square-integrable representation 7r : G ---+ U(X) ofG on a Hilbert space X. Then c<p = c,p.

Proof. By putting x = y = 'l/J in (9.1), we get

c<p,'" = fa ('l/J, 7r(g)<p) (7r(g)'l/J, 'l/J)dp,(g).

By (9.2) and (9.12), we getc,p,<p = c<p,,,,.

So, by Theorem 9.3, (9.13) and the fact that c<p is real-valued,

So,

(9.12)

(9.13)

c<p = c,p

if (<p, 'l/J) =J O. Now, suppose that (<p, 'l/J) = O. Let w E X be such that Ilwll = 1,(w, <p) =J 0 and (w, 'l/J) =J O. By Theorem 6.6, w is also an admissible wavelet forthe square-integrable representation 7r : G ---+ U(X). Thus, by what we have justshown, c<p = Cw and c,p = Cw. Hence c<p = c'" and the proof is complete. 0

For unimodular groups, the following theorem gives us a decomposition of awavelet transform into an orthogonal direct sum of wavelet transforms. It amountsto saying that any signal F on G, which comes as a wavelet transform, is a superposition of signals obtained as wavelet transforms associated to admissible waveletswhich form an orthonormal basis for X.

Theorem 9.5 Let {<pj : j = 1,2, ...} be an orthonormal basis for X. Then for all<p in X with 1I<p1l = 1,

<Xl

A<px = L EB(<p, <Pj )A<pj x, x E X.j=l

56 9. Wavelet Constants

Proof. For all x in X and all 9 in G, we get

00

L (cp, cpj)(A<pjx)(g).j=l

(9.14)

Thus, by (9.14), Theorems 9.1 and 9.3, the proof is complete. o

10 Adjoints

The left regular representation L : G -----t U(L2(G)) of a locally compact and Hausdorff group G given by

(L(g)u)(h) = u(g-lh), g, hE G,

for all u in L 2 ( G), has been playing a subsidiary role since its appearance inExample 5.7. In this chapter we show that it is an object of interest in its ownright. We are particularly interested in the adjoints of wavelet transforms for leftregular representations of unimodular groups.

In order to understand why the wavelet transform associated to the left regular representation of a group is important, let us recall that a filter in electricalengineering is a bounded linear operator from L 2 (G) into L 2 (G), which commuteswith L(g) for all 9 in G. That the left translation invariance should be an indispensable property of a filter is explained in, e.g., the book [29] by Gasquet andWitomsky.

Theorem 10.1 Let cp be an admissible wavelet for the left regular representationL : G -----t U(L2(G)) of G. Then the wavelet transform Acp : L2(G) -----t L 2(G) is afilter.

Proof. Using the definition of the wavelet transform Acp : L2(G) -----t L2(G), we get

g,hE G,

for all u in L 2 (G), and the proof is complete. 0

We have the following result, which tells us that if G is unimodular, thenadmissible wavelets for the left regular representation of G abound.

Theorem 10.2 Let L : G -----t U(L2(G)) be the left regular representation of a unimodular group G. Then every function cp in L1(G) n L2(G) with Ilcpll£2(G) = 1 isan admissible wavelet for L: G -----t U(L2(G)).


58 10. Adjoints

Proof. Using Minkowski's inequality in integral form, the unimodularity of thegroup G and 11<pllp(G) = 1, we get

1

{fa lfa <p(h)<p(g-lh)dJl(h) 1

2

dJl(9)} :2

1

< fa {fa 1<P(hW1<P(g-lhWdJl(9)} :2 dJl(h)

1

fa 1<p(h)1 {fa 1<p(g-lhWdJl(9)} :2 dJl(h)

II<pII£l(G)II<pII£2(G) = II<pII£l(G) < 00,


In order to compute the adjoint of a wavelet transform associated to anadmissible wavelet for a left regular representation of a unimodular group, weneed the following lemma.

Lemma 10.3 Let <p E L1(G) n L2(G) be such that II<pllp(G) = 1, where G is aunimodular group. Then so is the function <p* on G defined by

<p*(g) = <p(g-l), 9 E G.

The proof of Lemma 10.3 is so simple that we can omit it.

We can now give a formula for the adjoint of a wavelet transform associatedto an admissible wavelet for a left regular representation of a unimodular group.

Theorem 10.4 Let <p E Ll(G) n L2(G) be such that II<pII£2(G) = 1, where G is aunimodular group. Then for the left regular representation L : G ----; U(L2(G)) of

G, the adjoint of the wavelet transform Acp : L2(G) ----; L2(G) is equal to ~Acp. :

L 2 (G) ----; L 2(G).

Proof. Let U and v be in L2(G). Then, using the unimodularity of G, Young'sinequality and Fubini's theorem, we get

(Acpu, v)P(G) = fa (Acpu) (g)v(g)dJl(g)

= _1_ r{rU(h)(L(9)<P)(h)dJl(h)} v(g)dJl(g).;c; JG JG_1_ ru(h) { r(L(9)<P)(h)V(9)dJl(9)} dJl(h).;c; JG JG

-.-------_1_ ru(h) { r<p(g-lh)V(9)dJl(9)} dJl(h).;c; JG JG

10. Adjoints 59


An immediate corollary of Theorems 9.4 and 10.4 is the following result.

Corollary 10.5 If the left regular representation L : G ---. U(L2(G)) of a unimodular group G is irreducible, then for all cp in L1(G) n L2(G) with IIcpllp(G) = 1,the adjoint of the wavelet transform Aep : L2(G) ---. L2(G) is the wavelet transformAepo : L2(G) ---. L2(G).

An important corollary of Theorem 10.4 is the following characterizationof self-adjoint wavelet transforms for left regular representations of unimodulargroups.

Theorem 10.6 Let cp E L1(G) n L2(G) be such that Ilcpllp(G) = 1, where G is aunimodular group. Then for the left regular representation L : G ---. U(L2(G)) ofG, the wavelet transform Aep : L2(G) ---. L2(G) is self-adjoint if and only if cp = cp* .

Theorem 10.6 follows immediately from Theorem 10.4 and the followinglemma.

Lemma 10.7 Let cp and 'IjJ be two admissible wavelets associated to the left regularrepresentation L : G ---. U(L2(G)) of a locally compact and Hausdorff group Gsuch that cep = c,p and Aep = A",. Then cp = 'IjJ.

Proof. Using the definition of the wavelet transform and the fact that L : G ---.U(L2 (G)) is a unitary representation of G, we get

(L(g)U,cp)L2(G) = (L(g)u,'IjJ)P(G), 9 E G,

for all u in L2 (G). Thus, if we let 9 be the identity element in the group G, thenwe get cp = 'IjJ. 0

Remark 10.8 We end this chapter with the observation that if cp E L1(G)nL2(G) issuch that IIcpIIL2(G) = 1, where G is a unimodular group, then for the left regularrepresentation L : G ---. U(L2(G)) of G, the wavelet transform Aep : L2(G) ---.L2(G) associated to cp is in fact the convolution operator from L2(G) into L2(G)of which the kernel is the function ~~. Indeed,

yC<p

111(Aepu)(g) = /T(u, L(g)cp)p(G) = - u(h)cp(g-1h)dp,(h),yCep y'c; G

for all u in L 2 (G).

gEG,

11 Compact Groups

We look at left regular representations L : G --+ U(L2(G)) of compact and Hausdorff groups G in this chapter. Let <P E L2(G). Then, using Minkowski's inequalityin integral form, the unimodularity of the group G and Schwarz' inequality, we get

1

{fa lfa <p(h)<p(g-lh)dt-t(h)12

dt-t(9)} 2

1

< fa {fa [<p(hW l<p(g-lh) I2dt-t (g) } 2 dt-t(h)

1

= fa l<p(h) I{fa 1<p(g-lh)12dt-t(g)} 2 dt-t(h)

= 11<pII£l(G)II<pIIL2(G):::; t-t(G)!II<plli2(G) < 00.

Thus, every function <p in L2 (G) with 1I<p1IL2(G) = 1 is an admissible wavelet forthe left regular representation L : G --+ U(L2(G)) of G.

Theorem 11.1 Let <P E L2(G) be such that 11<p11L2(G) = I, where G is a compactand Hausdorff group. Then for the left regular representation L : G --+ U(L2(G))of G, the wavelet transform A"" : L2(G) --+ L2(G) is in the Hilbert-Schmidt classS2 and

Proof Let {<Pk : k = 1,2, ...} be an orthonormal basis for L2(G). Then, usingFubini's theorem, Parseval's identity, the fact that L(g) : L2(G) --+ L2(G) is aunitary operator for all 9 in G, and the fact that 1I<p1IL2(G) = 1, we get

00

L II A",,<pklli2(G)k=l

and hence, by Proposition 2.8, the proof is complete. oM. W. Wong, Wavelet Transforms and Localization Operators© Springer Basel AG 2002

11. Compact Groups 61

The significance of the Hilbert-Schmidt property of the wavelet transformAcp : L2(G) ....... L2(G) is revealed in the following theorem, which tells us how fastthe wavelet transform Acp : L2(G) ....... L2(G) can be approximated by the sequenceof finite truncations of its singular value decomposition.

Theorem 11.2 Let <p E L2(G) be such that 11<pIlL2(G) = 1, where G is a compactand Hausdorff group. Then for the left regular representation L : G ....... U(L2(G))of G, let

00

Acpu = L sk(Acp)(u, Uk)P(G)Vk, u E L2(G),k=1

be the singular value decomposition of the wavelet transform Acp : L2(G) ....... L2(G),where sk(Acp), k = 1,2, ... , are the singular values of the wavelet transform Acp :L2(G) L2(G), {Uk: k = 1,2, ...} is an orthonormal basis for L2(G), {Vk :k = 1,2, } is an orthonormal set for L2(G), and the convergence of the series isunderstood to be in L2(G). For N = 1,2, ..., let AN : L2(G) ....... L2(G) be the Nthtruncation of the singular value decomposition given by

N

ANu = L sk(Acp)(u, Uk)L2(G)Vk, u E L2(G).k=1

Then1

IIAN - AcpIIB(£2(G)) ~ { f (Sk(Acp))2}2

....... 0k=N+1

as N ....... 00, where 1IIIB(L2(G)~ is the norm in the C*-algebra of all bounded linearoperators from L2(G) into L (G).

Proof. We begin with the fact that

(11.1)

for all bounded linear operators A : X ....... X in 8 2 . Indeed, for all A in 8 2 , thecanonical form for compact operators given by Theorem 2.2 says that

00

Ax = LSj(A)(x,uj)Vj, x E X,j=1

(11.2)

where sj(A), j = 1,2, ... , are the positive singular values of A : X ....... X, {Uj : j =1,2, ...} is an orthonormal basis for N(A).L, {Vj : j = 1,2, ...} is an orthonormalset in X, and the convergence of the series is understood to be in X. Thus, by(11.2), the orthonormality of {Vj : j = 1,2, ...}, the normality of {Uj : j = 1,2, ...}

62

and the definition of 111182,

11. Compact Groups

00

IIAxl12 = L Sj(A)Sk(A)(x, Uj)(x, Uk)(Vj, Vk)j,k=l00

L(Sj(A))21(x,UjW:s IIAII1JxIl2, X E X,j=l

and hence (11.1) follows. Now, by (11.1), we get

00

IIAN - Aepll~(£2(G)) < IIAN - Aepl112 = L II Aepuklli2(G)k=N+1

00 00

as N --+ 00.

L II sdAep)uklli2(G):S L (sk(Aep))2 --+ 0k=N+1 k=N+1

oIt is interesting to note that Theorem 11.2 tells us that if G is a compact

and Hausdorff group, then for the left regular representation L : G --+ U(L2 (G))of G, the wavelet transform Aep : L2(G) --+ L2(G) associated to any function rpin L 2 (G) with Ilrpll£2(G) = 1 is compact and hence its range cannot be a closedsubspace of L2(G). Thus, the wavelet transform A", : L2(G) --+ L2(G) cannotbe an isometry. This is in sharp contrast with the theory of wavelet transformsassociated to irreducible and square-integrable representations given in Chapter 7.

12 Localization Operators

Let <p be an admissible wavelet for an irreducible and square-integrable representation rr : G -+ U(X) of a locally compact and Hausdorff group G on a Hilbert spaceX. In this chapter we introduce a class of bounded linear operators LF,tp : X -+ X,which are related to the wavelet transform Atp : X -+ L2(G) defined by (7.1), forall Fin P(G), 1 ~ p ~ 00. We first tackle this problem for F in L1(G) or LOO(G).In the case when p = 1, we do not need the assumption that the representationrr : G -+ U(X) be irreducible.

Let FE L1(G) U LOO(G). Then for all x in X, we define LF,tpx by

(LF,tpx, y) = 2- rF(g)(x, rr(g)<p) (rr(g)<p, y)dJ.l(g) (12.1)ctp JG

for all y in X. Then we have the following proposition.

Proposition 12.1 Let FE L1(G). Then LF,tp : X -+ X is a bounded linear operatorand

Proof. Let x and y be elements in X. Then, using (12.1), Schwarz' inequality,1I<p1l = 1 and the fact that rr(g) : X -+ X is unitary for all 9 in G, we have

I(x, rr(g)<p) (rr(g)<p, y)1 ~ IIxIIIIYII·Since FE L1(G), it follows from (12.1) and (12.2) that

1I(LF,tpX,y)1 ~ -IIFII£l(G)llxlillyll

ctp

and the proof of the proposition is complete.

We also have the following proposition.

(12.2)

o

Proposition 12.2 Let F E LOO(G). Then LF,tp : X -+ X is a bounded linear operator and

Proof. Let x and y be elements in X. Then, using the resolution of the identityformula in Theorem 6.1, we get

(12.3)


64

and

12. Localization Operators

(12.4)

Thus, using (12.3), (12.4), Schwarz' inequality, and the assumption that F E

LOO(G), we get1

!(LF,cpx,Y)I::; -llFllu"'(G)IIxlillYII,Ccp

and this completes the proof of the proposition. 0

We can now associate a localization operator L F,cp : X --+ X to every functionF in LP(G), 1 < P < 00, and prove that LF,cp : X --+ X is a bounded linearoperator. The precise result is the following proposition.

Proposition 12.3 Let F E LP(G), 1 < p < 00. Then there exists a unique boundedlinear operator L F,cp : X --+ X such that

(12.5)

and LF,cpx is given by (12.1) for all x in X and all simple functions F on G forwhich

lL{g E G: F(g) ~ O} < 00.

To prove Proposition 12.3, we need a recall of the Riesz-Thorin theoremgiven in, e.g., Chapter 10 of the book [102] by Wong. It is in fact a special case ofTheorem 2.10 in Chapter 2.

Theorem 12.4 (The Riesz-Thorin Theorem) Let (X,lL) be a measure space and(Y, v) a u-finite measure space. Let T be a linear transformation with domain Vconsisting of all simple functions f on X such that

lL{s EX: f(s) ~ O} < 00

and such that the range of T is contained in the set of all measurable functions onY. Suppose that a1, a2, /31 and /32 are numbers in [0,1] and there exist positiveconstants M 1 and M 2 such that

IITfil i; ::; Mjllfll -L , f E V, j = 1,2.L j (Y) L"'j (X)

Then for 0 < () < 1,

a = (1 - (})a1 + (}a2

we have

and /3 = (1 - (})/31 + ()/32,

12. Localization Operators 65

Proof of Proposition 12.3. Let U : X ~ L2(~n) be a unitary operator betweenX and L2(~n). Let F E L1(G). Then, by Proposition 12.1, the linear operatorLF,rp : L2(~n) ~ L2(~n), defined by

is bounded and

(12.6)

(12.7)

where 1IIIB(L2(lRn)~ is the norm in the C*-algebra B(L2(~n)) of all bounded linearoperators from L (~n) into L2(~n). If FE LOO(G), then, by Proposition 12.2, thelinear operator LF,rp : L2(~n) ~ L2(~n), defined by (12.6), is also bounded and

(12.8)

Let V be the set of all simple functions F on G such that

J.L{g E G : F(g) # O} < 00.

Let f E L2(~n) and T be the linear transformation from V into the set of allLebesgue measurable functions on ~n defined by

I

(12.9)

Then, by (12.7) and (12.8),

andIITFIIL2(Rn) s:; IIFI/LOO(G)llfll£2(lRn)

for all functions F in V. Thus, by Theorem 12.4,

1

I/ TF IIL2(lRn) s:; (c~) p 1IFIILP(G)llfll£2(lRn), FE V.

Therefore, by (12.9) and (12.10),

I/LF,rpfll£2(lRn) s:; (c~) f; IIFIILP(G) Ilfl/£2(lRn) , FE V.

Since (12.11) is true for arbitrary functions f in L2(~n), it follows that

(12.10)

(12.11)

(12.12)

66 12. Localization Operators

Let FE LP(G), 1 < p < 00. Then there exists a sequence {Fd~l of functions inV such that Fk -+ F in LP(G) as k -+ 00. By (12.12),

as j, k -+ 00. Therefore {iFk,'!'}~l is a Cauchy sequence in B(L2(lRn )). Usingthe completeness of B(L2(lRn )), we can find a bounded linear operator iF,,!, :L2(lRn ) -+ L2(lRn ) such that i Fk ,,!, -+ iF,,!, as k -+ 00. Since each i Fk ,'!' satisfies(12.12), it follows that iF,'!' also satisfies (12.12). Thus, the linear operator LF,'!' :X -+ X, where

-1-LF,,!, = U LF,,!,U,

is a bounded linear operator satisfying the conclusions of the theorem if F E LP(G),1 < P < 00. To prove uniqueness, let F E LP(G), 1 < P < 00, and suppose thatPF : X -+ X is another bounded linear operator satisfying the conclusions of thetheorem. Let Q : U(G) -+ B(X) be the linear operator defined by

Then, by (12.6),

1

IIQFII* ::; 2 C~) p IIFIILP(G» FE U(G).

Furthermore, QF is equal to the zero operator on X for all F in V. Thus, Q :LP(G) -+ B(X) is a bounded linear operator that is equal to zero on the densesubspace V of U(G). Therefore PF = LF,,!, for all functions F in U(G). 0

Remark 12.5 The bounded linear operators LF,'!' : X -+ X introduced in thischapter are dubbed localization operators in the paper [40]. The impetus for theterminology stems from the simple observation that if F(g) = 1 for all 9 in G,then the resolution of the identity formula in Theorem 6.1 implies that the corresponding linear operator is simply the identity operator on X. Thus, in general, the function F, which is also called the symbol of the localization operatorL F,,!, : X -+ X, is there to localize on G so as to produce a nontrivial boundedlinear operator on X with various applications in the mathematical sciences.

13 Sp Norm Inequalities, 1 < P < 00

We prove in this chapter that a localization operator LF,'P : X ---t X associated toa function F in LP(G), 1 ~ p ~ 00, and an admissible wavelet <p for an irreducibleand square-integrable representation of a locally compact and Hausdorff group Gon a Hilbert space X is in the Schatten-von Neumann class 8p , 1 ~ P ~ 00. Whenp = 1, the irreducibility of the representation n : G ---t U(X) can be dispensedwith.

The first result on the Schatten-von Neumann property of localization operators is given in the following proposition.

