The Discrete Calder on Reproducing Formula of Frazier and ...faculty.wwu.edu/benyia/papers/Benyi-Torres_AMSformat.pdf · our theory, the wavelet theory is immediately connected to

The Discrete Calderon Reproducing Formula of Frazier andJawerth

Arpad Benyi and Rodolfo H. Torres

Abstract. We present a brief recount of the discrete version of Calderon’s

reproducing formula as developed by M. Frazier and B. Jawerth, starting from

the historical result of Calderon and leading to some of the motivation andapplications of the discrete version.

Alberto P. Calderon’s genius has produced a plethora of deep and highly in-fluential results in analysis [6], and his reproducing formula certainly qualifies as agem among them. Also commonly referred to as Calderon’s resolution of identity,Calderon’s Reproducing Formula (CRF) is a strikingly elegant relation, inspired bythe simple idea of breaking down a function as a sum of appropriate convolutionsor wave like functions (see (1.1) below). Moreover, mathematicians working onwavelets consider nowadays Calderon as one of the forefathers of the theory.

The purpose of this expository note is to honor the memory of Bjorn Jawerthby presenting a brief account of some aspects of one of his major contributions tomathematics: the development in collaboration with Michael Frazier of what theycalled “the φ-transform” and how it relates to Calderon’s original formula. None ofwhat is presented here is new and we will repeat some arguments commonly found inthe literature; but some others that we shall present seem to be part of the folkloreof the subject or are hard to locate in the references. We will try to follow a partiallyhistorical and partially formal approach to this beautiful formula as we learned itfrom the horse’s mouth, in particular from Bjorn Jawerth himself, but also fromMichael Frazier, Richard Rochberg, Michael (Mitch) Taibleson, and Guido Weiss.1

The goal is to summarize here in a succinct way some simple motivations whilepointing to both classical as well as some not so well-known references, includingonly some technical details for completion purposes or to illustrate some concepts.Our hope is to convey, perhaps not to the experts but rather to a broader uninitiatedaudience, some of the profound contributions of Jawerth and collaborators, whichare sometimes overlooked in the continuing proliferation and rediscovery of resultsin the area of function space decompositions. We also want to insight the curiosityof those who may have not read numerous original works we shall mention, andmotivate them to further explore the reach existing literature.

2010 Mathematics Subject Classification. Primary: 42B20; Secondary: 42B15, 47G99.1In particular, in the development of some topics in this survey we have benefited a lot from

unpublished lectures notes of courses taught by Frazier and Jawerth at Washington University inthe 80’s.

1

2 A. BENYI AND R. H. TORRES

As it is today well-understood, the φ-transform is a way to represent functionsand even distributions as linear combinations of translates and dilates of a fixed func-tion. The representation is sometimes also referred to in the literature as the Frazier-Jawerth transform, almost orthogonal decomposition, or non-orthogonal waveletexpansion. In fact, like wavelets, the Frazier-Jawerth decomposition [24, 25] canbe viewed as a discrete version of the CRF. Moreover, in appearance (see (2.1)below) there is no difference between discrete wavelets and the φ-transform, buttheir origins and initial motivations for their developments were somehow different.While Frazier’s and Jawerth’s were rooted in the analysis of functions spaces, theirintrinsic features and atomic decompositions, and the classical operators acting onthem, the study of wavelets can be traced back to the rediscovery of CRF in anapplied context and the construction of orthonormal bases with certain particularproperties. We quote Frazier and Jawerth [25, p. 37]:

“Wavelets are a collection of functions similar to the representingfunctions in our decomposition, but which are mutually orthogo-nal. In fact, wavelets form an unconditional basis for the usualfunction spaces in harmonic analysis listed above. Thus, unlikeour theory, the wavelet theory is immediately connected to thevast literature on the construction of explicit unconditional basesfor various function spaces. However, for the applications thatwe have considered (not related to bases), our more elementarydecomposition has been sufficient. Thus, for reasons of simplicity(and perhaps stubbornness) we have presented our results withoutreference to the beautiful theory of wavelets. However, the readerwill readily note that our conclusions generally apply as well tothe wavelet decomposition.”

