Math. Nachr. 260, 14-20 (2003) / DOI l1.l}O2lmana.200310100

A matriceal analogue of Fejer's theory

Sorina Barza*1, Lars-Erik Persson**2, and Nicolae Popa***3I Dept. of Eng. Sciences, Physics and Mathematics, University of Karlstad, SE-651 88 Karlstad, Sweden2 Dept. of Mathematics, Lulei University of Technology, SE-971 87 Lulei, Sweden3 Institute of Mathematics of Romanian Academy, P.O. Box l-764,70700 Bucharest, Romania

Received 6 March 2001, revised 19 December 2002, accepted 29 December 2O02

Published online 28 October 2003

Key words Fourier series, Fejer's theory, Cesaro sums, infinite matrices, Toeplitz matrices, Schur multipliers

MSC (2000) 15 A57, 42A16, 42A45, 47 835

J. Arazy [1] pointed out that there is a similarity between functions defined on the torus and inflnite matrices.In this paper we discuss and develop in the framework of matrices Fejer's theory for Fourier series.

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1 Introduction

Let A - ("oi), l, j ) 0, be an inflnite matrix with complex entries and let n ) 0. The matrix ,4 is said to beof n-band type if crii : 0 for li - jl > n. This paper is concerned with questions of approximation of infinitematrices by matrices of band type. In particular, the approximation is considered in the space B(12) of boundedlinear operators on the one-sided scalar /2-space, and, in the space of Schur multipliers on B((.2).

We introduce some new Banach spaces of matrices which can be regarded as extensions of the classical Banachspaces of functions C(11) and trl(11). In our opinion these Banach spaces together with B(12) and the space ofSchur multipliers M(12) may be of interest in order to develop at least some results extending known theoremsof classical harmonic analysis in the framework of matrices. We intend to exploit further these notions and ideas.

The aim of the present paper is to extend in the framework of matrices Fejer's theory for Fourier series (see

t3 l).The main results in our opinion are Theorem 4.2, Theorem 4.3, Example 4.8 and Theorem 4.10.

For k : O, +1, +2,. . . , let us deflne ,46 : (alr), where

, lo,, it j_ i: k,n.. : <"tr

|. 0 otherwise.

Ar is called the Fourier coeficient of k-order of the matrix, Gee [4]). We have now a similarity between the

expansion in the Fourier series / - D* ox"'k" of a periodical function / on the torus lf and the decompositionA:Drr, Ar.

This type of similarity between functions defined on the torus 1l and the inflnite matrices was remarked for thefirst time by J. Arazy [A], and exploited further by A. Shields [4]. Our main tool is an important characterizationof Schur multipliers given by G. Bennet [2].

Moreover, there is a similarity between the convolution product / *, g of two periodical functions and the Schurproductof two matrices AandB,C : A*B,wherethematrixChavetheentriesc;7 : at.j.btj,forA: (",,)..and B : (a;3)

,_ lsee also [4]).Now we mention the following results obtained by Fejer, which have been guiding for our investigations:

* e-mail: sorina.barza@kau.se, Phone: +46 54 7001888, Fu: +46 54 7001851Conesponding author: e-mail: larserik@ sm.luth.se, Phone: +46 9ZO 49lll7 , Fax: +46 920 491073

*** e-mail: Nicolae.Popa@ imar.ro, Phone: +4O 2 l25ll}, Fax: +40 2 125126

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Math. Nachr. 260 (2003) / www.interscience.wiley.com

(A)Afunctionf (0):Dxezor".ikq iscontinuousonT(that;s/e C(11)), if andonlyif theCesarosums

k=

o^(r): E-"r('-#)"'r'converge unifurmly on T to f .

(B) A function f (0) : D*1vmpeikq € 11 (T) if and only if

ll".U) -.fllz'(r) - 0 as ?? + oo.

The paper is organized in the following way: In order not to disturb our discussions later on we present somepreliminaries in Section 2.

In Section 3, we derive some properties and relations between the basic sp aces B (1.2) and C (!.2) of independentinterest.