Proposition 13.1 Let FELl (G). Then the localization operator LF,'P : X ---t X isin 81 and

(13.1)

Proof. First we assume that the function F : G ---t C is nonnegative and realvalued. Then, by (12.1), we get

for all x in X, i.e., the localization operator LF,'P : X ---t X is positive. Let{<Pk : k = 1,2, ...} be any orthonormal basis for X. Then, using (12.1) and thefact that n : G ---t U(X) is a representation, we get

(13.2)

Hence, using (13.2), Fubini's theorem, Parseval's identity, 11<p11 = 1 and the factthat neg) : X ---t X is unitary for all g in G, we get

(13.3)

Hence, by (13.3) and Proposition 2.4, the localization operator LF,'P : X ---t X isin 8 1 . Furthermore,

(13.4)


68 13. Sp Norm Inequalities, 1 :S p :S 00

Thus, if {'l/lk : k = 1,2, ...} is an orthonormal basis for X consisting of eigenvectorsof (L},<pLp,<p)! : X -+ X, we have, by (13.2) and (13.4),

(Xl

IILp,<plls, = '2)(L},<pLp,<p)!'¢k' 'l/lk)k=1

(Xl

"L(Lp,<P'¢k, '¢k)k=1

1-lfFll£l(G)·c<p

(13.5)

Now, if FE £l(G) is a real-valued function, then we write F = F+ - F_, where

F+(g) = max(F(g) , 0)

andF_(g) = - min(F(g), 0)

for all 9 in G. Then Lp,<p : X -+ X is in 8 1 and by (13.5),

Finally, let F E L 1(G) be a complex-valued function. Then we write F = F1 +iF2 ,

where F1 and F2 are the real and imaginary parts of F respectively. Then LF,<p :X -+ X is a localization operator and by (13.6),

and the proof of Proposition 13.1 is complete. o

Remark 13.2 The very elementary proof of Proposition 13.1 gives us the constant...!.. in the estimate (13.1). By Proposition 12.1, it is natural to expect that theC<p

best constant should be ...!.. instead of .!. That this is indeed the case is proved inC<p <p

Chapter 14 using more elaborate techniques.

A consequence of Proposition 13.1 is the following result.

13. Sp Norm Inequalities, 1 :::; P :::; 00 69

Proposition 13.3 Let F E LP(G), 1 :::; p < 00. Then the localization operatorLF,'P : X -+ X is compact.

Proof. We again denote by V the set of all simple functions F on G such that

p,{g E G : F(g) :f O} < 00.

Let {Fdf::1 be a sequence offunctions in V such that Fk -+ Fin LP(G) as k -+ 00.

Then, by (12.5),

1

IILFk,'P - LF,'PII* :::; (c~) ;; IlFk - FIILP(G) -+ 0

as k -+ 00, i.e., LFk,'P -+ LF,'P in B(X) as k -+ 00. Since, by Proposition 13.1,LFk,'P : X -+ X is in 81 and hence compact, it follows that LF,'P : X -+ X iscompact. 0

Remark 13.4 That Proposition 13.3 is false for p = 00 can be seen easily by takingthe function F on G to be such that

F(g)=I, gEG.

For then, by the resolution of the identity formula in Theorem 6.1, L F,'P : X -+ X isthe identity operator on X. In view of the hypothesis that X is infinite-dimensional,LF,'P : X -+ X is not compact.

Now, we can come to the main result on the Schatten-von Neumann propertyof localization operators.

Theorem 13.5 Let F E LP(G), 1 :::; p :::; 00. Then the localization operator LF,'P :X -+ X is in 8p and

Proof. By Proposition 13.1,

(13.7)

and, by Proposition 12.2,

(13.8)

So, by (13.7), (13.8), Theorems 2.10 and 2.11, the proof is complete. 0

We end this chapter with a formula for the trace tr(LF,'P) of the localizationoperator LF,'P : X -+ X when F E L 1(G).

70 13. Sp Norm Inequalities, 1 ~ P ~ 00

Theorem 13.6 Let F E L1(G). Then tr(LF,cp) = c~ fa F(g)dJL(g).

Proof. Let {CPk : k = 1,2, ...} be an orthonormal basis for X. Then, using thedefinition of the trace given in Chapter 2, the definition of LF,cp : X -> X, Fubini'stheorem, Parseval's identity, Ilcpli = 1 and the fact that n(g) : X -> X is a unitaryoperator for all 9 in G, we get

00

tr(LF,cp) = ~)LF'CPCPk' CPk)k=l

o

14 Trace Class Norm Inequalities

As a sequel to Chapter 13, we prove in this chapter that the constant ...!. inc."

Proposition 13.1 can be improved to ...!.. and obtain a lower bound for the normc."

II LF,cp II Sl of the localization operator L F,cp : X -+ X in 8 1 in terms of the functionFcp on G defined by

Fcp(g) = (LF,cp1r(g)cp, 1r(g)cp) , 9 E G. (14.1)

The function Fcp can be interpreted as the expectation value of the "observable"LF,cp : X -+ X in the coherent states 1r(g)cp, 9 E G. See, for instance, the book[2] by Ali, Antoine and Gazeau, the survey paper [3] by Ali, Antoine, Gazeau andMueller, and the book [4] by Berezin and Shubin for comprehensive accounts ofcoherent states and related topics.

An important result in this chapter is the following theorem on the normIILF,cplls1 of the localization operator LF,cp : X -+ X in 8 1 when F E L1(G).

Theorem 14.1 Let F E L 1(G). Then

1IILF,cplls1 ~ -IIFII£l(G)'Ccp

If, in addition, the square-integrable representation 1r : G -+ U(X) of G on X isirreducible, then

1-llFcpll£l(G) ~ IILF,cplls1'Ccp

Remark 14.2 To see that Fcp E L1(G), we use (12.1), (14.1), Fubini's theorem,IIcpll = 1, the fact that 1r : G -+ U(X) is a unitary representation, the left invarianceof p, and the definition of the wavelet constant Ccp to get

1 IFcp (g) Idp,(g) = 11 c~ 1 F(h)I(1r(g)cp, 1r(h)cp)12dp,(h)1dp,(g)

< c~ 1 (llF(h)I'(1r(g)cp, 1r(h)cp)12dP,(9)) dp,(h)

c~ 1,F(h)1 (1 I(1r(g)cp, 1r(h)CP)12dP,(9)) dp,(h)

c~ 1,F(h)! (1 I(1r(h- 1g)cp, CP)12dP,(9)) dp,(h)

1IF(h),dP,(h) < 00.


72 14. Trace Class Norm Inequalities

The following connection between F and Fcp is also interesting in its ownright.

Proposition 14.3 Let F E L1 (G). Then

Proof. By (12.1), (14.1), Fubini's theorem, Ilcpll = 1, the fact that 7r : G -+ U(X)is a unitary representation, the left invariance of J.L and the definition of the waveletconstant ccp, we get

fa Fcp(g)dJ.L(g) = c~ fa (fa F(h)I(7r(g)CP,7r(h)CP) 12 dJ.L(h)) dJ.L(g)

c~ fa F(h) (fa I(7r(g)cP, 7r(h)cp)12dJ.L(9)) dJ.L(h)

c~ fa F(h) (fa 1(7r(h-1g)CP,cp) 12 dJ.L(9)) dJ.L(h)

fa F(h)dJ.L(h).

o

Remark 14.4 By Theorem 13.6, we know that

If F is a real-valued and nonnegative function in L1(G), then LF,cp : X -+ X is apositive operator and hence IILF,cpllsl = tr(LF,cp). Hence, by Proposition 14.3,

Thus, the estimates in Theorem 14.1 are sharp.

Proof of Theorem 14.1. We note that, by Proposition 13.1, LF,cp E S1' Then,using the canonical form for compact operators given in Theorem 2.2, we get

00

LF,cpx = L sk(LF,cp)(x, CPk)'l/Jk, x E X,k=1

(14.2)

where sk(LF,cp), k = 1,2, ..., are the positive singular values of LF,cp : X -+ X,{CPk : k = 1,2, ...} is an orthonormal basis for N(LF,cp).l.., {'l/Jk : k = 1,2, ...} is an

14. Trace Class Norm Inequalities 73

orthonormal set in X and the convergence of the series is understood to be in X.By (14.2),

00 00

'2)LF,<pCPj , 1/Jj) = 2:>j(LF,<p)'j=l j=l

So, using the definition of IllIsl and (14.3),

00

IILF ,<pllsl = L)LF,<pCPj, 1/Jj).j=l

(14.3)

(14.4)

(14.5)

Thus, by (12.1), (14.4), Fubini's theorem, Schwarz' inequality, Bessel's inequality,IIcpll = 1 and the fact that 1f : G -+ U(X) is a unitary representation, we get

00

IILF ,<pllsl = 2: (LF,<pCPk , 1/Jk)k=l00 1 rf; c<p J

eF(g)(CPk, 1f(g)cp) (1f(g)cP, 1/Jk)dJl(g)

00 1 r< f; c<p Je IF(g)II(CPk,1f(g)cp)I(1f(g)cP,1/Jk)ldJl(g)

1 r 00

c<p Je !F(g) If; I(cpk, 1f(g)cp) II(1f(g)cP, 1/Jk)ldJl(g)

1

< ~ lIF(g)1 {~I(l'j,«g)I')I'~ Il«g)l', ,p.)I' } , dM(9)

< ~ r IF(g)III1f(g)cpI12dJl(g)C<p Je

~ r !F(g)ldJl(g) = ~1!FII£l(e).C<p Je c<p

Also, by (14.1) and (14.2),

!F<p(g) I I(LF,<p1f(g)cP, 1f(g)cp) I00

= 2: Sk (LF,<p) (1f(g)cP, CPk)(1/Jk, 1f(g)cp)k=l

1~) 2 2S 2L. sk(LF,<p (I (1f(g)cP, CPk)1 + I(1/Jk ,1f(g)cp)I )k=l

for all 9 in G. So, using (14.5), Fubini's theorem, the irreducibility of the representation 1f : G -+ U(X) and hence the resolution of the identity formula in

74

Theorem 6.1, and

we get

14. Trace Class Norm Inequalities

lI<pkll = II'l/Jkll = 1, k = 1,2, ... ,

[ 1F<p(g)ldjL(g)

< ~ ~ sk(Lp,<p) {[ I(rr(g)<p, <Pk)12djL(g) + [I('l/Jk, rr(g)<p) 12djL(9)}

00

= c<p L sk(Lp,<p)'k=1

Thus, using the definition of 111181 and (14.6),

and the proof is complete.

(14.6)

o

An immediate consequence of Theorem 14.1 is the following improvement ofTheorem 13.5.

Theorem 14.5 Let F E LP(G), 1 ::; P ::; 00. Then the localization operator Lp, X is in 8p and

The proof of Theorem 14.5 is the same as the proof of Theorem 13.5 if wereplace the 8 1-estimate by

We are now in a position to derive optimal upper and lower bounds for thenorms in 8 1 of convex functions of self-adjoint localization operators. We beginwith a brief recall of convex functions and Jensen's inequality.

A real-valued function v defined on an interval (a, b), where -00 ::; a < b ::;00, is said to be convex if

v((l - A)X + AY) ::; (1 - A)V(X) + AV(Y)

for all x and Y in (a,b) and all A in [0,1]. A fundamental property of convexfunctions is provided by the following theorem.


Theorem 14.6 Let f E L 1 (n,J.L), where (n,J.L) is a probability measure space, i.e.,J.L(n) = 1. Let f be a real-valued function in L 1(n, J.L) such that the range of f iscontained in an open interval (a, b), where -(X) ~ a < b ~ 00. If v is a convexfunction on (a, b), then

v (in f(W)dJ.L(W)) ~ in (v 0 f) (w)dJ.L(w).

Remark 14.7 We allow a = -(X) or b = 00. The function v 0 f may fail to be inL 1(n,J.L), and in this case,

in (v 0 f)(w)dJ.L(w) = 00.

The inequality in Theorem 14.6 is known as Jensen's inequality. A proof can befound on pages 62 and 63 of the book [74] by Rudin.

Let F be a real-valued function in L 1(0). Then, by Proposition 13.1, thelocalization operator LF,cp : X ----+ X is in 81. Furthermore, it is self-adjoint. Indeed,for all x and Y in X, we can use (12.1) and get

~1F(g)(x, 1f(g)cp)(1f(g)cp, y)dJ.L(g)Ccp G

~1F(g)(y, 1f(g)cp) (1f(g)cp, x)dJ.L(g)Ccp G

(LF,cpY,x) = (x,LF,cpY)'

Hence, using the spectral theorem for compact and self-adjoint operators, we get

00

LF,cpX = LAj(X,cpj)cpj, x EX,j=1

(14.7)

where {cpj : j = 1,2, ...} is an orthonormal basis for X consisting of eigenvectorsof LF,cp : X ----+ X and Aj is the eigenvalue of LF,cp : X ----+ X corresponding to theeigenvector CPj, j = 1,2, ... , and the convergence is understood to be in X.

Let H : JR ----+ JR be a convex function. Then H is a bounded function oncompact subsets of JR. We define the bounded linear operator H(LF,cp) : X ----+ Xby

00

(H(LF,cp)x,y) = LH(Aj)(X,CPj)(CPj,y)j=1

(14.8)

for all x and y in X. That H(LF,cp) : X ----+ X is a bounded linear operator is easyto check. Indeed, we use the fact that H is a bounded function on compact subsets

76 14. Trace Class Norm Inequalities

of JR, Schwarz' inequality and Parseval's identity to get00

I(H(Lp,cp)x, yl :::; L IH(Aj )1I(x, <pj )11(<pj, y)1j=l

1 1

< AE~;TF)H(.\)1{t, I(x, 'P; )1'r{t, I(y, 'P; )1'rsup IH(A)llIxlillyll,

AE~(LF,,,,)

where E(Lp,cp) is the spectrum of Lp,cp : X --+ X.

Proposition 14.8 H(Lp,cp) : X --+ X is a self-adjoint operator.

Proof For all x and y in X, we get00

(H(Lp,cp)x,y) = LH(Aj)(X,<pj)(<pj,y)j=l00

L H(Aj )(y, <pj)(<Pj, x)j=l

(H(Lp,cp)y, x) = (x, H(Lp,cp)Y).

o

Proposition 14.9 For j = 1,2, ..., H(Aj) is the eigenvalue of H(Lp,cp) : X --+ Xcorresponding to the eigenvector <pj'

Proof Since {<Pk : k = 1,2, ...} is orthonormal, it follows that

00

(H(Lp,cp)<pj,x) = LH(Ak)(<pj,<Pk)(<Pk,x) = (H(Aj)<pj,X)k=l

for j = 1,2, ..., and all x in X. 0

We can now give a necessary and sufficient condition for the bounded linearoperator H(Lp,cp) : X --+ X to be in 81 in terms of the convex function H : JR --+ RIn the case when H(Lp,cp) : X --+ X is in 81 , we can give an upper bound and alower bound for the norm IIH(Lp,cp)lIs1 of H(Lp,cp) : X --+ X in 8 1 in terms of thefunctions H 0 F : G --+ JR and H 0 Fcp : G --+ JR respectively, where the function Fcpis defined by (14.1).

Theorem 14.10 The bounded linear operator H(Lp,cp) : X --+ X is in 81 if andonly if

00

L IH(Aj)1 < 00.

j=l


Proof By Proposition 14.9 and the spectral mapping theorem, IH(Aj)l, j =1,2, ..., are the singular values of H(Lp,<p) : X --+ X. Thus, using the definition of81 , the proof is complete. D

Now, we define IH(Lp,<p)lsl by

IH(Lp,<p)lsl = {IIH(LooP,<p)IIS1 if L:j:1 IH(Aj)1 < 00, (14.9)

if L:j:1IH(Aj)1 = 00,

and we have the following theorem.

Theorem 14.11 Let F E L1(G) be a real-valued function, and let H : JR --+ JR be anonnegative and convex function. Then

where F<p : G --+ JR is defined by (14.1).

Remark 14.12 If H(A) = IAI for all A E JR, then Theorem 14.11 is a special case ofTheorem 14.1. Thus, by Remark 14.4, Theorem 14.11 is sharp.

Remark 14.13 The inequalities in Theorem 14.11 are refined analogues of the inequalities (2.74) in Chapter 5 of the book [4] by Berezin and Shubin for localizationoperators. The origin of these inequalities can be traced back to the study of aFeynman inequality in quantization. See the paper [91] by Symanzik for details.

Proof of Theorem 14.11. Let x E X be such that IIxll = 1. Then, by (14.7), (14.8)and Jensen's inequality,

00

(14.10)

(H(Lp,<p)x,x) L H(Aj)l(X, CPjWj=1

> H (t, Aj I(x, I'j)1') ~ H«LF,.X, x)).

In (14.10), we let x = n(g)cp, 9 E G, integrate on G and use (14.1). Then we get

L(H(Lp,<p)n(g)cp, n(g)cp)dj.t(g) > LH((Lp,<pn(g)cp, n(g)cp))dj.t(g)

LH(F<p(g))dj.t(g). (14.11)

Now, by (14.8), (14.9), Fubini's theorem, the resolution of the identity formula inTheorem 6.1 and the fact that

IIcpjll = 1, j = 1,2, ... ,

78

we get

14. Trace Class Norm Inequalities

fa (H(LF,ep)1r(g)cP, 1r(g)cp)dJ.L(g)

< 1f IH(Aj)II(1r(g)cp, cpjWdJ.L(g)G j=l

f IH(Aj)lll(1r(g)cP, cpjWdJ.L(g)j=l G

00 00

L IH(Aj)1 Cep IIcpjll2 = Cep L IH(Aj)1j=l j=l

cepIH(LF,ep)lsl·

So, by (14.11) and (14.12),

1-IIH 0 Fepll£l(G) ~ !H(LF,ep)!Sl'cep

Next, by (12.1) and (14.9),

(14.12)

(14.13)

00 00

IH(LF,ep)lsl = L IH(Aj)1 = L IH(LF,epcpj, cpj)1j=l j=l

~ IH (C~ fa F(g)l(cpj,1r(g)cp) 12 dJ.L(9)) ,. (14.14)

SinceIlcpjll = 1, j = 1,2, ... ,

it follows from the resolution of the identity formula in Theorem 6.1 again that

~11(cpj, 1r(g)cp)12dJ.L(g) = IIcpjl12 = 1 (14.15)cep G

for j = 1,2, . ... Thus, by (14.14), (14.15), Jensen's inequality and Fubini's theorem,

(14.16)

So, by (14.16), Parseval's identity, the fact that 1r : G --+ U(X) is a unitaryrepresentation and Ilcpll = 1, we get

IH(LF,ep)lsl ~ ~ I IH(F(g))III1r(g)CPI12dJ.L(g) = ~ IIH(F(g))ldJ.L(g). (14.17)~ G ~ G

Thus, by (14.13) and (14.17), the proof is complete. 0

15 Hilbert-Schmidt Localization Operators

Let rr : G ---. U(X) be an irreducible and square-integrable representation ofa locally compact and Hausdorff group G on a Hilbert space X. Then for allfunctions F in LP(G), 1 :S p :S 00, and all admissible wavelets t.p for rr : G ---.U(X), Proposition 12.3 ensures that we can get a unique bounded linear operatorL P,<p : X ---. X such that

and

(Lp,<px, y) = ~1F(g)(x, rr(g)t.p)(rr(g)t.p, y)dJ.L(g)C<p G

for all x and y in X whenever F is a simple function on G for which

J.L{g E G : F(g) :f O} < 00.

(15.1 )

(15.2)

Furthermore, Theorem 14.5 tells us that Lp,<p : X ---. X is in the Schatten-vonNeumann class 8p and

(15.3)

These results entail the use of the Riesz-Thorin theorem and Theorems 2.10and 2.11 in the theory of interpolation. However, when p = 2, we can give anexplicit formula, i.e., (15.2), for the localization operator Lp,<p : X ---. X for all Fin L 2 (G) and prove that it is a bounded linear operator. If, in addition, the groupG is unimodular, then we can prove that L P,<p : X ---. X is in the Hilbert-Schmidtclass 82 satisfying (15.1) and (15.3) without using interpolation theory.

Let us begin with the following proposition.

Proposition 15.1 The localization operator Lp,<p : X ---. X defined by (15.2) for allx and y in X and all F in L2 (G) is a bounded linear operator and


80 15. Hilbert-Schmidt Localization Operators

Proof For all x and y in X, the function H : G --t C defined by

H(g) = (x, 1f(g)<p)(1f(g)<p, y), 9 E G, (15.4)

is in L 2 (G). Indeed, using Schwarz' inequality, 11<p11 = 1 and the fact that 1f: G --t

U(X) is a unitary representation of G on X, we get

I(1f(g)<p, y)1 ::; Ilyll, 9 E G.