Likewise, we will not attempt the almost impossible task to provide a compre-hensive account of wavelet theory, but just implicitly point out to some commonfeatures with the Frazier-Jawerth decomposition and a few references along the way.(For a more authoritative account of the subject of wavelets, the reader is referred tothe works of Y. Meyer [45], Hernandez and Weiss [34], and the references therein.)Nor shall we describe in its fullness the connection of the φ-transform to the veryactive theory of frames; in particular, frames generated by the action of a group(like the ax+b group of translations and dilations) on a fixed collection of functions.

In the derivation of the formula of Frazier and Jawerth one gives up the orthog-onality of the analogous wavelet formula for simplicity in the construction. Suchconstruction is astonishingly elementary, and this only adds to its beauty. At a the-oretical level, this lack of perfect orthogonality is of no consequence in most uses.In addition, the redundancy and flexibility of frame representations is nowadayspreferred in some numerical applications, while in others the perfect orthogonalityof wavelets still becomes crucial.

The potential of “almost orthogonal” or “quasiorthogonal” decompositions inapplications has been observed early on in the development of all these and relatedexpansions. For example, I. Daubechies, A. Grosmann, and Y. Meyer stated in [17,p. 1273]:

“We believe that tight frames and the associated simple (pain-less!) quasiorthogonal expansions will turn out to be very usefulin various questions of signal analysis, and in other domains

DISCRETE CALDERON’S REPRODUCING FORMULA 3

of applied mathematics. Closely related expansions have alreadybeen used in the analysis of seismic signals” [the authors referredto the works in [27], [30], [31]].

While in their work they were focusing on the construction of frames based onthe Weyl-Heinseberg group, the authors further mentioned in [17, p. 1273]:

“...the construction of tight frames associated with the Weyl-Heisenberg group is essentially the same as that of tight framesassociated with the ax+b group. Tight frames associated with theax+ b group were first introduced in a different context closer topure mathematics. In Ref. 13(b) [Frazier-Jawerth [24]] one canfind a definition of ‘quasiorthogonal families’ very close to ourtight frames, and a short discussion of the similarities betweena ‘quasiorthogonal family’ and an orthonormal basis. For themany miraculous properties of this orthonormal basis, see Ref.14 [Meyer [42]].”

As already mentioned, we will focus mainly on the relation of the φ-transformto the CRF and we want to keep this presentation not too extensive, hence leavingout many wonderful connections to other topics. For more details on the interplaybetween wavelets, frames, sampling, signal analysis, CRF, and much more, as well ashistorical accounts and relevant contributions, we refer to the delightful introductionby J. Benedetto to the compendium of articles in [33].

1. Continuous Calderon’s Reproducing Formula

The formula in its “modern” form can be briefly stated as

(1.1) u =

ˆ ∞0

u ∗ φt ∗ φtdt

t,

for an appropriate function φ and with convergence also understood in an appro-priate sense. Like several other of his profound contributions, the origins of thereproducing identity of Calderon are traced back to his 1964 paper [4] on complexinterpolation. The abstractness and the generality of the presentation do not makethis article the easiest of reads even for the experts. In his review of this work, J.Peetre says [46]:

“The presentation is not very clear, mainly owing, in the re-viewer’s opinion, to the unfortunate subdivision into two parts,the first giving the definitions and main results, the second givingthe detailed proofs; the reader has to spend quite a lot of time justsearching for the relevant passage for the proof of each particularstatement.”,

whereas C. Fefferman and E. Stein [19] eloquently summarize this masterpiece ofCalderon as follows:

“This important and lengthy paper of Calderon contains signif-icant conceptual insights of broad interest, but at the same timerequires a number of ingenious and tricky technical devices forexecutions.”