The main results are presented in Section 4 and Section 5 is reserved for some concluding remarks and results.

2 Preliminaries

In view of Fejer's result (A) it is natural to give the following deflnition:

Deflnition 2.1 Let,4 be a matrix corresponding to an operator from B(12). We denote by o.(A) the Cesaro

sum associated to S.(A) 19 tt-, Al",thatis o^(A) : D}:-*oo(t #) ,n* we say rhat.4 is acontinuous matrix if

Let us denote by C (lz) the vector space of all continuous matrices and consider on it the usual operator norm.Amatrix U : (*rt)r,i iscalled aSchurmultiptieriff M 'r A< B((.2) wheneverA e B(!.2).

The space of all Schur multipliers will be denotedby M(12) and the Schur multiplier norm of M will be, bydefinition:

llMll.^,t<t,l : sup llM ', AllBg,1.llAll,eil<,

Then it is known (see [2]) that M(12) is a Banach space which is a commutative unital Banach algebra withrespect to Schur product.

Moreover,if Mis aToeplitz matrix M, i.e. amatrix withtheentriesmli : ffij-,i.,foralli., j € N, thenthefollowing statement holds (see [2, Theorem 8.1]):

(l) The Toeplitz matrix M is a multiplier if and only if there exists a bounded, complex, Borel measure p, eM(T) with the Fourier coeficients

t"(n) : *., n : 0,*1, +2,...

Moreove4 we have

llMll,'ru; : llpll.nzrrt .

We also mention the following well-known fact (see e.g. [5]):The Toeplitz matrix M represents a linear and bounded operator on 1.2 if and only if there exists a function

/ € I-(T) with Fourier cofficients i@) : mn for all n € Z.Moreover, we have

llMllau : ll/llr-or.

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t6 Barza, Persson, and Popa: A matriceal analogue ofFejer's theory

3 Some properties of the space C (12)

First of all let us observe the following fact:

Remark 3.1 By Fejer's theorem (A) we have that a Toeplitz matrix ? : (t*)*ez € C((.2) if and only iffu@) Ej Drrutr"urt € C(T), and in this way we can see that the notion of a continuous matrix may be

regarded as an analogue of that of a continuous function.

Now let C- denote the space of all matrices defining compact operators.

Proposition 3.2 C((.2) is a proper closed ideal of B(1.2) with respect to Schur muhiplicationwhich, in itstum, conlains C* properly.

Proof. We have:

rr n 'r illtlo,(A)lts1r,1 : ll r er (t - # )ll < llM^|.u'o,rltAilarr,r .

llk=-n 'llo(tr)

where Mn is the n-band type Toeplitz matrix with the entries

_11-V ') ir lj_ il<rt,mil : ( n+]I O otherwise.

Hence, by [2, Theorem 8.1] (see also (1)), we have:

tto-(A)ttae,)= ll-i (' - #)"'-'ll-,, ttAtta..,t : tttttarn,t

Hence C (1.2) is a closed subspace ot B (1.2) .

Now we observe that for A,B € C((.2), o"(Ax B) : o.(A),, B and then we have for A e C(12),B e B(t2),

llA * B - o.(A* B)llsu,t : ll te - ".(A)j * Bllutn,t

< ll,4 - o.(A)llB<u>.llBllue,t< llBllarns'llA - o*(A)llsq,1 .

Here we have used the simple fact that

llBll.uu,t : llB * Lll,v.<r,t < llBlla<r,t' ll\ll,ttrut : llBlle<r,t,

where A.;i : 1 for all i., j € N, and llAlly U) : l.Herce C (12) is a closed ideal of B((.2) with respect to Schur multiplication.Next we note that C (1.2) is a proper ideal ot B(!.2).Denoting by e;7 the matrix whose single non-zero entry is 1 on the i-th row and on the j-th column, we consider

the matrix A : X*.s .41", where At: e*+t,z*+r, k ) 0, which belongs to B(12), since (AA*)1/2 : I (I is

the identity matrix). Moreover,

ll- r ' ll r k \|",(A) _ A|ar*t :

llE Ak +;+1I*o_ll : (fgf h)v 1 : l

llk>n k:0 ll A(r)

for all n and thus A / C(1.2).Now let A € C*. Denoting by

(o,, i.i--n.P"(A)(i,i) - )

I O otherwise.