So, using (15.5) and the resolution of identity formula in Theorem 6.1,

(15.5)

LIH(g)12dfk(9) LI(x, 1f(g)<p) (1f(g)<p, yWdfk(9)

< (L I(x, 1f(g)<P)12dfk(9)) lIyll2

c<p Ilx1l211y112. (15.6)

Thus, by (15.2), (15.4), (15.6), Schwarz' inequality and the resolution of the identity formula in Theorem 6.1,

and the proof is complete. D

Proposition 15.2 Let G be a unimodular group. Then for all F in L 2 (G), the localization operator L F,<p : X --t X is in the Hilbert-Schmidt class 82 and

Proposition 15.2 is definitely much weaker than Theorem 14.5 in which theunimodularity of the group G is not needed. However, it is instructive to see aproof of the Hilbert-Schmidt property of localization operators from first principles without using interpolation theory. The price for this elementary proof isunimodularity.

To prove Proposition 15.2, we use the following lemma, which follows from(15.2) immediately.

Lemma 15.3 The adjoint of LF,<p : X --t X is Lp,<p : X --t X.

Proof of Proposition 15.2. Let F E L1(G) n L 2 (G). Let {<Pk : k = 1,2, ...} bean orthonormal basis for X. Then, by (15.2), Lemma 15.3, Fubini's theorem and

Parseval's identity,

15. Hilbert-Schmidt Localization Operators 81

00

2: II L F,<p<PkI12

k=1

00

2:(LF,<p<Pk' LF,<p<Pk)k=1

00 1 1(; C<p G F(g)(<pk' 1r(g)<p) (1r(g)<p, LF,<p<Pk)d/-L(g)

00 1 1(; C<p G F(g)(<Pk, 1r(g)<p) (Lp ,<p1r(g)<p, <Pk)d/-L(g)

~1F(g)(Lp ,,,1r(g)<p, 1r(g)<p)d/-L(g). (15.7)c<p G 'r

(15.9)

Hence, by (15.5), (15.7), FUbini's theorem, Schwarz' inequality, the unimodularity of G, the fact that 1r : G --+ U(X) is an irreducible and square-integrablerepresentation, and II xII = 1, we get

00

L II L F,<p<PkI12

k=1

(C~) 2 fa F(g) (fa F(h)I(1r(h)<P,1r(g)<p)12d/-L(h)) d/-L(g)

(C~) 2 fa F(g) (fa F(h)I(1r(g-lh)<P,<p)!2d/-L(h)) d/-L(g)

(C~) 2 fa F(g) (fa F(gh)I(1r(h)<p, <P)12d/-L(h)) d/-L(g)

(C~) 2 fa 1(1r(h)<p,<pW (fa F(9)F(9h)d/-L(9)) d/-L(h)

1 2< -IIFIIL2(G)' (15.8)

c<p

Thus, by (15.8), we get

1

IILF,<pIIS2 :S (C~) 2 1IFIIL2(G)

for all F in LI(G) n L2(G). Now, let F E L2(G). Then there exists a sequence{Fl}~1 of functions in LI(G) n L2(G) such that

(15.10)

in L 2 (G) as l --+ 00. By (15.10) and Proposition 15.1,

(15.11)

82 15. Hilbert-Schmidt Localization Operators

in B(X) as l ~ 00. By (15.9) and (15.10),

IILFL,'P - AII s2 ~ 0 (15.12)

for some A in 82 as l ~ 00. Now, we claim that LFI''P ~ A in B(X) as l ~ 00.

Assume that this is true for a moment. Then, by (15.11),

LF,'P = A.

Hence LF,'P : X ~ X is in 82 and, by (15.9), (15.10), (15.12) and (15.13),

(15.13)

It remains to prove that LFI,'P ~ A in B(X) as l ~ 00. By Schwarz' inequality,Fubini's theorem and Parseval's identity,

II(LFz ,'P - A)x1l 2

(LF". - A) t,(X, 'P; )'P;, (LF". - A) f.(X, 'Pk)'Pk )

00 00

< L L I(x, ipj)ll(x, ipk)III(LFz,'P - A)ipjllll(LFI''P - A)ipkllj=lk=l

(t, I(x, 'P; )111 (LF". _ A)'P; II),

< (t, I(x, 'P; )1') (t, II(LF". - A)'P; II')IILFI''P - AII~211x112

for all x in X, and hence

(15.14)

for l = 1,2, .... Thus, by (15.12), (15.13) and (15.14), LFI,'P ~ A in B(X) asl ~ 00. 0

We end this chapter with a formula for the norm IILF ''Plls2 of the HilbertSchmidt localization operator LF,'P : X ~ X. To this end, we let F'P be thefunction on G defined by

(15.15)

15. Hilbert-Schmidt Localization Operators

In fact, for all 9 in G,

83

Thus, the square of the function Fr.p can be considered as the expectation value ofthe observable L'F,r.pLF,r.p : X --t X in the coherent states 1r(g)cp, 9 E G.

Theorem 15.4 Let F E L2 (G). Then

Proof. By Theorem 14.5, LF,r.p E 82. Using the canonical form for compact operators given by Theorem 2.2, we can write

00

LF,r.px = L Sk (LF,r.p) (x, CPk)'l/Jk, X E X,k=l

(15.16)

where sk(LF,r.p), k = 1,2, ..., are the positive singular values of LF,r.p : X --t X,{CPk: k = 1,2, ...} is an orthonormal basis for N(LF,r.p)J.., and {'l/Jk: k = 1,2, ...}is an orthonormal set in X. Thus, by (15.15) and (15.16), we get

00

(Fr.p(g))2 = L(Sk(LF,r.p))21(1r(g)CP,CPkW, 9 E G.k=l

So, by (15.17),

fc (Fr.p(g))2dp,(g) = ~(Sk(LF,r.p))2 fc I(1r(g)cp, CPkWdp,(g).

(15.17)

(15.18)

Hence, applying the resolution of the identity formula in Theorem 6.1 to the righthand side of (15.18), we complete the proof. 0

16 Two-Wavelet Theory

The results hitherto given are for localization operators L F,<p : X --t X definedin terms of one admissible wavelet cp for the square-integrable representation rr :G --t U(X) of G on X. In this chapter we introduce the notion of a localizationoperator LF,<p,,p : X --t X, which is defined in terms of a symbol F in L1(G)and two admissible wavelets cp and 'ljJ for the square-integrable representation rr :G --t U(X) of G on X. It is proved in this chapter that LF,<p,,p : X --t X is in8 1 and a formula for the trace of LF,<p,,p : X --t X is given. These results extend,respectively, the corresponding results in Chapter 12 and Chapter 13 from theone-wavelet case to the two-wavelet case. We also give in this chapter the traceclass norm inequalities for the localization operator LF,<p,,p : X --t X. In order toobtain a lower bound for the norm IILF,<p,,pllsl of LF,<p,,p : X --t X, we need theformula (9.1), which is an analogue of the resolution of the identity formula (6.3)for two admissible wavelets for an irreducible and square-integrable representationrr : G --t U(X) of G on X.

Let cp and 'ljJ be two admissible wavelets for an irreducible and square-integrable representation rr : G --t U(X) of G on X such that the two-wavelet constantc<p,,p defined by (9.2) is nonzero. Then we have

(x, y) = _1_ r (x, rr(g)cp) (rr(g)'ljJ, y)dJ1-(g) , x, y E X.c<p,,p JG

We call this formula the resolution of the identity formula for the irreducible andsquare-integrable representation rr : G --t U(X) of G on X corresponding to theadmissible wavelets cp and 'ljJ.

Let cp and 'ljJ be admissible wavelets for the square-integrable representationrr : G --t U(X) such that c<p,,p i= O. Let F E L1(G). Then we define the localizationoperator LF,<p,,p : X --t X by

(LF,<p,,px, y) = -1-1 F(g)(x, rr(g)cp)(rr(g)'ljJ, y)dJ1-(g) , x, y E x.C<p,,p G

(16.1)

Theorem 16.1 The localization operator LF,<p,,p : X --t X is in the trace class 81

and

('ljJ, cp) 1tr(LF,<p,,p) = -- F(g)dJ1-(g).C<p,,p G


16. Two-Wavelet Theory 85

Proof. By (16.1), Schwarz' inequality, the fact that 7r : G --+ U(X) is a unitaryrepresentation, and IIcpli = II'¢II = 1, we get

I(LF,cp,,px, y)1

::; -I1-lllxllllyli1IF(g) I117r(g)cpli 117r(g)'¢IIdJL(g) = -Il-,IIFII£l(G)llxli IIyllCcp,t/J G ccp,t/J

for all x and y in X. Thus, LF,cp,t/J : X --+ X is a bounded linear operator. Next,let {CPk : k = 1,2, ...} be an orthonormal basis for X. Then, by (16.1), Fubini'stheorem, Schwarz' inequality, Parseval's identity, the fact that 7r: G --+ U(X) is aunitary representation, and IIcpli = II'¢II = 1, we get

<Xl

L IILp,cp,t/JCPkI1 2

k=l

<Xl

L(LF,cp,t/JCPk, LF,cp,t/JCPk)k=l

1 <Xl 1- L F(g)(CPk, 7r(g)cp)(7r(g)'¢, LF,cp,t/JCPk)dJL(g)ccp,t/J k=l G

1 <Xl 1- L F(g)(CPk, 7r(g)cp) (L'F,cp,t/J7r(g),¢, CPk)dJL(g)ccp,t/J k=l G

-1-1 F(g) (L'F,cp,t/J7r(g)'¢, 7r(g)cp)dJL(g)Ccp,t/J G

< -I1-11IF(g)!II7r(g)cpIIIL'F,cp,t/JII*"7r(g)'¢lldJL(9)Ccp,t/J G

Ic;,t/JIIIL'F,cp,t/JII*IIFII£l(G) < 00, (16.2)

where Lp,'P',p : X --+ X is the adjoint of LF,cp,t/J : X --+ X. So, by (16.2) andProposition 2.8, LF,cp,t/J : X --+ X is in 82 and hence compact. Now, let {CPk :k = 1,2, ...} and {'¢k : k = 1,2, ...} be orthonormal sets in X. Then, by Fubini'stheorem, Schwarz' inequality, Bessel's identity, the fact that 7r : G --+ U(X) is aunitary representation, and IIcpll = II'¢II = 1, we get

<Xl

L I(LF,cp,t/JCPk,'¢k)1k=l

1 <Xl 1< Iccp,t/JI ~ G!F(g)!I(cpk, 7r(g)cp) I(7r(g)'¢, '¢k)!dJL(g)

1 1

< !c;,t/JI LIF(g) I{~I(CPk' 7r(g)cp) 12 }:2 {~1(7r(9)'¢' '¢kW } :2 dJL(g)

< -I1-11 IF(g) I 117r(g)cpllll7r(g),¢lldJL(g)Ccp,t/J G

1-I-IIIFII£l(G) < 00. (16.3)ccp,t/J

86 16. Two-Wavelet Theory

Thus, by (16.3) and Proposition 2.5, LF.<p.,p : X ---t X is in 81. Finally, let {<Pk : k =1,2, ...} be any orthonormal basis for X. Then, by Fubini's theorem, Parseval'sidentity, the fact that 7r : G ---t U(X) is a unitary representation, and II<pII = II'¢II =1, we get

00

'l)LF.<p.,p<Pk, <Pk)k=1

1 00 1= c<p.,p £; G F(g)(<pk' 7r(g)<p)(7r(g),¢, <Pk)dj.L(g)

1 1 00c<p.,p G F(g) £;(<pk, 7r(g) <p) (7r(g),¢, <Pk)dj.L(g)

-1-1 F(g)(7r(g),¢,7r(g)<p)dj.L(g)C<p.,p G

(,¢, <p) 1F(g)dj.L(g) ,C<p.,p G


Let F<p.,p be the function on G defined by

F<p.,p(g) = (LF.<p.,p7r(g)<p,7r(g),¢), 9 E G.

o

(16.4)

The function F<p.,p plays in the two-wavelet theory the role of the function F<pdefined by (14.1). It can be interpreted as the correlation of the filtered signalsLF.<p.,p7r(g)<p, 9 E G, generated by the admissible wavelet <P with the other family7r(g),¢, 9 E G, of signals generated by the admissible wavelet '¢.

We have the following analogue of Theorem 14.1 on the trace class norminequalities for two-wavelet localization operators.

Theorem 16.2 Let F E Ll(G). Then

If, in addition, the square-integrable representation 7r : G ---t U(X) of G on X isirreducible, then

Remark 16.3 To see that F<p.,p E L1(G), we use Fubini's theorem, the left invariance of j.L, the definition of one-wavelet constants, and the fact that 7r : G ---t U(X)

16. Two-Wavelet Theory

is a unitary representation to get

1IFcp,,p(g) IdJL(g)

11-1-1 F(h)(7r(g )<p, 7r(h) <p)(7r(h )1jJ, 7r(g)1jJ )dJL(h)1dJL(g)G ccp,,p G

< -,l-111IF(h)II(7r(g)<p, 7r(h)<p)1I(7r(h)1jJ, 7r(g)1jJ)ldJL(h)dJL(g)ccp,,p G G

< 2Ic:,,p,l,F(h), (1 1(7r(g)<p, 7r(h)<PW dJL(g)) dJL(h)

+2Ic:,,p1 fc IF(h)1 (fc 1(7r(h)1jJ, 7r(g)1jJ)12dJL(9)) dJL(h)

< 21:;,,p1 fc IF(h)ldJL(h) + 2IZ,,p1 fc IF(h)ldJL(h)

Ccp+C,p1= 2Iccp,,p1 G IF(h)ldJL(h) < 00.

87

Remark 16.4 If X is a positive operator. So, by Proposition 2.7 and Theorem13.6,

-1-1 Fcp,,p(g)dJL(g) = II LF,cp,,pllsl = 2-1 F(g)dJL(g).Ccp,,p G Ccp G

Thus, the estimates in Theorem 16.2 are sharp.

Proof of Theorem 16.2. By Theorem 16.1, the localization operator LF,cp,,p : X ---->

X is in 81, Using the canonical form for compact operators given in Theorem 2.2,we get

<Xl

LF,cp,,px = L~>k(LF,cp,,p)(x, Uk)Vk, X E X,k=1

(16.5)

where sk(LF,cp,,p), k = 1,2, ... , are the positive singular values of LF,cp,,p : X ----> X,{Uk: k = 1,2, ...} is an orthonormal basis for N(LF,cp,,p).1., {Vk: k = 1,2, ...} isan orthonormal set in X and the convergence of the series is understood to be inX. By (16.5),

So, by (16.6),

<Xl <Xl

Z)LF,cp,,puj,Vj) = LSj(LF,cp,,p).j=1 j=1

<Xl

IILF,cp,,pllsl = L(LF,cp,,puj, Vj)'j=1

(16.6)

(16.7)

88 16. Two-Wavelet Theory

Thus, by (16.1), (16.7), Fubini's theorem, Parseval's identity, Bessel's inequality,Schwarz' inequality, Ilcpll = II'¢II = 1 and the fact that 7r : G --+ U(X) is a unitaryrepresentation, we get

IILF,cp,,pllsl = 1~(LF'CP,,pUk' Vk)1

< ~ Ic;,,p fa F(g)(Uk' 7r(g)cp) (7r(g),¢, vk)dj.L(g) I

00 1 1< f; ICcp,,p1 G IF(g)II(Uk' 7r(g)cp) II(7r(g),¢, vk)ldj.L(g)

1 1 00ICcp,,p1 G IF(g)1 f; I(Uk, 7r(g)cp) II(7r(g),¢, Vk)ldj.L(g)

1 1

< Ic;,,pl fa IF(g)1 {~I(Uk, 7r(g)cp) 12 } 2 {~1(7r(9)'¢' VkW } 2 dj.L(g)

< -I1-IIIF(g)III7r(g)cpllll7r(g)'¢lIdj.L(g)Ccp,,p G

-I1-II IF(g)ldj.L(g) = -I1-IIIFII£l(G)'

Ccp,,p G Ccp,,p

Also, by (16.4) and (16.5),

IFcp,,p (g) I I(L F,cp,,p7r(g)cp, 7r(g),¢) I

I~ sk(LF,cp,,p)(7r(g )cp, Uk)(Vk, 7r(g),¢)I1~ I 2 2< '2 L,.;sk(LF,cp,,p)( (7r(g)cp,uk)1 + I(Vk,7r(g)'¢)1 )

k=l

(16.8)

for all 9 in G. So, by the irreducibility of the representation 7r : G --+ U(X)and hence the resolution of the identity formula in Theorem 6.1, (16.8), Fubini'stheorem and the fact that

we get

fa IFcp,,p(g)ldj.L(g)

< ~ ~ sk(LF,cp,,p) {fa I(7r(g)cp, uk)1 2dj.L(g) + fa I(Vk, 7r(9)'¢Wdj.L(9)}

16. Two-Wavelet Theory

1 00

< 2'(ccp + c",) L sk(LF,cp,,,,)'k=l

Thus, by (16.9),

: rIFcp,,,,(g)!djL(g) ::; IILF,cp,,,,llsllCcp 41e


89

(16.9)

o

17 The Weyl-Heisenberg Group

We show in this chapter that localization operators on the Weyl-Heisenberg groupare the same as the linear operators studied by Daubechies in the paper [12] onsignal analysis. We begin with a detailed study of the Weyl-Heisenberg group.

Let IRn x IRn = {(q,p) : q, p E IRn } and let Z be the set of all integers. Let(WH)n = IRn x IRn x 1R/21TZ. Then we define the binary operation on (WH)nby

(ql, PI, td . (q2, P2, t2) = (ql + q2, PI + P2, tl + t2 + ql . P2) (17.1)

for all points (ql, PI, tl) and (q2, P2, t2) in (WH)n, where ql . P2 is the Euclideaninner product of ql and P2 in IRn ; tI, t2 and tl +t2+ql 'P2 are cosets in the quotientgroup 1R/21TZ in which the group law is addition modulo 21T. It is easy to provethe following proposition and we omit the proof.

Proposition 11.1 With respect to the multiplication· defined by (17.1), (WH)nis a non-abelian group in which (0,0,0) is the identity element and the inverseelement of (q, P, t) is (-q, -P, -t + q. p) for all (q, p, t) in (WH)n.

Remark 11.2 To simplify the notation a little bit, we identify IRn x IRn with en.Thus, (WH)n = en x 1R/21TZ, which can also be identified with en x [0,21T] =IRn x IRn x [0, 21T].

Proposition 11.3 The Lebesgue measure dq dp dt on IRn x IRn x [0,21T] is the left(and right) Haar measure on (WH)n.

Proof. To prove left invariance, let f be an integrable function on (WH)n. It ishelpful to think of f as a function on IRn x IRn x IR such that f(q,p,·) is a periodicfunction with period 21T for fixed but arbitrary q and p in IRn . Then for all (z', t')in (WH)n, we get

! f((z',t')·(z,t))dzdt(WH)n

r21r r r f(q' + q, p' + p, t' + t + q' . p)dqdpdtlo lIRn lIRn

121r+ t' +q'.p1

f(z, s)dzdst'+q'.p en

= r21r r f(z, s )dz dslo len


17. The Weyl-Heisenberg Group

= 1 f(z, t)dzdt.(WH)"

The proof for right invariance is similar.

91

o

Remark 17.4 With respect to the multiplication· defined by (17.1), the set (WH)nbecomes a locally compact and Hausdorff group on which the left (and right) Haarmeasure is the Lebesgue measure on lRn x lRn x [0, 27f]. We call (WH)n the WeylHeisenberg group. In light of the existence of a left (and right) Haar measure on(WH)n, (WH)n is unimodular.

Let 7f: (WH)n --+ U(L2 (lRn)) be the mapping defined by

(7f(q,p,t)J)(x) = ei(p.x-q.p+t) f(x - q), x E lRn , (17.2)

for all points (q,p,t) in (WH)n and all functions f in L2(lRn).

Proposition 17.5 7f : (WH)n --+ U(L2(lRn)) is a representation of (WH)n onL2 (lRn ).