Though highly appreciated as one of the foundation rocks on which moderninterpolation theory was built upon, the article indeed contains many other results,


including the reproducing formula. Frazier, Jawerth, and Weiss, point in [26] to[4, §34] as the birthplace of Calderon’s identity. And it is there, in §14 in fact, ahidden treasure waiting to be discovered by the inquisitive eye of the reader.

At first, the formula in its great generality in [4] is not so easy to identify.Nonetheless, Calderon’s idea, reduced to a particular case, is as follows. For agiven function ϕ on Rn, and t > 0, let us write as usual ϕt(x) = t−nϕ(t−1x).τyϕ(x) = ϕ(x− y) denotes the translation operator, and ∗ denotes the operation ofconvolution of two functions. Letting u ∈ L2(R), and ϕ ∈ C∞(R) be a sphericallysymmetric function having vanishing moments up to a prescribed order, Calderonfirst introduces (in [4, §14]) the L2(R)-valued function of t

F (t) = Tu =

ˆR

(τyu)t−1ϕ(t−1y) dy,

where the integral is to be interpreted as Riemann vector valued. In other words,F (t)(u) = Tu(t) = u ∗ ϕt. Letting now ψ1, ψ2 ∈ C∞, spherically symmetric andwith compact support, he further defines the operator

S(F, u) = S(Tu, u) = u ∗ ψ2 + Cu,where

Cu =

ˆ ∞1

ˆRτyF (t−1)ψ1(ty)dy dt.

Note now that, by selecting ψ1 = ϕ, and further writing φt = ϕt−1 we have in fact

Cu =

ˆ ∞1

u ∗ φt ∗ φtdt

t.

Calderon claims and later proves (in §34) that it is possible to select ψ1 and ψ2 torecover u through the operator S: u = S(Tu, u). It follows that

u = u ∗ ψ2 +

ˆ ∞1

u ∗ φt ∗ φtdt

t,

which bears a strong resemblance to what is nowadays commonly referred to asCalderon’s reproducing formula.

The formula, even in a more modern form, appears also to have been known toCalderon and others around him. But surprisingly, the next published version of amore detailed CRF that we are aware of did not appear until 10 years later in anarticle by N. J. H. Heideman in 1974, [32]. Therein the author writes in page 68:

“We first prove a theorem of A.P. Calderon in [3] [here Calderon[4]] and a seminar at the University of Chicago...”

The theorem he is referring to translates into the following.

Theorem 1.1. Let µ be an L1(Rn) function so that its Fourier transform µ(ξ) =´Rn µ(x)e−ix·ξ dx verifies that, as a function of t, µ(tξ) is not identically zero forξ 6= 0. Then, there exist a function φ in the Schwartz space S whose Fouriertransform is compactly supported outside the origin and another function ψ ∈ S sothat

(1.2) u =

ˆ ∞0

u ∗ µt ∗ φtdt

t,

and

(1.3) u = u ∗ ψ +

ˆ 1

0

u ∗ µt ∗ φtdt

t,


for each tempered distribution u ∈ S ′.

Actually Heideman stated the theorem more generally, the dilations by 1/treplaced by a certain one parameter group of operators Tt and µ a Borel measurewith an appropriate Tauberian condition replacing the non-vanishing condition inthe statement above. Heideman implicitly uses in his proof the existence of theimproper integral in (1.2) as a distribution, allowing him to freely interchange theintegration with the action of the Fourier transform.

That this was the first explicit version in print of the formula in a form moresimilar to how it is used today seems to be corroborated by the comments in the1981 article by S. Janson and M. Taibleson [37]. In fact, the authors state on page29:

“Il teorema di rappresentazione e stato formulato per la primavolta nel corso di un seminario alla University of Chicago tenutoda Alberto Calderon intorno al 1960 ed e stato usato (implicita-mente) da Calderon in [4]. In forma esplicita compare nell’articolodi N. Heideman [2] [here [32]] e in molti altri successive.”