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we have

llP.(A)-AlleO,l _-0, as n+@.

But, by Bennett's theorem, we have for k > n:

ll -:- vt ll

llP"(A) - or(&(/))llauil : ll L tr"(A)), F;lllllt:-" '"''lle(,)il n tot ll

< ll )- -t(L"uell .llP,(Arlls,u,

-0.- lltL,k+l ll-,r, k-o

Hence P.(A) € C(1.2) for all n, € N and consequently C* c C((2).Since it is easy to see and well-known that a Toeplitz matrix does not represent a compact operator, by Remark

3. 1 it follows that C* is a proper subspace of C (!.2). The proof is complete. n

4 The main results

We will give another characterization of the space C(12 ) by using continuous vector-valued functions but first wenote the following simple fact:

Remark 4.1 Consider the function f a:T - B(!.2) givenby f e(t) : A',, (ei(L-k)t );,616. Th.,

llf e(0)llaw> : llAllav,t for all 0 €T.

Indeed, by Bennett's multiplier theorem, we have that

lVe(e)llew,t < llAllnrq lla-all : llAllarut,

wheredB € M(T) denotestheDiracpointmass at0 eT. Similarly,sinceA : fe(il,r (e-t'(l-*)e1 ,k>0,

we

have that llAlla<ut < llf e(0)llaus.An easy consequence of this remark is that /a is continuous on 1l if and only if it is continuous at one single

point.Now we ask ourselves how the matrix ,4 should be in order that the function /a shall be continuous.The answer to this question is as follows:

Theorem 4,2 l,et A be an infinite matrix. Then f a is a B(42)-valued continuous function if and only ifA e C(1.2), with equality of the corresponding norms.

P r o o f. By Remark 4. I it follows that

ll"-Ue) - f ellc$,e@,)) : llo"(.4) - AllB<r,t.

Now reasoning as in the proof of Fejer's result (A) (see for instance [3]) we get that for a continuous function

f t:T + B((.2) it follows that

ll".@)-Allue,t - 0,

as r, + oo. Thus A e C({.2).The converse implication follows easily from Remark 4.1 . D

Now we shall study the following question: What can we say about subspaces of M(12) in connection withthe multiplier property ?

The following theorem gives a justification of introducing C ((.2) and also a partial answer to the above ques-

tion. It is the matriceal analogue of [6, Theorem 1 1. 10, Chap. IV].

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18 Barza, Persson, and Popa: A matriceal analogue of Fejer's theory

Theorem 4.3 The Toeplitz matrix M : (m*)*ez is a Schur multiplier from B (12) into C ((.2) itr

D*r"nn'€ r1(T)KtL

Proof. If weidentify/ € r-(T) withitscorrespondingToeptitzoperator \: (i(i - k))3,.20 inB(12),then it is straightforward to see that a Schur multiplier M : (m1-n) ,,6>s mapping B(1.2) lnto C(/2) induces a

Fourier multipter sequence ^ : {^.}ff:-- mapping ,- (T') into C(T') which is known to correspond, in the

manner indicated in the statement above, to a function from tr1 (11) (see [6]). The converse follows also by thesame lines.

Guided by [2] we propose for matrices a similar notion (in our opinion) to that of Lebesgue integrable func-tions.

Definition 4.4 We say that an infinite matrix A is an integrable matrLr if o.(A) ,= e in the norm of

M(lr). Thespaceof allsuchmatrices,endowedwiththenorminducedby M(lz),will bedenotedby Lt(12).

Of course Ll (l.r) is a Banach space.

Remark 4.5 It A e L1((.2), then it follows that A* B e C(1.2) for all B € B(1.2).