Proof. Let (qI,PI, h) and (q2,P2, t2) be points in (WH)n. Then for all functionsfin L2 (lRn ), by (17.2),

(7f(ql, pI, tl)7f(q2, P2, t2)J)(X)ei(PI'X-ql'PI+tl)(7f(q2,P2, t2)J)(X - ql)ei(PI'X-ql'PI HI) ei(P2'(X-qt)-q2'P2H2) f(x - q2 - qI)

ei((PI +P2)'X-(PI+P2)·ql +tt H2-q2'P2) f(x - (ql + q2)) (17.3)

and

(7f((ql,PI, tl) . (q2,P2, t2))J)(X)

(7f(ql + q2,PI + P2, tl + t2 + ql . P2)J)(X)ei((Pl +P2)'X-ql'Pl -Q2'PI-q2'P2+tlH2) f(x - (ql + q2)) (17.4)

for all x in lRn . Hence, by (17.3) and (17.4),

7f(qI, PI, tI)7f(q2, P2, t2) = 7f((ql, PI, tI) . (q2, P2, t2))

for all points (qI,PI,tl) and (q2,P2,t2) in (WH)n.It remains to prove that 7f(q,p,t)f --+ f in L 2(lRn ) as (q,p,t) --+ (0,0,0) for

all functions fin L2 (lRn). But

117f(q,p, t)f - fIlI2(1R")

1" lei(p.x-q'PH) f(x - q) - f(x)12

dx

1" lei(p.x-q.p+t){f(X - q) - f(x)} + {ei(p.x-q.p+t) - I} f(x)12

dx

< 2 r {If(x - q) - f(x)1 2 + l(ei(p.x-q·PH) -l)f(xW}dx (17.5)JR."

92 17. The Weyl-Heisenberg Group

for all (q,p,t) in (WH)n and all functions f in L2(JRn). By the L2-continuity oftranslations,

r If(x - q) - f(xWdx ...... °JIRn

as q ...... 0. For almost all x in JRn,

I(ei(p.x-q.p+t) -1) f(x)12

...... °as (q,p,t) ...... (0,0,0), and

I(ei(p'X-q'PH) -1) f(x)12~ 4If(xW.

Hence, by (17.7), (17.8) and the Lebesgue dominated convergence theorem,

in I(ei(p'X-q'PH) -1) f(x)12

dx ...... °as (q,p, t) ...... (0,0,0). Hence, by (17.5), (17.6) and (17.9),

117T(q,p, t)f - fll£2(IRn) ...... °

(17.6)

(17.7)

(17.8)

(17.9)

(17.10)

as (q,p, t) ...... (0,0,0), and the proof is complete. D

We call 71' : (WH)n ...... L2 (JRn) the Schrodinger representation of the WeylHeisenberg group (WH)n on L 2 (JRn) and the following theorem gives us all theinformation that we want to know about it.

Theorem 17.6 For all functions f and 9 in L2(JRn), we have

1 IU,7T(Z,t)g)L2(IRn)12dzdt = (27T)n+lllfll~,2(lRn)lIglli2(lRn).(WH)n

Remark 17.7 The proof of Theorem 17.6 requires some basic knowledge of Fourieranalysis, which we assume. Standard references include the books by Goldberg [32],Stein and Weiss [88] and Wong [103].

Proof of Theorem 17.6. We begin with the case when both f and 9 are in theSchwartz space S. If we denote the left-hand side of (17.10) by lU, g), then

lU,g) = r27r r r Ir f(x)e-i(p.x-q·P+t)g(x - q)dXl2 dqdpdtJo JIRn JIRn JIRn

271' r r Ir e-ip.xf(x)g(x - q)dXl2 dqdpJIRn JIRn JIRn

(27Tt+1 r r !(27T)-lj r e-ip'x f(X)(T_ qg)(X)dX I

2dqdp

JIRn JIRn JIRn

(27Tt+l r r IUT_ qg)"'(p)1 2dqdp, (17.11)JIRn JIRn

17. The Weyl-Heisenberg Group 93

where (T_qg) = g(x - q), x, q E ~n. So, by (17.11), Plancherel's theorem andFubini's theorem,

I(J,g) (271"t+l r r If(x)g(x-q)1 2dxdqJJRn JJRn

(271")n+l In If(xW (In Ig(x - q)12dq) dx

(271")n+lll flli2(JRn) Ilglli2(JRn).

Now, for f, 9 E L2(~n), let {!k}~1 and {gk}~l be sequences in S such thatfk ---+ f in L2(~n) and gk ---+ 9 in L2(~n) as k ---+ 00. Then, by what we have justshown,

(17.12)

as k ---+ 00. Also,

(17.13)

for all (z, t) in (WH)n as k ---+ 00. Furthermore, for all j and k,

l(Jj, 71"(x,t)9j)£2(JRn) - (Jk,71"(z,t)gk)£2(JRn) 12

l(Jj -!k, 71"(Z, t)gj)L2(JRn) + (!k, 71"(Z, t)(gj - gk))£2(JRn) 12

< 2J(Ji - fk' 71"(Z, t)gj )£2 (JRn) 12+ 21 (!k, 71"(Z, t)(gj - gk) )£2 (JRn) /2,

and hence, using (17.10) for functions in S, we get a positive constant C such that

1 I(fJ, 71"(z, t)gj )£2(JRn) - (ik, 71"(z, t)gk)£2(JRn) /2dzdt(WH)n

< C (11Ji - fklli2(JRn) + Ilgj - gklli2(JRn)) ---+ 0

as j, k ---+ 00. So,(Jk, 71"(" ·)gk)£2(JRn) ---+ h (17.14)

for some h in L 2( (WH)n) as k ---+ 00. Therefore there exists a subsequence of

{(!k, 71"(-, ·)gk)L2(JRn)}~1' again denoted by {(!k, 71"(" ·)gk)£2(JRn)}~l' such that

(Jk, 71"(" ·)gk)£2(JRn) ---+ h

a.e. on (WH)n as k ---+ 00. Thus, by (17.13)-(17.15),

I(Jk, gk) ---+ r I(J, 71"(z, t)9)£2(JRn) 12dzdt

J(WH)n

as k ---+ 00. Hence, by (17.12) and (17.16), the proof is complete.

(17.15)

(17.16)

o


Corollary 17.8 rr : (WH)n -+ U(L2(lRn)) is an irreducible and unitary representation of (WH)n on L2 (lRn).

Proof. That rr : (WH)n -+ U(L2 (lRn)) is a unitary representation of (WH)non L 2 (lRn ) is an immediate consequence of (17.2). Let M be a nonzero andclosed subspace of L2 (lRn ) which is invariant with respect to the representationrr: (WH)n -+ U(L2(lRn)). Let 9 be a nonzero function in M. Then

{rr(z, t)g : (z, t) E (WHt} ~ M.

Let f E L 2 (lRn ) be such that f is orthogonal to M. Then, by (17.17),

Then, by Theorem 17.6 and (17.18),

(17.17)

(17.18)

and hence f = O. So, M is a dense subspace of L 2 (lRn ). Since M is also a closedsubspace of L2(lRn ), it follows that M = L2(lRn ) and the proof is complete. 0

Corollary 17.9 rr : (WH)n -+ U(L2 (lRn)) is a square-integrable representation of(WH)n on L2(lRn).

Proof. Let 'P be any nonzero function in L2 (lRn ). Then, by Theorem 17.6,

and this completes the proof.

(17.19)

o

Corollary 17.10 Every function 'P in L2(lRn ) with lI'PII£2(lRn) = 1 is an admissiblewavelet for the representation rr : (WH)n -+ U(L2(lRn)) of (WH)n on L2(lRn) and

(17.20)

Corollary 17.10 is an immediate consequence of (17.19).

We can now study localization operators on the Weyl-Heisenberg group(WH)n. To this end, let 'P be any function in L2 (lRn) such that 11'PIIL2(lRn) = 1,and let F be any function in L1(lRn x IRn ) UV'O(lRn x IRn ). Let F" be the functiondefined on (WH)n by

F"(q,p, t) = F(q,p), (q,p, t) E (WH)n.

17. The Weyl-Heisenberg Group 95

Then, by (17.2) and (17.20), the localization operator Lp~,<p : L2(lRn ) ---+ L2(lRn )is given by

(Lp~ ,<pf, g) L2(JRn)

= ~ f21r f f F(q,p)(J, rr(q,p, t)cp)£2(JRn) (rr(q, p, t)cp, g)£2(JRn)dq dpdtc<p Jo JJRn JJRn

= (2rr)-n f f F(q,p)(J,CPq,p)£2(JRn)(cpq,p,g)£2(JRn)dqdp (17.21)JJRn JJRn

for all functions f and gin L2(lRn), where CPq,p is the function on IRn given by

(17.22)

for all q and p in IRn . The localization operator Lp~,<p : L2(lRn) ---+ L2(lRn) is thenexactly the same as the linear operator Dp,<p : L2(lRn) ---+ L2(lRn) given by

(Dp,<pf,g)L2(JRn) = (2rr)-n f f F(q,p)(J,CPq,p)£2(JRn)(cpq,p,g)£2(JRn)dqdpJJRn JJRn

(17.23)for all functions f and g in L 2 (lRn ), where cPq,p is the function defined by (17.22).The linear operator Dp,<p : L2(lRn) ---+ L2(lRn) is the localization operator firststudied in the paper [12] by Daubechies in the context of signal analysis, andhence we call Dp,<p : L2(lRn) ---+ L2(lRn ) the Daubechies operator associated to thesymbol F and the admissible wavelet cp. See also Section 2.8 of the book [13] byDaubechies in this connection. By (17.20), (17.21), (17.23) and Theorem 14.5, wehave the following result.

Theorem 17.11 Let F E LP(lRn x IRn ), 1 :::; P :::; 00. Then there exists a uniquelinear operator Dp,<p : L2(lRn ) ---+ L2(lRn) in Sp such that

(17.24)

and, for all junctions f and g in L2(lRn), (Dp,<pf,g)£2(JRn) is given by (17.23)for all simple functions F on IRn x IRn such that the Lebesgue measure of the set((q,p) E IRn x IRn : F(q,p) #- O} is finite.

Proof. We only need to check the inequality (17.24). But, by (17.20), (17.21),(17.23) and Theorem 14.5,

But, by a simple computation,

11F~IILP((WH)n) = (2rr)t 1IFIILP(JRnxJRn).

Thus, by (17.25) and (17.26), (17.24) follows.

(17.25)

(17.26)

o


As a sharp contrast to the Weyl-Heisenberg group (WH)n, we end this chapter by showing that the Heisenberg group Hn introduced in Chapter 8 of the book[1021 by Wong is one on which every irreducible and unitary representation of Hnon L2 (JRn) is not square-integrable, or equivalently, the set AW(1r) of all admissible wavelets for any irreducible and unitary representation 1r : Hn

-t U(L2 (JRn ))

of Hn on L 2 (JRn) is empty.

The Heisenberg group H n is the non-abelian group en x JR in which the grouplaw . is given by

(Z, t) . (w, s) = (z + w, t + s + 2Im(z . w))

for all (z,t) and (w,s) in en x JR, where

n

z·w = LZjWj.

j=1

The Heisenberg group Hn is a unimodular group on which the left (and right)Haar measure is the Lebesgue measure dz dt on en x R

According to the Stone-von Neumann theorem, every irreducible and unitaryrepresentation 1r : Hn -t U(L2 (JRn)) of Hn on L2 (JRn) is, up to unitary equivalence,given by

(1r(Z, t)f)(x) = ei>.(q.x+~q.p+~t) f(x + p), x E JRn ,

for all functions f in L2(JRn), where>. E JR and (z,t) = (q,p,t).

(17.27)

Theorem 17.12 Every irreducible and unitary representation of Hn on L 2(JRn) isnot square-integrable.

Proof. Let 1r : H n-t U(L2 (JRn )) be an irreducible and unitary representation of

Hn on L2 (JRn). Suppose that 1r : Hn -t U(L2 (JRn)) is given by (17.27). Then forall cP in L2 (JRn),

J OO r l(cp,1r(z,t)cp)p(lRnlI2dzdt = JOO (r I(CP,CP>.,q,p)p(lRnlI2dqdP) dt,-ookn -00 knwhere

Thus,

unless

17. The Weyl-Heisenberg Group

or equivalently,

I: <p(x)e-iAq.Xep(x + p)dx = 0, q, P E IRn .

But (17.28) is valid if and only if

<p(x)ep(x +p) = 0

97

(17.28)

for almost all x and pin IRn . Thus, <p(x) = 0 for almost all x in IRn . Indeed, if<p(x) ;/:. 0 for all x in a set S with positive measure. Then for all x in S, <p(x+p) = 0for almost all p in IRn , and this is a contradiction. Hence the representation 7f :

Hn --+ U(L2 (lRn)) of Hn on L2 (lRn) is not square-integrable. 0

(18.1)

18 The Affine Group

We study in this chapter the affine group U, the Hardy space Hi (JR), and anirreducible and unitary representation 1r : U ....... U(Hi(JR)) of U on Hi(JR) forwhich the set AW(1r) of all admissible wavelets for the representation 1r : U .......U(Hi (JR)) is a proper subset of the unit sphere with center at the origin in Hi (JR).

Let U be the upper half plane given by

U = {(b, a) : b E JR, a> OJ.Then we define the binary operation . on U by

(bl, al) . (b2, a2) = (b1 + a1b2, ala2)

for all points (b1,ad and (b2,a2) in U.

Proposition 18.1 With respect to the multiplication . defined by (18.1), U is anon-abelian group in which (0,1) is the identity element and the inverse elementoj (b,a) is (-~,~) jorall(b,a) inU.

Proof. Let (b1,al) and (b2,a2) be points in U. Then, by (18.1),

((b1 , ad· (b2, a2)) . (b3, a3) (b1 + a1b2, ala2) . (b3, a3)

= (b1 + a1b2 + ala2b3, ala2a3)

and

(b1, al) . ((b2, a2) . (b3, a3)) (bl, al) . (b2 + a2b3, a2a3)

= (b1 + a1b2 + ala2b3, ala2a3).

Thus, the associative law is valid. For all (b,a) in U, by (18.1),

(b, a) . (0,1) = (b, a) and (0,1). (b,a) = (b,a).

Thus, (0,1) is the identity element. Finally, let (b,a) E U. Then, by (18.1),

(b,a).(-~,~) =(0,1) and ( -~,~) . (b,a) = (0,1).

Hence the inverse element of (b, a) is (- ~, ~). Therefore U is a group with respectto the multiplication· defined by (18.1). That the group U is non-abelian is easyto check and hence omitted. 0


18. The Affine Group

Proposition 18.2 The left and right Haar measures on U are given by

99

respectively.

and dv = dbdaa

Proof. To prove left invariance, let f be an integrable function on U with respectto dJ-L. Then for all (b', a') in U, we get

1 1=1= ~~f((b', a') . (b, a))dJ-L = f(b' + a'b, a'a)-2-'U 0 _= a

Let (3 = b' + a'b and a = a'a. Then, by (18.2),

1 1=1= d(3da 1f((b',a')· (b,a))dJ-L = f((3,a)-2- = f(b,a)dJ-L.U 0 -= a U

(18.2)

To prove right invariance, let f be an integrable function on U with respect to dv.Then for all (b', a') in U, we get

1 1=1= ~~f((b,a)· (b',a'))dv = f(b+ab',aa')--.U 0 _= a

Let (3 = b + ab' and a = aa'. Then, by (18.3),

1 1=1= d(3da 1f((b,a)· (b',a'))dv = f((3,a)-- = f(b,a)dv.U 0 -= a U

(18.3)

o

Remark 18.3 With respect to the multiplication' defined by (18.1), U is a locallycompact and Hausdorff group on which the left Haar measure is different from theright Haar measure. Thus, U is a non-unimodular group, which we call the affinegroup.

Let H~(JR.) be the subspace of L2 (R) defined by

H~(R) = {f E L 2 (R) : supp(J) ~ [0, co)},

where supp(j) is the set of every x in R for which there is no neighborhood of xon which j is equal to zero almost everywhere. Similarly, we define H~ (R) to bethe subspace of L 2 (R) by

We call H~(R) and H:(R) the Hardy space and the conjugate Hardy space respectively.

100 18. The Affine Group

Proposition 18.4 H~(I~) and H':'(lR) are closed subspaces of L2(lR).

Proof That H~(lR) is a subspace of L2(lR) is obvious. Let {fd~l be a sequencein H~(lR) such that !k ---. f in L2(lR) as k ---. 00. Then, by Plancherel's theorem,

in L 2 (lR) as k ---. 00. Thus, there exists a subsequence of {fd~l' again denotedby {!k}~l' such that

(18.4)

a.e. on lR as k ---. 00. Using the definition of H~(lR) and the definition of supp(fk),

we get !k = °a.e. on (-00,0] for k = 1,2, .... Thus, by (18.4), ! = °a.e. on(-00,0]. Hence f E H~(lR). Therefore H~(lR) is a closed subspace of L2(lR). Theproof that H':' (lR) is a closed subspace of L2 (lR) is similar. 0

To be specific, only the Hardy space H~ (lR) is considered. The discussion isequally valid for the conjugate Hardy space H':' (lR).

Let n : U ---. U(H~(lR)) be the mapping defined by

1 (x -b)(n(b,a)f)(x) = ...;af -a- , x E lR, (18.5)

for all points (b, a) in U and all functions f in H~(lR).

Proposition 18.5 n: U ---. U(H~(lR)) is a representation of U on H~(lR).

To prove Proposition 18.5, we use the subspace W of H~(lR) defined by

W = {f E H~(lR) : ! E 00 (0, oo)}.

Lemma 18.6 W is a dense subspace of H~(lR).

Proof Let f E H~(lR). Then supp(J) ~ [0,00). Let {cpd~l be a sequence offunctions in 0 0 (0,00) such that

cpk ---. f (18.6)

in L2(lR) as k ---. 00. For k = 1,2, ... , let fk be the function in L2(lR) such that

(18.7)

Then fk E W, k = 1,2, ... , and, by (18.6), (18.7) and Plancherel's theorem, fk ---. fin L2(lR) as k ---. 00. Therefore W is a dense subspace of H~(lR). 0

18. The Affine Group 101

Proof of Proposition 18.5. Let (b1, al) and (b2, a2) be points in U. Then, by (18.5),we get, for all functions f in Hi(JR),

(18.8)

and

(18.9)

for all x in R Hence, by (18.8) and (18.9), 7f : U ~ U(Hi(JR)) is a group homomorphism. It remains to prove that 7f(b, a)f ~ f in L2(JR) as (b, a) ~ (0,1) for allfunctions f in Hi (JR). But, by Plancherel's theorem and the elementary propertiesof the Fourier transform, we get, for all functions f in W,

117f(b, a)f - f1112(lll) I: I.)af (x: b) - f(X)12

dxI: lvae-ib~ j(a~) - j(~)12~

< 2I: lvae-ib~(j(~) - j(~)W~

+ 2I: I(vae-ib~ -l)j(~W~· (18.10)

For all ~ in JR,(18.11)

as (b, a) ~ (0,1) and

(18.12)

for all bin JR and all a in (0,2). By (18.11), (18.12) and the Lebesgue dominatedconvergence theorem,

(18.13)

as (b,a) ~ (0,1). For all ~ in JR,

(18.14)


as a ~ 1, andIj(a~) - j(~)1 ::; 2 sup Ij(~)lxR(~)

eEIR

for all a in (!, 2), where R is a fixed positive number such that

j(~) = 0, ~ > R,

(18.15)

and XR is the characteristic function on [0, 2R]. Thus, by (18.14), (18.15) and theLebesgue dominated convergence theorem,i: Iv'ae-ibe(j(a~) - j(~))12~ ~ 0

as (b,a) ~ (0,1). So, by (18.10), (18.13) and (18.16),

n(b, a)f ~ f

(18.16)

(18.17)

in L2 (1R) as (b, a) ~ (0,1) for all f in W. Let f E H~(IR). Then, by Lemma 18.6,we can find a sequence {fd~l of functions in W such that fk ~ f in L2 (1R) ask ~ 00. Then for any positive number €, let ko be the positive integer such that

2€Ilfko - fll£2(IR) < 3· (18.18)

So, by (18.17), (18.18) and the obvious fact that n(b, a) : H~(IR) ~ H~(IR) is aunitary operator for all (b, a) in U, there exists a positive number a such that

Iln(b, a)f - fll£2(IR)

< Iln(b,a)(f - fko) II L2 (IR) + IIn(b,a)fko - fkoll£2(IR) + IIfko - fll£2(IR) < €

whenever (b, a) is within a-distance of (0,1). Thus, n(b, a)f ~ f in L2 (1R) for allf in H~(IR) as (b, a) ~ (0,1) and the proof is complete. 0

Proposition 18.7 n: U ~ U(H~(IR)) is an irreducible and unitary representationof U on H~(IR).