Like those around Calderon in Chicago learning about his formula in seminars inthe 60’s, those around Washington University in the 80’s (such as the second namedauthor of this article) learned about Janson and Taibleson’s paper directly fromMitch. The authors in [37] made explicit the implicit weak convergence argumentin Heideman’s paper looking at the limit of truncated integralsˆ R

ε

u ∗ µt ∗ φtdt

t

in the sense of distributions. They rigorously showed, in particular, that given thefunction µ one can always construct the associated φ so thatˆ ∞

0

µ(tξ)φ(tξ)dt

t= 1

for all ξ 6= 0, and ψ ∈ S,

ψ(ξ) =

ˆ ∞1

µ(tξ)φ(tξ)dt

t

for ξ 6= 0 and ψ(ξ) ≡ 1 in a neighborhood of the origin, such that (1.3) holds foru ∈ S ′ and in the sense of distributions. However, as observed by M. Wilson in [63](where we also find a very useful survey of some of the history of CRF), Janson andTaibleson seem to be the first to state that the continuous homogeneous formula(1.2) cannot actually hold for an arbitrary distribution. The reason is simple: since

φ vanishes at the origin, convolution with φ is completely blind to distributionswhose Fourier transforms are supported at the origin. Such distributions are theinverse Fourier transforms of finite linear combinations of the delta function and itsderivatives at the origin, hence they are polynomials. Janson and Taibleson estab-lished then the validity of the homogeneous formula (1.2) for tempered distributionsbut with weak convergence modulo polynomials. An earlier study of this modulopolynomial convergence, but in a simpler discrete version of the CRF, is in the bookby Peetre [47] (we will come back to this convergence in the next section).

Presumably, some of the “several subsequent” works referred to by Janson andTaibleson in the above quote from their work implicitly include Calderon’s three


articles on parabolic Hp spaces, two in collaboration with A. Torchinsky, [7], [8]and [5]. These works were published in the period 1975-1977, hence before [37]. Inparticular, it is indicated in [8, pp.130-131] that the formula converges pointwise forsmooth functions u with compact support, and in [5, pp. 219-220] that it convergeswhen paired against test functions if u ∈ Hp.

The 80’s saw a proliferation of very important results in harmonic analysis thatwere proved taking advantage of the CRF. Some relevant appearances of the formulathat should be mentioned are in the works of Chang-Fefferman [9] (where the for-mula is stablished by formally exchanging Fourier transform and integration), andUchiyama [60] and Wilson [62], where the CRF plays a central role in constructingatomic decompositions. During this time, the formula started to be presented morefrequently as

u =

ˆ ∞0

u ∗ φt ∗ φtdt

t,

for an appropriately normalized function φ (see Theorem 1.2 below).Another crucial use of the formula in the same decade is in the celebrated proof

of the T1-theorem of David and Journe in [18]. In that work, the authors also provethe convergence in L1 of the truncated integrals in the CRF for functions in L1 withmean zero. In their own proof of the T1-theorem, Coifman and Meyer [12] made avery nice use of the so-called “Pt−Qt formula” (which goes back to their work withMcIntosh [11]) where one can recognize traces of Calderon’s own formula, exceptthis time used to decompose an operator instead of a function. In an application tostudy the minimality of the Besov space B0,1

1 , Meyer [44] proved the convergenceof the formula in such space norm.

In the mid 80’s the CRF was rediscovered within the mathematical physicscommunity; it is studied in the article of Grossman and Morlet [30] and otherrelated works which helped jump start much of the wavelets revolution. In fact, theCRF can be written as

(1.4) u =

ˆ ∞0

ˆRn〈u, φt,y〉φt,y(x)

dydt

tn+1,

where now φt,y(x) = t−n/2φ((x− y)/t) and the last expression is nowadays referredto as the continuous wavelet representation of u.