Indeed, for B € B(12), inview of the fact that

llo^(A,' B) - A* Bllae) < ll".(A) - All.ue; .llBllsu"> *, 0,

we get A,, B e C(1.2),

Now it is clear that L1(1.2) is a closed ideal of M(1.2) with respect to the Schur product.We have the following analogue of the Riemann-Lebesgue Lemma:

Lemma 4.6 Let M e L1(12).Then

,lim llMelllr1r,1 : 0.IKI-.@

Proof. ItisclearfromDefinition4.4thatforanye >0,thereisan(e) suchthat,forlkl >n(e),itfollowsthat llMkllL, A,; ( e and the proof is complete. tr

Remark 4.7 It is easy to see that for a diagonal matrix Ap, k e Z, wehave:

llArlla<ut : llAe,llu<r,t .

Thus, in Lemma 4.6 we can take llMullBeS instead of llMrll",eS.In view of Theorem 4.2 and Remark 4.5 it is natural to ask: ff A is a Schur multiplier which maps B($) into

C (1.2) does it follow that A e Ll (1.2) ?

The answer to the above question is negative, in fact we have:

Example 4,8 Let A be the following matrix:

tr

,:(lii)

The matrix A is a Schur multiptier which maps B ((.2) into C (1.2) btt it does not belong to L' (lr).Infact,AisaSchurmultiplierwiththepropertythatA*B €C(1.2) forallB e B((.2) sincethematrixAxB

has rank 1 and therefore represents a compact operator and consequently it belongs to C ((.2). Moreover, -4 doesnot belong to Lt (!.2) by Lemma 4.6.

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Therefore the Banach space M (B((.2), C(tz)) of all multipliers from B(!.2) into C(1.2) is different from bothM($) and L'(b). Thus, it seems to us that this space deserves to be studied in more detail.

On the other hand the space M(C(1.2),C([z)) of all infinite matrices A such that A* B e C([.2) for allB e C(tr) can be described easily. More precisely we have:

Theorem 4.9 M(C(1.2),C({.2)) is exactly the space M(12) of all Schur multipliers.

Proof. Sinceo,(A * M): o"(A)x Mitfollowseasilythat M e M(C({.2),C(!z))if M € M(12)andA e C(!.2).

Conversely, assuming that MeM(C(!.2),C({.2)), we have for Ae B(!.2) thatllM ', o^(A)lla1z,;(const. xll"-@)llB<r;.

Moreover, o.(A) + A in the weak topology of operators in B(1.2), that is (a.(,4) r,a) - (Ar,y) for allr,y € t2, where (., .) is the scalar product in /2 (use./2-seQuences with a flnite number of nonzero components

and a standard approximation argument). This yields that llM *,4116(rz) ( const. llAlleUS, that is M is a Schur

multiplier. The proof is complete. !

Now we give a characterization of an integrable matrix in the spirit of Theorem 4.2:

Theorem4.L0 Let A e M({.2) and f a(0) : A*("'(t-r)t) i,orofor0 €T.Then f is apointwisewell-defined

function / : 1l * M(12) suchthatllf (0)lly<ul: llAll.utt),for all0 e T. Furthermore, f e C(T,M(lz)),that is, f is continuous, if and only if A e Lt (12).

Proof. Forp€M(T') letusintroducethenotationT"fortheToeplitzmatrixwithsymbolp,thatis

r, : (t"(i - k))3,*20 , where [,(n) : | "-0"' appl

is the n-th Fourier coefficient of p. Note that f (0) * A * T5", where d6 € M(11) denotes the unit point mass at

d e 11. By Bennett's multiplier theorem (Theorem 8.1 in [1]) we have that

llf (e)ll.u<t,t < llAll.na<ut.

Similarly, since ,4 : I @) * T;e, we have that lll4ll7'2.<n,l < llf @)llufn,t.Assume next that A e Ll ((.2) . We then have

ll"r,,(/)(B) - f@)ll.uu,t : ll"r(,a) - All"vet + 0.