Proof. That n(b,a): H~(IR) ~ H~(IR) is a unitary operator for all (b,a) in U iseasy to check and has actually been used in the proof of Proposition 18.5. Let Mbe a nonzero and closed subspace of H~(IR) such that M is invariant with respectto n : U ~ U(H~ (1R)). Let 9 be a nonzero function in M. Then

{n(b,a)g: (b,a) E U} ~ M.

Let f E H~(IR) be such that f is orthogonal to M. Then for all points (b, a) in U,

Joo (x -b)-00 f(x)g -a- dx = 0,

18. The Affine Group

and hence, by Plancherel's theorem,

Thus, by (18.19),j(~)g(a~) = 0

103

(18.19)

(18.20)

for almost all ~ in R Suppose that j(~) =f. 0 for all ~ in a set S with positivemeasure. Then for all ~ in S, by (18.20), we get

g(a~) = 0

for all positive numbers a. Thus, g= 0 and hence 9 = O. This is a contradiction.o

To get more information on the irreducible and unitary representation 1r :U - U(H~(lR)), we need the following subspace A of H~(lR) given by

A = {f E H~(lR) :100

Ij(?12

~ < oo}.Theorem 18.8 For all f in H~(lR) and all 9 in A,

1001002dbda 100

A 2 100Ig(~)121(1, 1r(b, a)g)L2(1R) I -2 = 21r If(~)1 ~ -c-d~.

o -00 a 0 0 '>

Proof. Let fEW and 9 E A. Then, by (18.5), Plancherel's theorem and theelementary properties of the Fourier transform, we get

(18.21)

where(18.22)


Thus, by (18.21), (18.22) and Fubini's theorem,

100 100 2 db dal(f, n(b, a)g)L2(IR) 1-2o -00 a

2n100 (100Ij(~)12~) Ig(a~W daa

2n roo 1j(~W~ roo Ig(~W ~10 10 ~

(18.23)

for all f in Wand all 9 in A. Now, let f E H~(JR) and 9 E A. Then, by Lemma18.6, there exists a sequence {A}~l of functions in W such that

(18.24)

in L2 (1R) as k ~ 00. For k = 1,2, ..., we get, by (18.23), (18.24) and Plancherel'stheorem,

100100

~~I(Ii, n(b, a)g )£2(IR) - (A, n(b, a)g )£2(IR) 1

2-2-

o -00 a

2n roo Ijj(~) _ jk(~)12~ roo Ig(~W ~ ~ 010 10 ~

as j, k ~ 00. So, {(fk, n(·, ·)9)£2(IR)}~1 is a Cauchy sequence in L2(U). Hencethere exists a function h in L2(U) such that

(18.25)

in L2(U) as k ~ 00. Therefore there exists a subsequence of {(A, n(-, ·)g)L2(IR)}~1'again denoted by {(fk, n(·, ·)9)£2(IR)}~1' such that

(fk, n(·, ·)9)£2(IR) ~ h

a.e. on U as k ~ 00. But, by (18.24),

(18.26)

(fk, n(b, a)g)L2(IR) ~ (f, n(b, a)g)£2(IR) (18.27)

for all (b, a) in U as k ~ 00. So, by (18.25)-(18.27),

100 100 db da 1001

00db daI(A, n(b, a)g)£2(IR) 12 - 2- ~ l(f, n(b, a)g)£2(IR) 1

2- 2-o -00 a 0 -00 a(18.28)

as k ~ 00. But, by (18.23), (18.24) and Plancherel's theorem,

100 100 2db~ 100• 2 100

Ig(~WI(A,n(b,a)g)£2(IR)1 -2 ~ 2n If(~)1 ~ -c-~ (18.29)o -00 a 0 0."

as k ~ 00. Hence, by (18.28) and (18.29), the proof is complete. o

18. The Affine Group 105

Corollary 18.9 1r : U ---t U(H~(lR)) is a square-integrable representation of U onH~(lR).

Proof. Let <p E A. Then, by Theorem 18.8,

< 00

and this completes the proof.

(18.30)

o

Corollary 18.10 Every junction <p in A with 11<pIIL2(R) = 1 is an admissible waveletfor the representation 1r : U ---t U(H~(lR)) of U on H~(lR) and

Corollary 18.10 is an immediate consequence of (18.30).

Remark 18.11 Corollary 18.10 tells us that the set AW(1r) of all admissible wavelets for the representation 1r : U ---t U(H~(lR)) of U of H~(lR) is nonempty. ThatAW(1r) is a proper subset of {f E H~(lR) : Ilfll£2(R) = I} is illustrated by thefollowing example.

Example 18.12 Let X be the characteristic function on [0,1) and let fa be thefunction in L2 (lR) such that fo = x. Then fo E H~(lR). Using the calculations inthe derivation of (18.23), we get

100 /00 2 dbda 100A 2 100

Ifo(~WIUo, 1r(b, a)fo)£2(R) I -2 = Ifo(~)1 ~ ~~.a -00 a 0 0

But 100Ifo~~)12 ~ =11

Zd~ = 00.

Thus, by (18.31) and (18.32), the function <p on lR defined by

<p = fo/llfoIIL2(R)

is in {f E H~(lR) : IlfIlL2(R) = I}, but not in AW(1r).

(18.31)

(18.32)

Using Theorem 14.5 and Corollary 18.10, the following result on localizationoperators on the affine group is immediate.


Theorem 18.13 Let ep E A and F E V(U), 1 :::; p :::; 00. Then the localizationoperator Lp,'P : H~(lR) --+ H~(IR) given by

for all f and g in H~(IR), where

is in Sp and

19 Wavelet Multipliers

Let C1 E Loo(lRn). Then we define the linear operator T(7 : L2 (lRn) ---+ L2 (lRn) by

T(7u = F- 1C1Fu, u E L2 (lRn),

where F and F-1 are the Fourier transformation and the inverse Fourier transformation respectively. The Fourier transform Fu, sometimes denoted by fL, of afunction u in L 2 (lRn ) is given by

Fu = lim (XRU)'''R-+oo

where XR is the characteristic function of the ball with center at the origin andradius R,

and the convergence of (XRuy' to Fu is understood to be in L 2 (lRn). It is aconsequence of Plancherel's theorem that T(7 : L2 (lRn) ---+ L2 (lRn) is a boundedlinear operator.

Let <p be any function in L2 (lRn)nL4 (lRn)nLOO(lRn) such that 11<p11£2(IRn) = 1.The aim of this chapter is to make precise the definition of the linear operator<pT(7tp : L2 (lRn) ---+ L2 (lRn), where C1 is a function in LP(lRn), 1 ::; p ::; 00, and toprove that the resulting bounded linear operator is in the Schatten-von Neumannclass Spo To this end, we first prove that if C1 E Loo(lRn ), then the bounded linearoperator <pT(7tp : L2 (lRn) ---+ L2 (lRn) can be realized as a wavelet multiplier (tobe explained) associated to a unitary representation 7r : lRn ---+ U(L2 (lRn )) of theadditive group lRn on L2 (lRn ). This connection explains the impetus for the study ofthe linear operator <pT(7tp : L2 (lRn) ---+ L2 (lRn) and also reveals that the techniquesdeveloped in Chapters 12-14 can be exploited.

Let 7r : lRn ---+ U(L2 (lRn )) be the unitary representation of the additive grouplRn on L 2 (lRn ) defined by

(7r(~)u)(x) = eiXo{u(x), x, ~ E lRn, (19.1)

for all functions u in L 2 (lRn ).

Proposition 19.1 Let <p be any function in L2 (lRn)nLOO (lRn) such that 11<p11£2(IRn) =1. Then for all functions u and v in S,

(27r)-n r (u, 7r(~)<p)L2(IRn)(7r(~)<p, V)£2(IRn)d~ = (<pu, <pV)£2(IRn). (19.2)}IRn


108 19. Wavelet Multipliers

Proof. Using Plancherel's theorem and the fact that

where(T-d)(x) = f(x - ~), x E jRn,

for all measurable functions f on jRn, we get

(19.3)

and

for all ~ in jRn, where

(19.4)

(19.5)

and

for all functions f in S. Thus, by (19.3)-(19.5), Plancherel's theorem and the factthat

(19.6)

we get


{ (u * ~)(~)(f; *~)(~)~JRn(271")n In u(~)'l/J(~)v(~)'l/J(~)~

(271")n (cpu, cpv) £2(lRn ) ,

o

Remark 19.2 Formula (19.2) can be considered as an analogue of the resolutionof the identity formula (6.3) for the unitary representation 71" : jRn -+ U(L2 (jRn))of jRn on L2 (jRn).

Proposition 19.3 Let a E Loo(jRn) and let cp be any function in L2 (jRn) n Loo(jRn)such that IlcpIIL2(lRn) = 1. If for all functions u in S, we define Pu,<pu by

for all functions v in S, then

19. Wavelet Multipliers

Proof. By (19.3)-(19.5), we get

(Pu,cpu, V)£2(lRn) = (27l")-n r a(~)(u * ~)(~)(v *~)(~)~JlRn

for all functions u and v in S. But, by (19.6) and (19.8),

(Pu,cpU,V)£2(lRn) = r a(~)(u'I/J)I\(O(V'lp)I\(~)~, u, v E S.JlRn

109

(19.8)

(19.9)

Thus, by (19.5), (19.9), Plancherel's theorem and the definition of Tu : L2(lR.n )

---+ L2(lR.n ), we get

for all functions u and v in S.

(Tu('l/Ju), 'l/JV)£2(lRn)

((ij;Tu'I/J)u, V)L2(lRn)

((epTu'P)u, V)£2 (IRn)

o

Remark 19.4 By Proposition 19.3, the linear operator epTu'P : L2(lR.n ) ---+ L2(lR.n )

associated to a in LCXl(lR.n ) and ep in L2(lR.n ) n LCXl(lR.n ) with the condition thatlIepll£2(lRn) = 1 is reminiscent of a localization operator studied in Chapter 12. See,in particular, formula (12.1) for the analogy. Had the "admissible wavelet" ep in(19.7) been replaced by the function epo on lR.n given by

epo(x) = 1, x E lR.n ,

we would have obtained

(PuU,V)£2(lRn) = (TuU,V)£2(lRn), U, v E S,

i.e., Pu,cp would have been a "constant coefficient" pseudo-differential operator,or a Fourier multiplier, studied in, say, the book [103] by Wong. In view of thefact that the function ep in the linear operator epTu'P : L2(lR.n ) ---+ L2(lR.n ) playsthe role of the admissible wavelet in the linear operator Pu,cp : L2(lR.n ) ---+ L2 (lR.n ) ,

it is reasonable to call the linear operator epTu'P : L2(lR.n ) ---+ L2(lR.n ) a waveletmultiplier.

In order to define the linear operator epTu'P : L2(lR.n ) ---+ L2(lR.n ), where a isa function in P(lR.n ), 1 ~ P < 00, and ep is a function in L2(lR.n ) n LCXl(lR.n ) withIlepll£2(lRn) = 1, we need some preparation.

Proposition 19.5 Let a E L1 (lR.n ) and let ep be any function in L2(lR.n ) n LCXl(lR.n )

such that Ilepll£2(lRn) = 1. Iffor all functions u in L2(lR.n ), we define Pu,cpu by (19.7)for all functions v in L2(lR.n ), then Pu,cp : L2(lR.n ) ---+ L2(lR.n ) is a bounded linearoperator and

IIPu ,cpIIB(L2(lRn» ~ (21T)-nlla ll£l(lRn), (19.10)

where 1111 B (£2 (IRn» is the norm in the C* -algebra of all bounded linear operatorsfrom L2(lR.n ) into L2(lR.n ).


Proof. By (19.1), (19.7), Schwarz' inequality and the assumption that 11<p11£2(IRn)= 1,

I(P",cpu, V)£2(IRn) I :::; (271") -n r la(~) II (u, 7I"(~)<p)£2 (IRn) II (7I"(~)<p, V)£2(IRn) Id~JIRn

:::; (271")-nllall£l(IRn) lIull £2 (IRn) Ilvll £2(IRn)

for all functions u and v in L 2 (lRn ). 0

Proposition 19.6 Let a E LP(lRn), 1 < p < 00, and let <P be any function inL2 (lRn) n Loo(lRn) such that 1I<pIlL2(IRn) = 1. Then there exists a unique boundedlinear operator P",cp: L2 (lRn) --+ L 2 (lRn) such that

IIP",cpll B(£2(IRn)) :::; (271")-nIPII<pIl~!'(IRn) lIall£l'(IRn),

and for all functions u and v in L2 (lRn ), (P",cpU,V)£2(IRn) is given by (19.7) forall simple functions a on IRn for which the Lebesgue measure of the set {~ E IRn :

a(~) :f; O} is finite.

Proof. Let a E LOO(lRn). Then, by (19.1), (19.3)-(19.5), (19.7), Schwarz' inequalityand the assumption that 1I<p1l£2(IRn) = 1, we get

I(P"U,V)£2(IRn)l:::; (271")-nll a Il L=(IRn) II it * ~1I£2(IRn)IIV * ~1I£2(IRn) (19.11)

for all functions u and v in L2 (lRn ). Using (19.5), (19.6), (19.11) and Plancherel'stheorem, we get

I(P",cpu, V)£2(IRn) I :::; lI a IlL=(IRn) II u'l/J II £2 (IRn) II v'l/J II £2 (IRn)

:::; lI a IlL=(IRn) II <p1I I=(IRn) II U Il£2(IRn) II V Il£2(IRn) (19.12)

for all functions u and v in L 2 (lRn ). So, by (19.12),

IIP",cpIlB(£2(IRn)) :::; 1I<PIII=(IRn)lIaIlL=(IRn), a E Loo(lRn). (19.13)

Thus, by (19.10), (19.13) and the interpolation argument used in the proof ofProposition 12.3, the proof is complete. 0

Remark 19.7 Propositions 19.5 and 19.6 allow us to define the wavelet multiplier<pT,,0 : L 2 (lRn) --+ L2 (lRn) for all functions a in U(lRn ), 1 :::; p < 00, and allfunctions <P in L 2 (lRn) n Loo(lRn) such that 1I<p1l£2(IRn) = 1, as the bounded linearoperator P",cp : L2 (lRn) --+ L2 (lRn).

We can now give the Schatten-von Neumann property of the wavelet multiplier <pT,,0 : L2 (lRn) --+ L 2 (lRn), where a E LP(lRn), 1 :::; p :::; 00, and <P isa function in L 2 (lRn) n Loo(lRn) such that 1I<p1l£2(IRn) = 1. The strategy is tolook at wavelet multipliers as localization operators associated to symbols a andadmissible wavelets <po However, not all functions <P in L2 (lRn) n Loo(lRn) with1I<p1l£2(IRn) = 1 can serve as admissible wavelets, which are characterized by thefollowing theorem.

19. Wavelet Multipliers 111

Theorem 19.8 The set of admissible wavelets for the unitary representation 7f :IRn -+ U(L2 (lRn )) defined by (19.1) consists of all functions 'P in L2 (lRn ) n L4 (lRn )

for which 11'PIIL2(lRn) = 1.

Proof. Let'P E L2 (lRn )nL4 (lRn ) be such that 11'PIIL2(lRn) = 1. Then, by Plancherel'stheorem,

Thus, 'P is an admissible wavelet for the unitary representation 7f of IRn on L2 (lRn ).

The same calculations show that every admissible wavelet for the unitary representation 7f : IRn -+ U(L2 (lRn

)) is a function 'P in L2 (lRn )nL4 (lRn ) with 11'PIIL2(lRn) = 1.o

Remark 19.9 Let 'P be an admissible wavelet for the square-integrable representation 7f : IRn -+ U(L2 (lRn )). Then from the proof of Theorem 19.8, we see that thewavelet constant Ccp is given by

and hencePa,cp = 1I'Plli4(lRn)La,cp,

where La,cp : L2 (lRn ) -+ L2 (lRn ) is the localization operator on the additive groupIRn associated to the symbol a and the admissible wavelet 'P.

In view of Remark 19.9, we can use Theorem 14.1 to obtain the followingtheorem.

Theorem 19.10 Let a E £l(lRn) and let 'P be any function in L2 (lRn ) n L4 (lRn ) n

L"'O(lRn) such that lI'PIIL2(lRn) = 1. Then the wavelet multiplier 'PTa'P : L2 (lRn ) -+

L2 (lRn ) is in 81 andII'PTa'Pllsl ::; (27f)-nll a ll£l(lRn).

We are now ready to give the Schatten-von Neumann property of waveletmultipliers.

Theorem 19.11 Let a E LP(lRn ), 1 ::; P ::; 00, and let 'P be any function in L2 (lRn )nL4 (lRn ) n LOO(lRn ) such that 11'PIIL2(lRn) = 1. Then the wavelet multiplier 'PTa'P :L2 (lRn ) -+ L2 (lRn ) is in 8p and

2

II'PTa'Pllsp ::; 11'Pllfoo(lRn) (27f)-*IlaIlLP(lRn).

Proof. If p = 1, then Theorem 19.11 follows from Theorem 19.10. If p = 00, thenTheorem 19.11 follows from (19.13). Thus, for 1 < P < 00, Theorem 19.11 followsfrom Theorems 2.10 and 2.11 in the theory of interpolation, and the endpoint casesp = 1 and p = 00. 0


We end this chapter with a trace formula for wavelet multiplers.

Theorem 19.12 Let a E Ll(lRn) and let rp be any function in L2(lRn) n L4 (lRn) nL=(lRn) such that IlrpIIL2(IRn) = 1. Then

tr(rpTu<p) = (2n)-n r a(~)d~.}IRn

Proof. Let {rpk : k = 1,2, ...} be an orthonormal basis for L2(lRn). Then, usingTheorem 19.10, the definition of the trace, Fubini's theorem, Parseval's identity,Ilrpll£2(IRn) = 1 and the fact that n(~) : L2(lRn) -> L2(lRn) is a unitary operator forall ~ in IRn , we get

=tr(Pu,rp) = L(Pu,rprpk, rpk)£2(IRn)

k=l

~(2n)-nIn a(~)I(rpk,n(~)rph2(IRn)12~

(2n)-n In a(~)~ l(rpk,n(~)rp)L2(IRn)12~

= (2n)-n r a(~)lIn(~)rplli2(IRn)~}IRn

(2n)-n r a(~)~.}IRn

o

Remark 19.13 It is interesting to note that the trace tr(rpTu<p) of the waveletmultiplier rpTu L2(IRn) is independent of the "admissible wavelet" rp.

(20.1)

20 The Landau-Pollak-Slepian Operator

We show in this chapter that the Landau-Pollak-Slepian operator arising in signalanalysis is in fact a wavelet multiplier. We begin with a discussion of the LandauPollak-Slepian operator.

Let nand T be positive numbers. Then we define the linear operators Po :L2 (lRn ) -+ L2 (lRn ) and QT : L2 (lRn ) -+ L2 (lRn ) by

(Pofr\(~) = {j(~), I~I ~ n,0, I~I > n,

and

(QTf)(X) = {f(X), Ixl ~ T,

0, Ixl >T,(20.2)

Proposition 20.1 Po : L2 (lRn ) -+ L2 (lRn ) and QT : L2 (lRn ) -+ L2 (lRn ) are selfadjoint projections.