It turns out that the CRF converges in a very strong sense, namely in L2. Thishas been folklore in the subject based on a formal computation using the Fouriertransform, but we could not find a published explicit rigorous argument for thisconvergence until the book by Frazier, Jawerth and Weiss [26]. This work waspublished in 1991 but was based on lectures at a CBMS conference in Alabama in1989. Within the wavelet literature, the L2 convergence can be traced to the bookof I. Daubechies [16, pp. 24-30], where pointwise convergence for some continuousfunction is also presented. Though not commonly found in the literature, the CRFconverges almost everywhere for arbitrary functions in L2, as stated in the followingtheorem.

Theorem 1.2. Assume that φ ∈ S(Rn) is real valued, radial, with Fouriertransform compactly supported away from the origin, and such that

(1.5)

ˆ ∞0

φ(tξ)2 dt

t= 1


for all ξ 6= 0. Then, for all u ∈ L2(Rn), we have

(1.6) u =

ˆ ∞0

u ∗ φt ∗ φtdt

t,

where the convergence

u = limε→0+

R→+∞

uε,R = limε→0+

R→+∞

ˆ R

ε

u ∗ φt ∗ φtdt

t

holds both in the L2 and the pointwise almost everywhere sense. Moreover, theconvergence also holds for u ∈ Lp for 1 < p <∞.

Proof. As mentioned, the statement for L2 appeared in this form in [26, §1],from where we repeat the proof. It is clearly not a challenge to construct a functionin S, real valued, radial, and with Fourier transform compactly supported awayfrom the origin. For any such function, it is easy to see that the integral in therighthand side of (1.5) is (a real) constant for any ξ 6= 0. Hence, normalizing anysuch function one obtains (1.5).

Working formally, the identity (1.6) follows by simply comparing the Fourier

transforms of each side of (1.6). In particular, since φt(ξ) = φ(tξ), using the integralcondition on φ, the Fourier transform of the right hand side simplifies to u. Therigorous justification of this is a nice exercise in real analysis which uses the Fubini,Plancherel, and Lebesgue dominated convergence theorems.

First, the argument for u ∈ L2 ∩ L1 goes as follows. One observes that in thiscase u ∗ φt ∗ φt ∈ L1(Rn) for all t > 0; in fact,

‖u ∗ φt ∗ φt‖L1 ≤ ‖u‖L1‖φ‖2L1 .

A similar reasoning shows that, for all 0 < ε < R , we have

‖uε,R‖L1 ≤ ln(R/ε)‖u‖L1‖φ‖2L1 ,

thus uε,R ∈ L1(Rn). Now, using Fubini’s theorem we get

uε,R(ξ) = u(ξ)

ˆ R

ε

φ(tξ)2 dt

t.

Therefore, combining Plancherel and Lebesgue’s dominated convergence theoremsleads to

‖uε,R − u‖2L2 = (2π)−n‖uε,R − u‖2L2

= (2π)−nˆRn|u(ξ)|2

(1−ˆ R

ε

φ(tξ)2 dt

t

)2

dξ → 0

as ε→ 0 and R→∞.Next, consider the case of a general function u ∈ L2(Rn) and let uj ∈ L2 ∩ L1

be such that ‖uj − u‖L2 → 0 as j →∞. Then

‖uε,R − u‖L2 ≤‖uj − u‖L2 + ‖(uj)ε,R − uj‖L2(1.7)

+∣∣∣∣∣∣ˆ R

ε

(uj − u) ∗ φt ∗ φtdt

t

∣∣∣∣∣∣L2.

Let τ > 0 be arbitrary. As explained in the L2 ∩L1 case above, we immediately seenow that

‖(uj)ε,R − uj‖2L2 = (2π)−n‖(uj)ε,R − uj‖2L2


. ‖uj − u‖2L2 +

ˆRn|uj(ξ)|2

(1−ˆ R

ε

φ(tξ)2 dt

t

)2

dξ

. ‖uj − u‖2L2 + τ,

for ε sufficiently small and R sufficiently large, since the L2 norms of the functionsuj are uniformly bounded. Also, using Minkowksi’s inequality, we can bound thethird term on the right hand side of (1.7) byˆ R

ε

‖(uj − u) ∗ φt ∗ φt‖L2

dt

t

≤ˆ R

ε

‖uj − u‖L2‖φt‖2L1

dt

t. ln(R/ε)‖uj − u‖L2 .