Thus, a,",.(/) + / uniformly and we obtain that f e C (T , M(lz)).Assume now that f € C(T, M(lz)). We consider thenthe M(42)-valued integral

fN(o) : 0eT,

where K17 is the -A/-th Fejer kemel. It is straightforward to see that f N - f in C(11, M(lz)) as ly' + oo. Aneasy computation yields that

rN(e) :,i (, - #) i(n)"u-,,

wtrere / (n) is the n-th Fourier coefficient of /. To compute the Fourier coefflcient /(n) we need only to observe

that the operation M + mjk of taking the (j, k)-th entry is abounded linear functionalon M(12). By this we

clearly have i@) : A-. Summing up we have shown that

t9

! [ to-t)KN(t)dt,z7r Jr

l//

,r,jl.i, (, - #) An.ino : r(o)

in C(11, M(lz)). For d : 0 this yields oN(A) - Ain M(!.2)). The proof is complete. u

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Barz4 Persson, and Popa: A matriceal analogue ofFejer's theory

We also remark that / is continuous on ll if and only if it is continuous at one single point. This is clear bythe flrst assertion of the above theorem.

The next remark is an easy consequence of Fejer's theory.

Remark4.l1 Let,4beaToeplitzmatrix. Then A e Lt((.2) if andonlyif itrepresentsafunctionfrom-L1(11).

5 Concluding remarks and results

We recall the following well-known result (see [3, Chapter 2]):(2) A function f on T belongs to -L- (11) if and only if

sup lia"(/)lla-1r; { oo.

We have the following matriceal analogue of (2):

Proposition 5.1 Let A be an infinire matrix. Then A belongs to B((.2) if and only ifsup lla.(,4)ll6 @,) { cn.

Proof. Assumethatsup. ll".U)lle@ ( oo.Then,byreasoningasintheproof of Theorem4.9,wegeteasilythatAeB(1.2).

The converse implication can be proved by using the same arguments as in the proof of Proposition 3.2. tr

Proposition 5.2 A e M(1.2) if and only ifsup lla.(,4)lly e,) I cn.

Proof. Assume that

supllo"(,4)lly(.,) I 6

and fix B e B (!.2) . We then have

llo*(B * A)lla<ts < llBlle<n,tsup lla"(,A)llq U) 1 (n.

Itnowfollowsthato"(B'oA) + B',Aintheweaktopologyof operatorsrnB(1.2). (Seetheproofof Theorem4.9.)lnparticular, B*AeB(12).SinceBisarbitrarythismeansthat,4isaSchurmultiplier.

Conversely, letA e M(lr). Theno"(A) : A*o"(M), where M: (*t)rczwithmi: l.Thus,byusing[2, Theorem 8.1], we f,nd that

ll" -@)ll.n trns < llAll uus' ll".(M )ll.vtrr"t < llAll ^a

@i .

The proof is complete.

Acknowledgements The first named author thanks the Royal Swedish Academy of Sciences and the Romanian Academy ofSciences for financial support which made this collaboration possible. She was partially supported by EUROMMAT ICAI-CT-2O00-70022 and KAW 2000.0048. The last named author wants to thank the members of the Dept. of Eng. Sciences,

Physics and Mathematics for hospitality during his visit at the University of Karlstad, Sweden. He was also partially supported

by EUROMMAT ICA|-CT-2000-7AO22. Finally, we thank the referee for some generous advices, which have essentially

improved this final version of the paper.

References

[1 ] J. Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces,

Integral Equations Operator Theory L, 453-495 (1978).[2] G. Bennett, Schur multipliers, Duke Math. J.44,603-639 (1977).[3] K. Hoffman, Banach Spaces of Analytic Functions (Englewood Cliffs, New Jersey, Prentice Hall, 1962).[4] A. L. Shields, An analogue of a Hardy-Littlewood-Fejer inequality for upper triangular trace class operators, MaIh. Z.

182,473484 (1983).[5] K. Zhu, Operator Theory on Banach Function Spaces (Marcel Dekker, New York, 1990).[6] A. Zygmund, Trigonometric Series I-II (Cambridge University Press, Cambridge, 1959).

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