Proof. By (20.1) and Plancherel's theorem,

(Pof,g)p(lRn) ((Poft,[;)L2(lRn) = r (Pofr\(~)g(~)~llRn

r j(~)g(~)~ = r j(~)(POg)A(~)~lBn lBnr j(~)(POg)A(~)~ = (j, (POg)A)p(lRn)

JlRn= (f, Pog)L2(lRn)

for all functions f and gin L2 (lRn ), where Bo is the ball in IRn with center at theorigin and radius n. Therefore Po : L2 (lRn ) -+ L2 (lRn ) is self-adjoint. By (20.2),

(QTf,g)L2(lRn) r (QTf)(x)g(x)dx = r f(x)g(x)dxJlRn lBT

= r f(X)(QTg)(x)dx = (j, QTg)£2(lRn)JlRn


114 20. The Landau-Pollak-Slepian Operator

for all functions I and gin L2 (lRn ), where BT is the ball in IRn with center at theorigin and radius T. Therefore QT : L2(lRn ) -t L2(lRn ) is self-adjoint. By (20.1),the fact that Pn : L2(lRn ) -t L2(lRn ) is self-adjoint and Plancherel's theorem,

(Pn/, Png)£2(JRn) = ((Pn/r', (Pngr')L2(JRn)

{ (Pn/)'''(~)(Png)!\(~)~ = ( j(~)g(~)~JJRn JBn{ (PnJ)/\(~)g(~)~ = ((PnJ)",g)£2(JRn)

JJRn(Pn/, 9)£2(JRn)

for all functions I and 9 in L2(lRn ). Thus, pJ = Pn and hence Pn : L2(lRn ) -t

L2(lRn ) is a projection. Finally, by (20.2) and the fact that Pn : L2(lRn ) -t L2(lRn )

is self-adjoint,

(QT I, QTg)£2(JRn) = { (QT J)(x) (QTg)(x)dxJJRn

{ I(x)g(x)dx = { (QTJ)(x)g(x)dx = (QTI,g)£2(JRn)JBT JJRn

for all functions I and 9 in L2(lRn ). Thus, Q} = QT and hence QT : L2(lRn ) -t

L2 (lRn ) is a projection. 0

In signal analysis, a signal is a function I in L2(lRn ). Thus, for all functions Iin L 2 (lRn ), the function QTPnl can be considered to be a time and band-limitedsignal. Therefore it is of interest to compare the energy IIQTPn/lli2(lRn) of the time

and band-limited signal QTPnl with the energy 1I/IIi2(JRn) of the original signalI. Using the fact that Pn : L2(lRn ) -t L2(lRn ) and QT : L2(lRn ) -t L2(lRn ) areself-adjoint and the fact that QT : L2(lRn ) -t L2(lRn ) is a projection, we get

{IIQTPn/lli2(JRn) 2 }

sup 1I/IIi2(JRn) : IE L (IRn

) , I of 0

{(QTPn/, QTPnJ)L2(JRn) IE L2 (lRn ), f of o}

sup IIflli2(JRn) :

{(PnQTPnf, J)£2(JRn) f E L2 (lRn ), f of o}

sup Ilflli2(JRn) :

sup {(PnQTPnf, J)L2(lRn): f E L2 (lRn), 1I/11£2(JRn) = I}. (20.3)

Since PnQTPn : L2(lRn ) -t L2(lRn ) is self-adjoint, it follows from (20.3) that

20. The Landau-Pollak-Slepian Operator 115

The bounded linear operator PnQrPn : £2(JRn) -+ £2(JRn) that we have justseen in the context of time and band-limited signals is called the Landau-PollakSlepian operator. See the fundamental papers [59, 60] by Landau and Pollak,[83, 84] by Slepian and [85] by Slepian and Pollak for more detailed information.

That the Landau-Pollak-Slepian operator is in fact a wavelet multiplier studied in Chapter 19 is the content of the following theorem.

Theorem 20.2 Let 0,(20.4)

where I-£(Bn) is the volume of Bn, and let a be the characteristic function on B r ,i.e.,

a(~) = {I, I~I ~ T,0, I~I >T.

(20.5)

Then the Landau-Pollak-Slepian operator PnQrPn : £2(JRn) -+ £2(JRn) is unitarily equivalent to a scalar multiple of the wavelet multiplier <pT(j<p : £2 (JRn) -+

£2(JRn). In fact,

(20.6)

Proof. By (20.4), 0,(20.9)

116 20. The Landau-Pollak-Slepian Operator

for all functions u in S, where u is the inverse Fourier transform of u. So, by (20.8),(20.9) and the Fourier inversion formula,

=

(20.10)

for all functions u in S. Hence, by (20.2), (20.5), (20.7), (20.10), Plancherel'stheorem and the fact that Pn : L2(1~n) --t L2(lRn) is self-adjoint,

((cpTacp)u, V)£2(lRn) p,(~n) Ln a(~)(Pnu)(~)(Pnv)(~)d~

(~ ) ( (Pnu)(~)(Pnv)(~)~p, n JBT

= p,(~n) Ln (QTPnu)(~)(Pnv)(~)~1

p,(Bn) (QTPnU,PnV)£2(lRn )

p,(~n) (PnQTPnu, V)£2(lR n )

1 1p,(Bn) (FPnQTPnF- u, V)£2(lRn )

for all functions u and v in S, and hence (20.6) is proved. 0

We have the following result on the trace of the Landau-Pollak-Slepian operator PnQTPn : L 2 (lRn ) --t L 2 (lRn ).

Theorem 20.3 tr(PnQTPn) = ar (~)} -2 (T2nr.

Theorem 20.3 is an immediate consequence of (20.6), Theorem 19.12, Theorem 20.2 and the fact that the volume of the ball in lRn with radius r is equal to7("n/2 r n

11r( 11r2 2

21 Products of Wavelet Multipliers

The wisdom of the preceding chapter is that a wavelet multiplier can be consideredas a filter which time and band-limits a signal. Thus, if we are interested in findinga filter that has the same effect as two wavelet multipliers arranged in series, weare actually seeking a formula for the product of two wavelet multipliers.

We give in this chapter two formulas for the product of two wavelet multiplierscpTa<fJ : L2 (lRn ) ---t L2 (lRn ) and cpTr<fJ : L2 (lRn ) ---t L2 (lRn ), where (j and Tarefunctions in L 2 (lRn ), and cp is a function in L 2(lRn)nV'O (lRn ) such that IIcpll£2(Rn) =1. To do this, we need a recall of some basic results without proofs on Weyltransforms from the book [26] by Folland, the books [94, 95] by Thangavelu andthe book [102] by Wong.

Let (j E L 2 (lRn xlRn). Then the Weyl transform associated to (j is the bounded

linear operator W a : L2 (lRn ) ---t L2 (lRn ) given by

for all functions f and 9 in L2 (lRn ), where W(f,g) is the Wigner transform of fand 9 defined by

That the Weyl transform so defined is the same as that defined by (1.1) is wellknown and can be found in, e.g., Chapter 4 of the book [102] by Wong.

It can be proved that

(21.2)

where the function V(f,g) on lRn x lRn is the Fourier-Wigner transform of f and9 defined by

and(p(q,p)f)(x) = eiq.x+!iq.p f(x + p), x E lRn .

(21.3)

(21.4)

For all Schwartz functions f and 9 on lRn , the functions V (f, g) and W (f, g)are Schwartz functions on lRn x lRn.


118 21. Products of Wavelet Multipliers

For all functions f and 9 in L2(J~n), the functions V(f,g) and W(f,g) are inL 2 (lRn x lRn

). Furthermore, we have

(21.5)

for all functions f and 9 in L2 (lRn ). That the same identity is true when W isreplaced by V follows from (21.2) and Plancherel's theorem.

Let h E L2(lRn x lRn ). Then for all functions f in L 2(lRn ), we define thefunction Khf on lRn by

(Khf)(x) = r h(x, y)f(y)dy, x E lRn.JRn

Then Kh ; L 2(lRn ) -+ L 2(lRn ) is a bounded linear operator and we call it theHilbert-Schmidt operator corresponding to the kernel h. The following result, obtained by Pool in [69], is the main ingredient in the derivation of the first productformula for two wavelet multipliers.

(J = (2?r)~ F2Th, (21.6)

where F2 is the Fourier transform on L2 (lRn x lRn) with respect to the second

variable and T is the linear operator on L2 (lRn x lRn ) defined by

Proposition 21.1 Let h E L 2(lRn x lRn ). Then the Hilbert-Schmidt operator corresponding to the kernel h is the same as the Weyl transform W u ; L 2(lRn ) -+

L 2(lRn ), and

(TJ)(x,Y)=f(x+~,x-~), x,yElRn,

for all functions f in L 2(lRn x lRn).

(21. 7)

We can now give the first formula for the product of two wavelet multipliers.

Theorem 21.2 Let (J and T be functions in L 2(lRn ) and let ep be any function inL 2(lRn ) n LOO(lRn ) such that lIepIIL2(Rn) = 1. Then the product of the wavelet multipliers epTurfJ: L 2(lRn ) -+ L 2(lRn ) and epTrrfJ : L 2(lRn ) -+ L 2(lRn ) is the same as thelinear operator epW.xrfJ : L2(lRn ) -+ L 2(lRn ), where W.x : L 2(lRn ) -+ L 2(lRn ) is theWeyl transform associated to >. and

(21.8)

for all x and ~ in lRn.

The following lemma will be used in the proof of Theorem 21.2.

Lemma 21.3 Let f and 9 be functions in L 2(lRn ). Then

W(j,g)(x,~)= W(f,g)(~, -x), x, ~ E lRn.

21. Products of Wavelet Multipliers 119

Proof Let f and g be functions in S. Then, by (21.3) and (21.4),

V(j,g)(q,p) = (2rr)-Jij(e~iq'PMqTpj,g)£2(JRn) (21.9)

for all q and pin !R.n, where

(Tpu)(x) = u(x + p), x E !R.n,

and(Mqu)(x) = eiq.xu(x), x E !R.n,

for all measurable functions u on !R.n. So, by (21.3), (21.4), (21.9) and Plancherel'stheorem,

V(j, g)(q, p) = (2rr)- Jij (e~iq'P(T_qMpf) v, g)L2(JRn)

(2rr)-Jij (e~iq·PT_qMpf, g)£2(JRn)

(2rr)-Jij r e~iq·Peip.(x-q) f(x - q)g(x)dx}JRn

= (2rr)-Jij r eip,x-~iq.p f(x - q)g(x)dx}JRn

= (2rr)-Jij r (p(p, -q)f)(x)g(x)dx}JRn

= V(f,g)(P, -q) (21.10)

for all q and p in !R.n. So, by (21.1) and (21.9),

W(j,g)(x,~) = (2rr)-n r r e-iq'X-ip·eV(j,g)(q,p)dqdp}JRn }JRn

(2rr)-n r r e-iq.x-ip·eV(f,g)(p, -q)dqdp}JRn }JRn

(2rr)-n r r eiq'X-ip·eV(f,g)(p,q)dqdp}JRn }JRn

W(f,g)(~, -x) (21.11)

for all x and € in !R.n. Thus, by (21.5), (21.10), (21.11), Plancherel's theorem anda limiting argument, the proof is complete. 0

Proof of Theorem 21.2. We begin with the observation that for all functions f inLl(I~n) n L2(!R.n),

(Tuf)(x) = (2rr)-Jij r eiX·e(J(~)j(O~ = (2rr)-Jij(a * f)(x) (21.12)}JRn

for all x in !R.n. Now,(21.13)

where(21.14)


and for all functions f in S, we get, by (21.12) and Fubini's theorem,

(TawTrf) (x) = (21T)-~ (0- *wTrf)(x)

= (21T)-n(0- *w(f * f))(x)

(21T)-n r o-(x - y)w(y)(f * f)(y)dyJJRn= (21T)-n in o-(x - y)w(y) (in f(y - Z)f(Z)dZ) dy

= (21T)-n in (in o-(x - y)w(y)f(y - Z)dY) f(z)dz

= r h(x, z)f(z)dz (21.15)JJRnfor all x in jRn, where

h(x, z) = (21T)-n r o-(x - y)w(y)f(y - z)dy, x, z E jRn.JJRn (21.16)

By Minkowski's inequality in integral form, Fubini's theorem, Plancherel's theoremand (21.14),

1

(in in Ih(x, ZWdXdZ) "21

(21T)-n (in in lin o-(x - y)w(y)f(y - Z)dyr dXdZ) "2

1

< (21T)-n r (r r 10-(x- y)w(y)f(y-z) 12 dxdz)"2 dyJJRn JJRn JJRn1

(21T)-n r Iw(y)1 (r r 100(x- y)f(y-ZW dXdz)"2 dyJJRn JJRn JJRn(21T) -n IIfPlli2(JRn) 110' II £2(JRn) II Til L2(JRn). (21.17)

So, by (21.15)-(21.17) and Proposition 21.1, TawTr : L2(jRn) -- L2(jRn) is a Weyltransform Wx : L2(jRn) -- L2(jRn), and by (21.6),

oX = (21T)~:F2Th. (21.18)

By (21.7) and (21.16),

(Th)(x, z) (21T)-n in 0- (x + ~ - y) w(y)f (y - X+ ~) dy

(21T)-n in 0- (x - y + ~) w(y)f G- (x - y)) dy

(21T)-n in 0- (x - y + ~) w(y)r (x - y - Ddy (21.19)


for all x and Z in IRn . For almost all x in IRn , we get, by (21.14), (21.19), Fubini'stheorem, Schwarz' inequality and Plancherel's theorem,

In 11n iT (x - Y+~) W(Y)T (x - y - ~)dyl dz

< In (1JiT (x - y + DIIw(y)1 IT (x - y -~) IdY) dz

In Iw(y)1 (1n liT (x - y + ~) liT (x - y - ~) IdZ) dy

< In'W(y)'(1nliT(x-y+DI2dZ)! (1nIT(x-y-~)12dz)!dy

2n 1I<p11i 2 (lRn) II 0"11 £2(lRn) II Til £2(lRn).

Thus, by (21.1), (21.14), (21.19), Fubini's theorem and Lemma 21.3,

(F2Th)(x,~)

(27T)-nlnW(y) {(27T)-n/21ne-iz'~iT (x - y + ~) T (x - y - ~)dZ} dy

= (27T)-n r w(y)W(iT,T)(x-y,~)dyllRn

(27T)-n r W(O",T)(~,y - x)w(y)dyllRn

for all x and ~ in IRn , and hence, by (21.13), (21.14) and (21.18), (21.8) follows.D

In order to give another formula for the product of two wavelet multipliers,we need a recall of a formula, in the paper [35] by Grossmann, Loupias and Stein,for the product of two Weyl transforms associated to functions in L2 (lRn x IRn ).

To this end, we need the notion of a twisted convolution.

As usual, we identify IRn x IRn with en and any point (q,p) in IRn x IRn withthe point Z = q + ip in en, and we define the symplectic form [ , ] on en by

[z, w] = 2Im(z· w), z, wEen,

where

andn

Z oW = LZjWj.j=l

Now, for any fixed real number .x, we define the twisted convolution f *), 9 of twomeasurable functions f and 9 on en by

U *), g)(z) = r f(z - w)g(w)ei),[z,w1dw, Z E en,len (21.20)


where dw is the Lebesgue measure on Cn , provided that the integral exists. Thefollowing theorem can be found in the paper [351 by Grossmann, Loupias andStein.

Theorem 21.4 Let a and T be functions in L2(Cn). Then the product of the Weyltransforms W" : L2(lRn) --+ L2(lRn) and W7" : L2(lRn) --+ L2(lRn) is the same asthe Weyl transform Ww : L2(lRn) --+ L2(lRn), where w is the function in L2(Cn)given by

w= (21r)-n(a- *1 f).4

A proof of Theorem 21.4 can be found in Chapter 9 of the book [1021 byWong.

Another ingredient in the derivation of another formula for the product oftwo wavelet multipliers is given in the following theorem.

Theorem 21.5 Let a E L2(lRn) and let <p be any function in L2 (lRn )nLOO(lRn ) suchthat II<pll£2(IRn) = 1. Then the wavelet multiplier <pT"tp : L2(lRn) --+ L2(lRn) is thesame as the Weyl transform W,,<p : L2(lRn) --+ L2(lRn), where

a'P(x,~) = (21r)-~ f W(<p,<p)(x,~ -1])a(1])d1], x, ~ E IRn.

}IRn

Proof By (21.12), we get, for all functions f in S,

«<pTqtp)f)(x) (21r)-~<p(x)(a * tpf)(x)

(21r)-~<p(x) f a(x - y)tp(y)f(y)dy}IRn

(21r)-~ f <p(x)a(x - y)tp(y)f(y)dy}IRn

= in h(x, y)f(y)dy

for all x in IRn , where

h(x, y) = (21r)-~<p(x)a(x - y)tp(y), x, Y E IRn.

Now, by (21.23), Fubini's theorem and Plancherel's theorem,

f f Ih(x, y)1 2dxdy}IRn }IRn

= (21r)-n f f 1<p(xWla(x - yWI<p(y)12dxdy}IRn }IRn

= (21r)-n In 1<p(y)12 (In 1<p(xWla(x - yWdX) dy

< (21r)-nll<plli,oo(lRn) (In 1<p(y)12dY) Ila lli,2(lRn)

= (21r)-nll<plli,oo(lRn)II<Plli,2(lRn)IIalli,2(lRn) < 00.

(21.21)

(21.22)

(21.23)

(21.24)


So, by (21.22)-(21.24), cpTuep : L2(jRn) --+ L2(jRn) is a Hilbert-Schmidt operatorwith kernel h, and hence, by Proposition 21.1, is the same as the Weyl transformWu", : L2(jRn) --+ L2(jRn) , where

(J<p(x,~) = (21r)~(J:·2Th)(x,~), X, ~ E jRn.

But, by (21.7) and (21.23),

(21.25)

(Th)(x,y)

and hence by (19.6),

(21r)-n kn e-i~.ycp (x + ~) a(y)ep (x - ~) dy

(21r)-n(w(cp, cp)(x,·) * (J)(~) (21.26)

for all x and ~ in jRn. Hence, by (21.25) and (21.26), the proof is complete. 0

We can now give another formula for the product of two wavelet multipliers.

Theorem 21.6 Let (J and T be functions in L2(jRn) and let cp be any function inL2(jRn) n LCXl(jRn) such that IIcpll£2(lRn) = 1. Then the product of the wavelet multipliers cpTaep : L2(jRn) --+ L2(jRn) and cpTrep : L2(jRn) --+ L2(jRn) is the same asthe Weyl transform W,\ : L2(jRn) --+ L2(jRn), and oX is the function in L2(]Rn X ]Rn)given by

~ = (21r)-n(o-<p *1 7'1')'4

where (J<p and T<p are defined by (21.21).

Theorem 21.6 is an immediate consequence of Theorems 21.4 and 21.5.

22 Products of Daubechies Operators

Let F E L2 (Cn) and let cp E L2 (lRn

) be such that IlcpIIL2(IRn) = 1. Then theDaubechies operator associated to the symbol F and the admissible wavelet cp isthe bounded linear operator DF,<p : L 2 (lRn ) ~ L 2 (lRn ) defined by (17.23) for allfunctions f and 9 in L 2 (lRn ). We give in this chapter a formula for the productof two Daubechies operators when the admissible wavelet cp is chosen to be thefunction given by

(22.1)

The starting point is the following theorem.

Theorem 22.1 Let cp be the function on lRn given by (22.1) and let A be the functionon Cn defined by

A(z) = 7r-ne-lzI2, Z E cn. (22.2)

Then for all functions F in L 2(Cn ), the Daubechies operator DF,<p : L 2 (lRn ) ~

L2 (lRn ) is the Weyl transform WF*A : L2 (lRn ) ~ L2 (lRn ).

Theorem 22.1 is Theorem 17.1 in the book [102] by Wong. It is importantto emphasize right at the beginning the fact that the results in this chapter arevalid only for the choice of cp defined by (22.1) as the admissible wavelet for theDaubechies operator DF,<p : L2 (lRn ) ~ L2 (lRn ).

For any fixed real number A, we define the new twisted convolution or theA-convolution f *A 9 of two measurable functions f and 9 on Cn by

(22.3)

provided that the integral exists. We have the following result.