All in all, we have obtained that for ε and 1/R sufficiently small depending on τbut fixed,

‖uε,R − u‖L2 . ‖uj − u‖L2(1 + ln(R/ε)) + τ.

Finally, for j large enough, we can then write

‖uε,R − u‖L2 . τ,

thus proving Calderon’s reproducing formula in this case as well.The pointwise almost everywhere convergence can be established in the follow-

ing way. Following [18, p. 376], we observe that if φ ∈ S is as above, then thereexists a function η ∈ S such that η(0) = 1 and

(1.8) − t ddt

(u ∗ ηt) = u ∗ φt ∗ φt.

We note that (1.8) can be viewed as an identity relating the families of operatorsQt(u) = u ∗ φt and Pt(u) = u ∗ ηt, namely Q2

t = −t ddtPt. To verify the conditionson η simply set

η(ξ) = 1−ˆ 1

0

φ(tξ)2 dt

t.

We note immediately that η(0) = 1. Then, since φ ∈ S, for all multi-indices α andall N ∈ N, we can write for ξ 6= 0

|ξ|N |∂αη(ξ)| ≤ˆ ∞

1

|ξ|N (∂α(φ)2)(tξ)t|α|−1 dt

. |ξ|−|α|−1 → 0 as |ξ| → ∞;

thus proving that η ∈ S and hence η ∈ S.

Moreover, using the fact that φ is radial we get that η(tξ) =´∞tφ(sξ)2 ds

s ,

which implies ddt η(tξ) = − 1

t (φt)2(ξ). The last equality gives the identity (1.8).

If u ∈ L2(Rn), we simply need to recall that for such η ∈ S(Rn) with η(0) = 1,Pt forms an approximation to the identity. That is, we have the almost everywhereconvergence of u ∗ ηt to u as t→ 0+. Note also that

(1.9) |u ∗ ηt(x)| ≤ ‖u‖L2‖ηt‖L2 . t−n/2‖u‖L2 ,

since ηt is L1-normalized, and hence u ∗ ηt → 0 as t → ∞. Therefore, using now(1.8), we see that

limε→0+

R→+∞

ˆ R

ε

u ∗ φt ∗ φt(x)dt

t= lim

ε→0+

R→+∞

−ˆ R

ε

d

dt(u ∗ ηt)(x)dt = u(x)


for a.e x.The method used for the pointwise convergence and again a simple limiting

argument give also the convergence in Lp for 1 < p < ∞. Indeed, adapting thearguments from [58], we first note that for u ∈ L2 ∩Lp, u ∗ ηt converges to u in Lp

as t→ 0+ and|u ∗ ηt(x)| .Mu(x),

where M is the Hardy-Littlewood maximal function. By dominated convergence,u ∗ ηt converges to 0 in Lp as t → ∞ and hence uε,R converges to u. Now, for ageneral u ∈ Lp, we can approximate it by functions uj ∈ L2 ∩ Lp. Observe alsothat, by Minkowksi’s inequality, uε,R makes sense in Lp and

‖uε,R − (uj)ε,R‖Lp .ε,R ‖M(u− uj)‖Lp .ε,R ‖u− uj‖Lp ,while

‖(u ∗ ηε − u ∗ ηR)− (uj ∗ ηε − uj ∗ ηR)‖Lp . ‖M(u− uj)‖Lp . ‖u− uj‖Lp .It follows that we also have, for arbitrary u ∈ Lp and any 0 < ε < R <∞,

uε,R = u ∗ ηε − u ∗ ηR.Finally, notice that in place of (1.9) we have

|u ∗ ηR(x)| ≤ ‖u‖Lp‖ηR‖Lp′ . R−n/p‖u‖Lp ,

where 1/p + 1/p′ = 1, and so the same arguments used in the case p = 2 give theconvergence for general p. �

We want to remark that the conditions we have imposed on the function φ areto some extent excessive and they could be relaxed quite a bit (though it is easy

to see that φ must be zero at the origin and infinity). Also, as done in [26], it ispossible and sometimes convenient to use φ with compact support instead of having

φ with such property. A lot has been and continues to be written about conditionson φ for the formula to hold and also about the converge of the CRF in manyother senses and various function spaces. For example, S. Saeki [50] proved thatfor u ∈ Lp(Rn), 1 < p <∞, the integralsˆ ∞

y

u ∗ φt ∗ φt(x)dt

t

converge nontangentially to u(x) at (x, 0) ∈ R(n+1)+, if x is a Lebesgue point of u.For 1 < p <∞, M. Wilson [63] proved convergence in the norm of Lp(w), where wis an Ap weight, allowing also more general truncations of the improper integral inthe formula. K. Li and W. Sun [41] established then pointwise almost everywhereconvergence in Lp(w). Wilson also obtained in [64] convergence in H1 and weak∗ inBMO (as the dual of H1). More recently, the second named author of this articleand E. Ward proved in the already cited article [58] the convergence of a naturalanalog of the CRF in mixed Lebesgue spaces LpLq(Rn × R). There is also a richtheory with versions of the CRF in more abstract group theoretic setting. One ofthe first works in this direction is the article by H. Feichtinger and K. Grochenig[20] involving square integrable representations. More references to the literaturedealing with wavelets generated through the action of other groups are given in thesurvey by E. Wilson and G. Weiss [61].

One may speculate that the continuous CRF converges in norm as a limit ofthe functions uε,R in any (homogeneous) space which admits a Littlewood-Paley


decomposition; in particular, in all the homogeneous Besov and Triebel-Lizorkinspaces Bα,qp and Fα,qp .2 However, we have not been able to locate an explicit refer-ence for this fact in the literature. Such a general result does exists for the discreteformula in the next section, as proved by Frazier and Jawerth [24, 25] (see also[44] and the references therein for discrete orthonormal wavelets). Nonetheless, inmany applications, the following very strong convergence in the appropriate com-mon dense subspace for most Bα,qp and Fα,qp spaces suffices to justify many uses ofthe CRF.

Proposition 1.3. Let

S0 = {u ∈ S :

û(x)xα dx = 0 for all multi-indices α}.

Then, if u ∈ S0, limε→0 uε,1/ε = u in the topology of S.

Proof. Note that since φ has compact support away from the origin, for eachε > 0 we have

u(ξ)

(1−ˆ 1/ε

ε

φ(tξ)2 dt

t

)= 0

unless |ξ| . ε or |ξ| & 1/ε. Also, for all α,M , we have

|∂αu(ξ)| .α,M (1 + |ξ|)−M

because u ∈ S. In addition, for all multi-indices α, β,

|∂αu(ξ)| .α,β |ξ||β|

because ∂αu(0) = 0 for all α. With these estimates, it is easy to see that

(1 + |ξ|)M |∂α(u(ξ)− uε,1/ε(ξ))| .α,M ε,

which proves the required convergence. �

Naturally, a simpler version of the reproducing formula also holds: supposingthat ψ is such that ˆ ∞

0

ψ(tξ)dt

t= 1,

then

u =

ˆ ∞0

u ∗ ψtdt

t.

But the advantage of having the double convolution φt ∗ φt in Calderon’s formulais that both convolution factors u ∗ φt and φt are with compact support on theFourier side, while in the simpler version above only one of the factors, ψt, has thisproperty. As it will become clear, having the double convolution proves to be acrucial ingredient in the discrete version of the formula by Frazier and Jawerth.

2We will not need the definition of these spaces for this survey. The norm of some of these

spaces, however, needs to be interpreted modulo polynomials of appropriate degrees, which isconsistent with the convergence modulo polynomials of the CRF for arbitrary distributions.