Theorem 22.2 Let F and G be in L 2 (Cn ). If there exists a function H in L 2 (Cn )

such that the Daubechies operator DH,<p : L2 (lRn ) ~ L2 (lRn ) is the same asthe product of the Daubechies operators DF,<p : L2 (lRn ) ~ L2 (lRn ) and Dc,<p :L2 (lRn ) ~ L2 (lRn ), then

(22.4)

Proof. By Theorem 22.1,

(22.5)


22. Products of Daubechies Operators 125

It follows from Theorem 21.4 and (22.5) that for all ( in Cn ,

By (22.2) and an easy computation, we get

A(O = (2n)-ne-~, (E cn. (22.7)

(2n)n{(PA) *1 (GA)}(()4

(2n)-n1 p(( - w)e- H=-W I2 G(w)e-~IWI2 e~i[(,w]dwen

= (2n)-n r p(( _ w)G(w)eH-I(-wI2-lwI2+i[(,w]}dwlen

Thus, by (22.5)-(22.7) and the definition of the twisted convolution given by(21.20), we get

fI(()e-~ =

(22.8)

for all ( in cn . So, by (22.8),

fI(O = (2n)-n1p(( - w)G(w)eHI(12-1(-wI2_lwI2+i[(,w]}dw, (E cn. (22.9)en

Now, for all ( and w in cn,

1(1 2- I( - wl 2

- Iwl 2 + i[C w]= 1(1 2

- 1(1 2 + 2Re((· w) -lwl 2- Iwl 2 + 2iIm((· w)

= 2((- w) - 21w1 2.

Therefore, by (22.3), (22.9) and (22.10), we get, for all ( in cn,

fI(() = (2n)-n1p(( - w)G(w)e!(.w-1WI2 )dw,en

(22.10)

and hence (22.4). 0

From the proof of Theorem 22.2, we get the following corollary.

Corollary 22.3 Let F and G be functions in L2 (Cn) such that P *! GE L2 (Cn).Then there exists a function H in L2 (Cn) such that fI = (2n)-n (p *! G) andthe Daubechies operator DH,<p : L2 (lRn) --+ L2 (lRn) is the product of the Daubechiesoperators DF,<p : L2 (lRn) --+ L2 (lRn) and Da,<p : L2 (lRn) --+ L2 (lRn).

Remark 22.4 In general, for functions F and G in L 2 (Cn ), it is not true thatP *! G E L2 (Cn ). So, the product of two Daubechies operators associated tofunctions in L 2(Cn ) need not be a Daubechies operator associated to a functionin L2 (Cn ). This can best be seen from the following example.

126 22. Products of Daubechies Operators

Example 22.5 Let W be the subset of IR x IR defined by

W = {(q,p) E IR x IR: 0::; q, p::; I}. (22.11)

We identify points wand (in C with points (q,p) and (x,~) in IR x IR respectively.Let F E L2 (<C) be defined by

(22.12)

where X is the characteristic function on [-1, 1], and let G E L2 (<C) be defined by

G(w) ~ {.!lwl 2

e 2 ,

0,

WEW,

W \l w.(22.13)

Then, by (22.11)-(22.13),

(F *~ G)(()

fw F(( - w)G(w)e-~lwI2 e!<Wdw

1111

e-llx-qlx(~ - p)e~CqX+P~)e~iCq~-pX)dqdp

(1 1

e-lIX-qlehx+~iq~dq) (11

x(~ - p)e~p~-~iPXdP) (22.14)

for all ( in C. But for x > 1 and °< ~ < 1, we get from (22.14)

(F *~ G)(() = (11

e-lXehCl+2()dq) (11

e-~iP(dP)

~:~~ (elCl+2() -1) ~i (e-~i( -1)

= 14:

l;( (el+~i~ _ e-~x) ~i (eHe-~iX - 1)

In view of Remark 22.4 and Example 22.5, it is a natural problem to seeksome subspace of L 2 (cn) such that the product of two Daubechies operators associated to functions in the subspace is indeed a Daubechies operator associatedto a function in L 2 (Cn ) .

For any nonnegative real number c, we denote by Sc the set of all measurablefunctions F on cn such that

22. Products of Daubechies Operators 127

for some function f in L2(Cn ). It is clear that Sc is a subspace of L2(Cn ) for allc ~ O. It is also clear that if c :::; d, then Sd ~ Sc.

We can now give a formula for the product of two Daubechies operatorsassociated to functions in Sc, where c > 1+815.

Theorem 22.6 Let F and G be functions in Sc, where c > 1+815. Then the productof the Daubechies operators DF, 0, andO<d<c'

Proof. Let f and 9 be functions in L2 (cn) such that

(22.15)

and(22.16)

for all ( in cn. Then, by (22.3), (22.15) and (22.16), we get, for all ( in cn,

I(P *! G)()I

I[n p( - w)G(w) e!«·w- 1WI2 )dw l

< I IP( - w)IIG(w)1 eH:llwl e-!lwI2dw

en

< 1 e-cl(-wI2If( - w)1 e-clwI2Ig(w)1 e i ([([2+lwI2) e-!lw!2 dwen

< e-(C-i),(,21If( - w)llg(w)1 e2cRe«.w) e-(2c+i)lwI2dw. (22.17)

en

But for all positive numbers c:, we have

2cRe(· w) < 2cl(· wi :::; 2cl(llwl

2c..fiI(!M..fi

< c (c:1(12+ ~lwI2)

for all ( and w in Cn . So, by (22.17) and (22.18), we have, for all ( in Cn ,

(22.18)

128 22. Products of Daubechies Operators

(22.20)

Since c> ¥, it follows from (22.19) that for any positive number e such that

c 1--<e<l--2c+ i 4c'

there exists a positive constant de; such that

(22.21)

where1

Ce; = C - 4 - ceo (22.22)

Since, for any e satisfying (22.20), the function Igle-d,I'12

is in L1(Cn ), it followsfrom Young's inequality that the function he; on Cn defined by

is in L 2 (Cn ). Thus, by (22.21) and (22.23),

I(F *! G)(()I ~ e-e,I< 12 he;((), (E cn,

(22.23)

(22.24)

for any e satisfying (22.20). Now, by Plancherel's theorem, let HE L2 (Cn ) be suchthat

if = (27r)-n(F *! G). (22.25)

Then, by (22.24) and (22.25), H E Sed and hence, by (22.20) and (22.22), H E

n Sd· That the Daubechies operator DH,cp : L2(IRn ) ~ L2(IRn ) is the productO<d<e'of the Daubechies operators DF,cp : L2(IRn ) ~ L2(IRn ) and Dc,cp : L2(IRn ) ~L 2 (IRn ) is then a consequence of (22.25) and Corollary 22.3. 0

(23.1)

23 Gaussians

As a sequel to the previous chapter, it is of interest to seek another subspace M ofL2 (Cn ) such that the product of two Daubechies operators with symbols in M is aDaubechies operator with symbol H in M. We prove in this chapter that M can betaken to be the subspace of L 2(Cn ) spanned by Gaussian functions. Furthermore,an explicit expression for H is given.

Let M be the subspace of L2 (Cn ) spanned by all functions of the form

where d is a positive number. So, in general, the functions in M are Gaussianfunctions of the form

where C1 , C2 , ... , Cm are complex numbers and d1 , d2 , ... , dm are positive numbers. Then we have the following theorem.

Theorem 23.1 Let F and G be functions in M, i.e.,

m

F(z) = L:Cke-dklzI2, z E cn,k=l

andI

G(z) = L::Cje-dj 1zI2, z E cn,j=l

where C 1 , C2 , ... , Cm; CL C~, ... , C{ are complex numbers, and d1 , d2 , ... , dm;d~, d~, ... , df are positive numbers. Then DF,<pDc,<p = DH,<p, where H is also inM and

m I

H(z) = L::L::CkCje-rk,JlzI2, z E cn,k=lj=l

where rk,j = dk + dj + 2dkdj .

To prove Theorem 23.1, we need some preparation. We again identify anypoints (q,p) and (x,~) in JR2 with the points z = q + ip and ( = x + i~ in Crespectively. The following lemma follows from the fact that the Fourier transform

2

of the function 'lj; given by 'lj;(x) = e- x2 , x E JR, is equal to 'lj;.


130 23. Gaussians

Lemma 23.2 Iocoe-

x2cos(2hx)dx = :{f-e-

h2, where h is any real number.

Theorem 23.3 Let

andG(z) = e-dlzI2, z E en,

where c and d are positive numbers. Then

1 (7r)n cdl 12(F *2 G)(Z) = -;: e-'" Z ,

(23.2)

where1

r=c+d+ 2·

Proof First, we consider the case when n = 1. By (22.3) and (23.2), for all z ine,

(F *~ G)(z)

[ e-c1z-(12 e-dl(12 e~(z(-1(12)d(

[ e-clzI2+2cRe(z()-cl(!2 e-dl (1 2 e~z(-~1(12 d(

[ e-clzl2 e-(c+d+!)1(12 e2cRe(z() e~z(d(

( e-c(q2+ p2) e-(c+d+~)(x2+e) e2c(qx+p€) e~[qx+P€+i(pX-q€)Jdx~JJR2( e-c(q2+p2) e-r (x2+e) e(2c+~)(qx+p€)ei~(pX-q€)dx~

JJR2

[~ ]( 2 2)1 ( )2 ( )2e 16r -c q +p e- ~c»q-,Jrx e- *p-,Jr€ ei!(pX-q€)dx~.

JR2

(23.3)

Leth = 4c+ 1

4y'r

Then, by (23.3) and (23.4),

(F *! G)(z)

e(h2-c)(q2+p2) ( e-(hq-,Jrx)2 e-(hp-,Jr€)2 ei(p~-q~)dx~JJR2

4e(h2-c)(q2+p2) { e-(hq-2,Jrx)2 e-(hp-2,Jr€)2 ei(pX-q€)dx~JJR2

(23.4)

23. Gaussians 131

4e(h2-c)(q2+p2) r e-(hq-2y'rx)2 e-(hp-2y'r~)2lnp

[cos(px - q~) + i sin(px - q~)]dx~

4e(h2-c)(q2+p2) r e-(hq-2y'rx)2 e-(hp-2y'r~)2cos(px - q~)dx ~

la2

+ i4e(h2-c)(q2+p2) r e-(hq-2y'rx)2 e-(hp-2y'r~)2sin(px - q~)dx ~

lJR24e(h

2-c)(q2+p2) h (z) + i4e(h

2-c)(q2+p2)h (z) (23.5)

for all z in C, where h(z) and I 2 (z) are, respectively, the first integral and thesecond integral on the second last line of (23.5). Then

II (z) r e-(hq-2y'rx)2 e-(hp-2y'r02 [cos(px) cos(q~) + sin(px) sin(q~)]dx~lJR2r e-(hq-2y'rx)2 e-(hp-2y'r~)2cos(px) cos(q~)dx~

la2

+ r e-(hq-2y'rx)2 e-(hp-2y'r~)2sin(px) sin(q~)dxd~lJR2i: e-(hq-2y'rx)2 cos(px)dx· i: e-(hp-2y'r02 cos(q~)~

+i: e-(hq-2y'rx)2 sin(px)dx .i: e-(hp-2y'r~)2 sin(q~)~ (23.6)

for all z in Co By Lemma 23.2,i: e-(hq-2y'rx)2 cos(px)dx

1 JOO 2 [p ]= 2ft -00 e-x

cos 2ft (x + hq) dx

= 2~i: e-x2

[cos (2~x) cos (2~hq) - sin (2~x) sin (2~hq)] dx

= ~ cos (2~hq)100

e-x2

cos (2~x) dx

= .Jff cos (hqp) e-l~rP2 (23.7)

2ft 2ft

and i: e-(hq-2y'rx)2 sin(px)dx

1 JOO _x2 . [ P ( )] d

= 2ft -00 e sm 2ft x + hq x

132 23. Gaussians

2~I: e-x2

[sin (2~x) cos (2~hq)

+ cos (2~x) sin (2~hq)] dx

~ sin (2~hq)100

e-x2

cos (2~x) dx

= 2v:r sin (;;r) e-l~rP2 (23.8)

for all z in C. So, by (23.7) and (23.8), (23.6) becomes

h(z) 7f 2 (hqp ) _--L(q2+p2) 7f. 2 (hqp ) _--L(q2+p2)-cos - e l6r + -SIn - e l6r

4r 2y'r 4r 2yr7f _ --L (q2 +p2)-e l6r

4r(23.9)

for all z in C. Similarly, by (23.7) and (23.8),

12 (z) ( e-(hq-2y'rx)2_(hp-2y'r~)2 [sin(px) cos(q~) - cos(px) sin(q~)]dx~JJR2

( e-(hq-2y'rx)2_(hp-2y'r~)2 sin(px) cos(q~)dxd~JJR2

_ ( e-(hq-2y'rx)2_(hp-2y'r~)2 cos(px) sin(q~)dx~JJR2I: e-(hq-2y'rx)2 sin(px)dx .I: e-(hp-2y'r~)2 cos(q~)~

_I: e-(hq-2y'rx)2 cos(px)dx·I: e-(hp-2y'r~)2 sin(q~)~

.!!.- sin ( hqp ) cos ( hqp ) e- l~r (q2 +p2)4r 2y'r 2y'r

_.!!.- cos ( hqp ) sin ( hqp ) e-l~r (q2+ p2)4r 2y'r 2y'r

= 0 (23.10)

for all z in C. So, by (23.5), (23.9) and (23.10), we have

(F *~ G)(z) = ~e(h2-c-l~r)(q2+p2) = ~e(h2-c-l~r)lzI2, z E C.r r

Furthermore, by (23.2) and (23.4),

1 (4c+l)2 1 Ch2 - C - - - c - - = -(2c + 1 - 2r)

16r 16r 16r 2rc cd

- (2c + 1 - 2c - 2d - 1) = - -.2r r

Therefore we get

23. Gaussians 133

1 7r cd I 12(F *2 G)(Z) = -e- r z, Z E e, (23.11)r

and the first step of the proof is complete. Next, setting Z = (ZI' Z2, ... , zn) and( = ((1, (2,·'" (n), and using (23.11), we have

(F *~ G)(z) = {e-c1z -<1 2 e-dl<1 2 e~(z'(-1<12)d(len

nII1e-c1Zj -<jI2 e-dl<jI2 e~(Zj(j-l<jI2)d(jj=1 e

n

II 7r -£4lz'1 2_ (7r)n -£4lzI 2-e r J _ - e r

r rj=1

for all z in en, and the proof is complete. 0

The following result is also a consequence of the fact that the Fourier trans-2

form of the function t/J given by t/J(x) = e-'T, x E JR, is equal to t/J.

Lemma 23.4 Let cp(z) = e-tlzI2, t > 0, z E en. Then $(() = (2t)-ne-itl<12

,

(E en.

Now, we are ready to prove Theorem 23.l.

Proof of Theorem 23.1. By Theorem 22.2 and Corollary 22.3, it is enough toprove (23.1). By Lemma 23.4,

and

G(i) = ~C,._l_ _ :q1<12

i E en.., L..J J (2d'-)n e ,.,

j=1 J

So,

(23.12)

where

and

134 23. Gaussians

where Sk = 4~k and tj = 4~'· By Theorem 23.3 and (23.12),3

where

Now,__1_ 4dk dj

16dk dj dj + dk + 2dk dj

1

Therefore, setting Tk,j = dj + dk + 2dk dj, we have

Then, by (22.4), (23.13) and Lemma 23.4 again, we get

(23.13)

H(z)

for all z in Cn , and this completes the proof. o

In fact, we can give a larger subspace of L 2 (Cn ) which contains the subspaceM and has the same property as M with respect to the composition of Daubechiesoperators.

Let M be the set of all series of the form

00

"'C'e-djlzI2 "....nL..J3 ,zEIL-,k=l

where Cl, C2 , ... are complex numbers, and dl, d2 , ... are positive numbers such

that the sequences {Cj }~1 and {~}oo are both in [1. We call any such function3 3=1

a Gaussian series. It is clear that every Gaussian series is absolutely and uniformlyconvergent on Cn .

Proposition 23.5 M is a subspace of L 2 (Cn ).

23. Gaussians 135

Proof. By Fubini's theorem, the assumption that the sequences {Cj }~1 and

{ ~ } 00 are both in [1, and the fact that3 J=l

(23.14)

200L Cje-djl'12j=l

1f: f: CjCke-(dj+dk)lzI2 dz

en j=l k=l

f: f: CjCk1 e-(dj+dk)l z I2dz

j=l k=l en00 00 n

~(;CjCk (dj : dk)n

< trn f: ICjl f: I~:I < 00

j=l k=l k

for all L:;:1 Cj e-dj /./2

in M. Thus, every series in M is in L2 (Cn ). To see that M

is a subspace of L 2 (Cn), let

00L Cje-djl'12 E Mj=l

and let a E C. Then

and

00 00 00L Cje-djlzl2 +L Cje-dj1zl2 = L Cj'e-dj'lzI2, z E cn,

j=l j=l j=l

whereC~j_1 = C j , d~j_1 = d j

for j = 1,2, .... Thus,

and

00 IGj'1L (d")nj=l J

~ IC~j_11 +~ IC;jlL.J (d". )n L.J (dll)nj=l 2J-1 j=l 2J

~ ICjl +~ IGjIL.J dn L.J (dl)n < 00.j=l J j=l J

136

Similarly, L:~1 ICj'1 < 00. Hence

23. Gaussians

That a L:~1 Cje-djl'12 E M is obvious. D

We have the following result, which is better than Theorem 23.1.

Theorem 23.6 Let F and G be in M, i. e.,

00

F(z) = L Cje-djlzI2, z E en,j=l

and00

G(z) = LC~e-d~lzI2, z E en,k=l

where C1 , C2 , ... ; C~, C~, '" are complex numbers and d1 , d2 , ... ; d~, d~, ...

are positive numbers such that the sequences {Cj }f= l' {~}00 , {CD~1 and

{~}:1 are all in [1. Then DF,cpDa,ep = DH,ep, where H i: a::

1

in M and

00 00

H(z) = LLCjCke-Tj,k/Z/2, z E en,j=lk=l

To prove this theorem, we need some preparation.

Lemma 23.7 Let F be in M, i.e.,

00

F(z) = LCje-djlzI2, z E en,j=l

(23.15)

where C1 , C2 , '" are complex numbers, and d1 , d2 , .. , are positive numbers such

that the sequences {Cj }f=1 and {~}~ are both in [1, and let Fm be the function3 J=l

on en given bym

Fm(z) = L Cje-djlzI2, z E en.j=l

Then

23. Gaussians 137

Proof. By Fubini's theorem, the assumption that the sequences {Cj}~l and

{ ~}~ are both in [1, and (23.14),3 J=l

2

=00L Cje-djl'12

j=m+1

= 1 f f CjCke-(dj+dk)lzI2 dzen j=m+1 k=m+1

f f CjCk r e-(dj+dk)l z I2dz

j=m+1 k=m+1 len

~m-oo. 0Lemma 23.7 shows that, as subs~ces of L 2 (Cn ), the subspace M spanned

by the Gaussian functions is dense in M.

Lemma 23.8 Every F in M is in Ll(<en).

Proof. Let FE M. Then

00

F(z) = LCje-djlzI2, z E <en,j=l

where C l , C2, ... are complex numbers, and d1 , d2, ... are positive numbers such

that the sequences {Cj}~l and {~}OO are in [1. Then, by Fubini's theorem and3 J=l

(23.14),

r IF(z)jdzlen

o

138

Lemma 23.9 Let F be in M, i.e.,

23. Gaussians

00

F(z) = LGje-djlzI2, z E cn,j=l

where G1 , G2 , ... are complex numbers, and d1 , d2 , ... are positive numbers such

that the sequences {Gj }~1 and {~}OO are both in ll. Then F E M and1 J=l

Proof. It follows from Lemma 23.7 that

m

Fm = L Gje-djl-12 ~ F

k=l

in L2 (Cn ) as m ~ 00. So, by Plancherel's theorem and Lemma 23.4,

F: = ~G__1_e-4~jl·12 ~ Fm L..J J (2d .)n

j=l J

(23.16)

in L2 (Cn ) as m ~ 00. Therefore there exists a subsequence of {Fm };;;=l, stilldenoted by {Fm };;;=l' such that

for almost all (in cn. Then, by (23.16) and (23.17),

F(() =~ G __1_e_~1<;12L..J J (2d.)nj=l J

(23.17)

(23.18)

for almost all ( in Cn . By Lemma 23.8 and the Riemann-Lebesgue lemma, F iscontinuous on cn. Thus, (23.18) is true for all ( in cn. That F is in M is nowobvious. 0

Now, we are ready to prove Theorem 23.6.

Proof of Theorem 23.6. Let F and G be in M. Then

00

F(z) = LGje-djlzI2, z E cn,j=l

and00

G(z) = L Gje-djlzI2, z E cn,j=l

23. Gaussians 139

where C 1 , C2, ... ; CL C2, ... are complex numbers and dl, d2, ... ; d~, d~, ... arepositive numbers such that the sequences

{ }oo

Cjdn ,

J j=l{CDk=l and {

C' }ood'~ k=l

(23.20)

are all in ll. Then, by (23.12), Lemmas 23.8 and 23.9, Fubini's theorem and theLebesgue dominated convergence theorem,

(23.19)

for all ( in cn. By (23.19) and the proof of Theorem 23.1,

(F*! G)(() = ~~CjC~ (r:k) n e-4r~,kl(12,

Thus, by (23.20) and Proposition 23.5, F *! G E L 2 (Cn ). By Theorem 22.2,Corollary 22.3 and the fact that the function H defined by (23.15) is in M, it isenough to prove that (22.4) is valid. But, by (23.15), Fubini's theorem and Lemma23.4,

00 00

if(() = (2n)-n1 e-iz'( L L CjC~e-rj,klzI2dzen j=l k=l

f f CjC~(2n)-n r e-iz'(e-rj,klzI2dzj=lk=l lenCXJOO 12

= L L CjC~(2rj,k)-ne - 4rj,k 1(1j=l k=l

(23.21)

for all z in cn. So, by (23.20) and (23.21), if = (2n)-n(F *! G) and the proof iscomplete. D

140 23. Gaussians

We give a remark on the commutativity of the product of Daubechies operators.

Remark 23.10 The product (or composition) of two Daubechies operators withsymbols F and G in M is commutative because the ~-convolutionof P and Giscommutative in view of (23.20). Thus, as explained in Section 1.1 in Chapter 1 ofthe book [4] by Be~zin and Shubin, any finite collection of N Daubechies operatorswith symbols in M can be considered as a collection of N compatible quantummechanical observables of which arbitrarily accurate simultaneous measurementscan be made.

24 Group Actions and Homogeneous Spaces

A compact account of group actions and homogeneous spaces, which we give inthis chapter, is helpful for a study oflocalization operators on homogeneous spacesin the next chapter. The book [2] by Ali, Antoine and Gazeau and the book [27]by Folland contain much more detailed material pertinent to this chapter.

Let 0 be a locally compact and Hausdorff topological space and let G be alocally compact and Hausdorff group. We say that G is a left transformation groupon 0 if there exists a continuous mapping G x 0 3 (g,w) f-+ gw E 0 such that forall g in G, the mapping 0 3 w f-+ gw E 0 is a homeomorphism of 0 onto 0,

(gh)w = g(hw), g, hE G, w E 0,

andew =w, wE 0,

where e is the identity element in G. The topological space 0 on which G acts iscalled a G-space and G is sometimes called a group action on O.

Let G be a left transformation group on 0 such that for all WI and W2 in 0,there exists an element g in G for which W2 = gWI. Then we say that the actionof G on 0 is transitive and we call 0 a homogeneous space.

Proposition 24.1 Let 0 be a homogeneous space on which G acts transitively. Letw E O. Then the set Hw defined by

Hw = {g E G : gw = w}

is a closed subgroup of G.

Proof. H w is nonempty because e E Hw . Let g and h be two elements in H w .

Then gw = wand hw = w. Thus,

(gh)w = g(hw) = gw = w.

So, gh E Hw . Also,w = ew = g-Igw = g-IW.

Thus, g-I E Hw . To see that Hw is closed, let g (j. H w . Then gw i- w. Since 0is Hausdorff, we can find a neighborhood UI of gw and a neighborhood U2 of wsuch that gw E UI , W E U2 and UI n U2 = ¢. Using the continuity of the mapping


142 24. Group Actions and Homogeneous Spaces

G X n 3 (g,w) 1-* gw E n, we can find a neighborhood N of 9 and a neighborhoodU3 of w such that N x U3 ~ U1 . Thus, hw =I- w for all h in N. Therefore h ¢ Hw

for all h in N. This proves that the complement of Hw in G is open and the proofis complete. 0

We call Hw the stability subgroup of G associated to w.

Example 24.2 Let H be a closed subgroup of G and let n = Gj H, where

GjH={gH:gEG},

and gH, 9 E G, is the left coset of gin H. Then the action of G on n defined by

G x n 3 (g,hH) 1-* (gh)H E n, g,h En,

is transitive. Hence Gj H is a homogeneous space.

Remark 24.3 All homogeneous spaces considered in this book are coset spacesgiven in Example 24.2.

Let n be a homogeneous space given by n = Gj H, where G is a locallycompact and Hausdorff group and H is a closed subgroup of G. Let 1/ be a Borelmeasure on n. Then we say that 1/ is left invariant if

1/(8) = l/(g8), 9 E G,

for all Borel subsets 8 of n, where g8 = {gw : wEn}. Thus, for all 9 in G,

in f(gw)dl/(w) = in f(w)dl/(g-lW) = in f(w)dl/(w)

for all Borel functions f on n. In view of Theorem 4.2, the group G always carries a left invariant measure, namely, the left Haar measure. Unfortunately, thehomogeneous space Gj H need not possess a left invariant Borel measure. A leftquasi-invariant measure, though, does exist on Gj H. A Borel measure 1/ on n issaid to be left quasi-invariant if 1/ and I/g are equivalent measures on n, where

I/g (8) = l/(g8), 9 E G,

for all Borel subsets 8 of n.We end this chapter with some notions that we need in the next chapter.

Let H be a closed subgroup of a locally compact and Hausdorff group G. Letn = Gj H. Then the mapping q : G - n defined by

q(g)=gH, gEG,

is called the canonical surjection of G on n. A mapping s : n - G is said to be asection on n if

q(s(w)) = w, wEn.

25 A Unification

Localization operators in the setting of homogeneous spaces are first defined inthis chapter. They are then shown to be in the trace class 8 1 and a trace formula for them is given. Localization operators on locally compact and Hausdorffgroups equipped with square-integrable representations, Daubechies operators andwavelet multipliers are then shown to be localization operators on homogeneousspaces. In this perspective, this chapter can be seen as a unification of the threeclasses of linear operators.

Let G be a locally compact and Hausdorff group and let H be a closedsubgroup of G. Let v be a left quasi-invariant measure on the homogeneous spaceo = G/ H. Let 1r be a unitary representation of G on a Hilbert space X. As usual,we denote the inner product and the norm in X by (, ) and IIII respectively. LetS : 0 --+ G be a Borel section. Suppose that there exists an element cp in X suchthat IIcpll = 1 and

In I(cp, 1r(s(w))cp)12 dv(w) < 00.

Then we say that 1r is a square-integrable representation of G on X with respect toHand s, and we call cp an admissible wavelet for 1r. If cp is an admissible waveletfor the square-integrable representation 1r of G on X with respect to Hand s,then we define the constant Cs,H,'fJ by

Cs,H,cp = In I(cp, 1r(s(w))cp)1 2dv(w).

Let FE £1(0). Then we define the linear operator LF,s,H,cp : X --+ X by

(LF,s,H,cpX,y) =~ ( F(w)(x, 1r(S(w))cp) (1r(s(w))cp, y)dv(w)s,H,'fJ in

for all x and y in X.

Proposition 25.1 LF,s,H,cp : X --+ X is a bounded linear operator and

IILF,s,H,'fJII* S _1_ ( IF(w)ldv(w).Cs,H,'fJ in


144 25. A Unification

D

Proof. Using Schwarz' inequality, Ilcpll = 1 and the fact that 1r(g) : X --+ X is aunitary operator for all 9 in G, we get

I(LF,s,H,<pX,y)1 = I~ rF(w)(x, 1r(S(w))cp)(1r(S(w))cp, y)dv(w)Is,H,<p in

::; _1_ rIF(w) II(x, 1r(S(w))cp) I1(1r(s(w))cp,y)ldv(w)Cs,H,<p in

::; cs,~,<p ~ IF(w)ldv(w)llxllllyll

for all x and y in X, and the proof is complete.

Remark 25.2 If

(x,y) = _1_ r (x, 1r(s(w))cp)(1r(s(w))cp, y)dv(w)Cs,H,<p in

for all x and y in X, then the identity operator I on X can be written as

1= _1_ r(', 1r(s(w))cp)1r(s(w))cpdv(w).Cs,H,<p in

(25.1)

Thus, in this case, (25.1) plays the role of the resolution of the identity formulafirst formulated in (6.3). The role of the symbol F : n --+ C is again to localize onthe homogeneous space n so as to produce a nontrivial bounded linear operatorLF.s,H,<p : X --+ X with applications in the mathematical sciences. So, we callthe bounded linear operator LF,s,H,<p : X --+ X a localization operator on thehomogeneous space n.

We are now in a position to prove that localization operators on homogeneousspaces are in 8 1 and compute their traces.

Proposition 25.3 The localization opemtor LF,s,H,<p : X --+ X is in 81 .

Proof. As in the proof of Proposition 13.1, it is enough to prove the propositionfor the case when F is a nonnegative and real-valued function on n. Let {CPk : k =1,2, ...} be an orthonormal basis for X. Then, using Fubini's theorem, Parseval'sidentity, Ilcpll = 1, and the fact that 1r(g) : X --+ X is a unitary operator for all 9in G, we get

00

'L(LF,s,H,<PCPk, CPk)k=l

1 00 rCs,H,<p (; in F(w)l(CPk' 1r(s(w))cp)1

2dv(w)

1 r 00

< Cs,H,<p inlF(w)1 (; I(cpk, 7l"(s(w))cp)12dv(w)

< _1_ rIF(w)ldv(w) < 00, (25.2)Cs,H,<p in

and the proof is complete in view of Proposition 2.4. D

25. A Unification 145

Theorem 25.4 The trace tr(LF,s,H,<p) of the localization operator LF,s,H,<p : X --+ Xis given by

tr(LF,s,H,<p) = ~ rF(w)dv(w).s,H,<p in

Proof. Let {CPk : k = 1,2, ...} be any orthonormal basis for X. Then, usingFubini's theorem, Parseval's identity, Ilcpll = 1 and the fact that 11"(g) : X --+ X isa unitary operator for all g in G, we get

00

tr(LF,s,H,<p) = ~(LF,s,H'<PCPk' CPk)k=l

00 1 r{; Cs,H,<p in F(w)l(cpk,11"(S(w))cp)1

2dv(w)

1 r 00

Cs,H,<p in F(w) {; I(cpk, 11"(s(w))cp)12dv(w)

= _1_ rF(w)II11"(s(w))112dv(w)Cs,H,<p in_1_ rF(w)dv(w),Cs,H,<p in


We can now give three examples of localization operators on homogeneousspaces.

Example 25.5 (Locally Compact and Hausdorff Groups) Let cP E X be an admissible wavelet for a square-integrable representation 11" : G --+ U(X) of a locallycompact and Hausdorff group G on X. Let H = {e}, where e is the identity element in the group G. Then, of course, G/ H = G and JL is a left invariant measureon the homogeneous space G/ H, which is the same as G in this situation. Thesection S : G/ H --+ G can be taken to be the identity mapping on G. Thus, thelocalization operator LF,<p : X --+ X defined by (12.1) can be considered as alocalization operator LF,s,H,<p: X --+ X on the homogeneous space G/H.

Example 25.6 (Daubechies Operators) Let H be the subgroup of the Weyl-Heisenberg group (WH)n defined by

H = {(O, 0, t) : t E 1R/211"Z}.

Then, of course, H is the center of (WH)n, and the homogeneous space (WH)n / His simply the Euclidean space IRn x IRn. The Lebesgue measure on IRn x IRn is a leftHaar measure on the homogeneous space (WH)n / H = IRn x IRn. Let s : IRn x IRn --+

(WH)n be the section defined by

s(q,p) = (q,p,O), (q,p) E IRn x IRn.

146 25. A Unification

Then for all functions cp in L2 (lRn ) with Ilcpll£2(lRn) = 1, we get, by (17.19),

for all q and p in IRn . Thus, every function cp in L2 (lRn ) with Ilcpll£2(lRn) = 1 is anadmissible wavelet for the square-integrable representation 7r of (WH)n on L 2 (lRn )with respect to Hand s, and

Cs,H,<p = (27r)-n.

Let F E L1(lRn x IRn). Then the localization operator LF,s,H,<p : L2(lRn) ----; L2(lRn)on the homogeneous space (WH)n / H is exactly the same as the Daubechies operator DF,<p : L2(lRn) ----; L2(lRn) defined by (17.23).

Example 25.7 To see that the wavelet multipliers introduced in Chapter 19 arein fact localization operators on a homogeneous space, we let H be the subgroupof IRn given by H = {O}, where 0 is the additive identity of the group IRn . Then,obviously, IRn/ H = IRn and the Lebesgue measure on IRn is a left invariant measureon the homogeneous space IRn / H, which is simply the same as IRn . The section s :IRn/ H ----; IRn is taken to be the identity mapping on IRn. Let cp E L2 (IRn )nL4 (IRn)be such that Ilcpll£2(lRn) = 1. Then

Cs,H,<p = r I(cp, 7r(s(Ocp)£2(lRn)12~ = r I(cp, 7r(~)cp)£2(lRn)12d~JlRn JlRn

kn Ikn eix'~ICP(XWdXI2 ~ = (27r)nllcplli4(lRn).

Thus, every function cp in L2(lRn) n L4 (lRn) with IIcpll£2(lRn) = 1 is an admissiblewavelet for the unitary representation of IRn on L2 (lRn ) with respect to Hand s.Furthermore, let a E L1(lRn ). Then

(La,s,H,<pU, V)L2(lRn)

_1_ r a(~)(u, 7r(~)cp)£2(lRn)(7r(~)CP, V)L2(lRn)d~Cs,H,<p JlRn

(27r)-nllcpIILf(lRn) r a(~)(u, 7r(~)cp)£2(lRn)(7r(~)CP, V)£2(lRn)~JIRn

II cpll Lf(lRn) (Pa,<pu, V)L2(IRn)

for all u and v in L2(lRn). Thus, the localization operator La,s,H,<p : L2(lRn) ----;L2(lRn) on the homogeneous space IRn/ H is a scalar multiple of the wavelet multiplier Pa,<p : L2(lRn ) ----; L2(lRn).

26 The Affine Group Action on IR

The aim of this chapter is to show that localization operators on lR considered as ahomogeneous space under the action of the affine group U are wavelet multipliers.

First, we look at the mapping U x lR 3 ((b,a),x) t--+ ax+b E R It is obviousthat U acts on lR transitively. In other words, lR is a homogeneous space on whichthe transitive group action is given by the affine group U. Let H be the closedsubgroup of U given by

H = {(O, a) : a > O}.

Proposition 26.1 The homogeneous space U/ H is isomorphic to lR as topologicalgroups.

Proof For all (b,a) E U, we get

(b,a) = (b, 1)· (O,a).

Thus,(b, a)H = (b, l)H, (b, a) E U.

Hence the mapping lR 3 b t--+ (b, l)H E U/ H is bijective. That the mapping is ahomeomorphism is obvious. Finally, for all b1 and b2 in lR, we get

and hence the mapping is also a group homomorphism. o

Proposition 26.2 The Lebesgue measure on lR is a left quasi-invariant measure onlR considered as a homogeneous space under the group action U.

Proof Let v be the Lebesgue measure on R For all (b, a) in U, we have

dV(b,a)(x) = dv(ax + b) = adv(x),

and hence the measures v and V(b,a) are equivalent. 0

Let s : lR --+ U be the global section, i.e., s(x) = (x, 1), x E R Let 7r be theunitary representation of U on L2 (lR) defined by

1 (x -b)(7r(b,a)u)(x) = ,jau -a- , x E lR,


148 26 The Affine Group Action on R

for all (b, a) in U and all u in L2(1R). Let cP be any function in L2(1R) such thatIIcpllp(lR) = 1 and cj; E L 4 (1R). Then, using Plancherel's theorem, we get

Cs,H,I{) = I: I(cp, 7T(S(X))cp)p(IR)/2dx

= I: I(cp, 7T(X, 1)cp)p(IR)12dx

I: I(cp, T_ x cp)L2(1R)1 2dx

I: II: eix~Icj;(~W~12 dx

27T1I cj;lIi4(1R)'

where (T-xcp)(b) = cp(b - x), bE R

Let a E L1 (1R). The localization operator La,s,H,I{) : L2(1R) --> L2(1R) on 1R,considered as a homogeneous space under the group action U, is defined in Chapter25. In order to understand the localization operator La,s,H,I{) : L2(1R) --> L2(1R)better, we suppose that a is also in LOO(IR). Then for all u and v in Co(IR), we canuse Plancherel's theorem and a change of variables to obtain

for all x in R Thus, for all u and v in L 2 (1R), we get

(La,s,H,I{)U, v)P(IR)

1Icj;IILt(lR) i: a(x)~(x)(~)(x)dx

1Icj;IILt(lR)I: (Ta(~u))(x)(cj;v)(x)dx

II cj;1I Lt(lR) (Ta(~U), ~V)P(IR)

1Icj;1I Lt(lR) (cj;Ta~u, v)P(IR)

II cj;1I Lt(lR) (1:-1 (cj;Ta~)Fu,v )P(IR)'

where a(x) = a( -x), x E RTherefore the localization operator La,s,H,I{) : L2(1R) --> L2(1R) is unitarily

equivalent to the linear operator 1Icj;1I4"4pa,rp : L2(1R) --> L2(1R), where Pa,rp :L2(1R) --> L2(1R) is the wavelet multiplier associated to the symbol a and theadmissible wavelet cj; studied in Chapter 19.

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Index

admissibility condition, 39admissible wavelet, 39affine group, 99anti-Wick symbol, 9asymptotic expansion, 2

canonical form, 11canonical surjection, 142coherent state, 71compatible, 19complete quadratic form, 43conjugate Hardy space, 99convex function, 74convolution operator, 59

Daubechies operator, 95degree, 34dimension, 34

expectation value, 71extended Schur's lemma, 42

filter, 57form domain, 43Fourier multiplier, 109Fourier transform, 2Fourier transformation, 107Fourier-Wigner transform, 117

Gaussian function, 129Gaussian series, 134group action, 141

Haar measure, 25Hardy space, 99Hausdorff group, 22Heisenberg group, 96

Hilbert-Schmidt class, 13Hilbert-Schmidt operator, 118homogeneous space, 141

intermediate space, 19interpolation space, 19interpolation theory, 19inverse Fourier transformation, 107irreducible, 34

Jensen's inequality, 75

kernel, 118

A-convolution, 124Landau-Pollak-Slepian operator, 115left Haar measure, 25left invariant measure, 142left invariant quasi-invariant

measure, 142left regular representation, 38left transformation group, 141localization operator, 66

modular function, 30

Plancherel's theorem, 49positive quadratic form, 43product, 117pseudo-differential operator, 2

quadratic form, 43quantization, 3

representation space, 34reproducing kernel, 49resolution of the identity, 39

156

Riesz-Thorin theorem, 64right Haar measure, 25

sampling theorem, 52Schatten-von Neumann class, 12Schr6dinger representation, 92Schur's lemma, 34Schwartz space, 2second representation theorem, 43section, 142a-compact, 29signal, 114singular value, 11singular value decomposition, 61spectrum, 5square-integrable representation, 39stability subgroup, 142Stone-von Neumann theorem, 96symbol, 66symbolic calculus, 9symmetric quadratic form, 43symplectic form, 121

time and band-limited signal, 114topological group, 21trace, 16trace class, 13transitive, 141twisted convolution, 121two-wavelet constant, 54

unimodular, 31unitary representation, 34

wavelet constant, 39wavelet multiplier, 109wavelet transform, 48Weyl-Heisenberg group, 91Weyl transform, 117Wigner transform, 117

Index

Documents

Wavelet Transforms and Localization Operators