2. Discrete Calderon’s Reproducing Formula

A full discretization of Calderon’s identity through appropriate Riemann sumsmay be used to motivate the Frazier-Jawerth φ-transform or discrete wavelet de-compositions. We first work at a formal level.

Suppose that φ is as in the statement of Theorem 1.2. For ν ∈ Z, k ∈ Zn,let Q = {Qνk}ν,k be the collection of dyadic cubes, where Qνk has side lengthl(Q) = 2−ν and lower left corner xQ = 2−νk.

By making the change of variable t = 2−µ in (1.6), we see that, modulo amultiplicative constant,

u(x) =∑ν∈Z

ˆ ν+1

ν

ˆRnu ∗ φ2−µ(y)φ2−µ(x− y)dy dµ.

The integrand in the last expression depends smoothly on µ so we can roughlyapproximate the integration in µ by the value of the integrand at say µ = ν. Hence,braking the integral in Rn into dyadic cubes of side length 2−ν , we get

u(x) ≈∑ν∈Z

∑k∈Zn

ˆQνk

u ∗ φ2−ν (y)φ2−ν (x− y)dy

Note now that the Fourier transforms of both functions u ∗ φ2−ν and φ2−ν aresupported around frequency of the order 2ν , and so the functions themselves cannotoscillate too much on a cube of side length 2−ν . We can then attempt to write theRiemann sum approximation

u(x) ≈∑ν∈Z

∑k∈Zn

u ∗ φ2−ν (2−νk)φ2−ν (x− 2−νk)2−νn.

Of course, we have ignored issues about convergence and have not precisely quan-tified any of the approximation, but this simple reasoning clearly suggests the pos-sibility of a discrete φ-transform or wavelet expansion

(2.1) u =∑ν,k

〈u, φνk〉φνk,

where the functions3

(2.2) φνk(x) = 2νn/2φ(2νx− k) = 2−νn/2φ2−ν (x− 2−νk)

are translated and dilated version of single function φ.A first very natural and rigorous discretization of the CRF, which appeared

many times over the years and sometimes independently of Calderon’s identity, hasbeen extensively used in the context of function spaces on Rn and their Littlewood-Paley characterizations; see the books by Stein [51], Peetre [46], Triebel [59], andthe many references therein for historical accounts. This discretization is sometimes

stated in the following form. Let φ ∈ S be radial, real valued and such that φ is

supported in the annulus π/4 < |ξ| < π. Assume also that |φ(ξ)| is bounded awayfrom zero on a smaller annulus π/4 + ε ≤ |ξ| ≤ π − ε, and

(2.3)∑ν∈Z

φ(2−νξ)2 = 1, ξ 6= 0.

3For simplicity, we are assuming φ to be radial.


Such a function φ is sometimes referred to as “admissible”. The existence of afunction ψ which satisfies all the above properties except possibly (2.3) is of noconcern. Noticing that any such function satisfies for some 0 < c < C <∞

c <∑ν∈Z

ψ(2−νξ)2 < C,

one can define φ as desired by letting

φ(ξ) =ψ(ξ)(∑

ν∈Z ψ(2−νξ)2)1/2

.4

Then again, in various appropriate senses,

(2.4) u =∑ν∈Z

u ∗ φ2−ν ∗ φ2−ν .

For example, we have the following result.

Proposition 2.1. If u ∈ L2, then the convergence of (2.4) holds in L2 sense,while, if u ∈ S0, then the convergence is in S.

In fact, the proof of the convergence in L2 is rather trivial while the one forfunctions in S0 can be obtained in a similar fashion to Proposition 1.3. Also,although the righthand side of the above formula makes sense for an arbitrarydistribution u ∈ S ′, there are issues with the convergence since, yet again, the

functions φ2−ν are completely blind to distributions supported at the frequencypoint ξ = 0. As mentioned earlier, Peetre [46, pp.52-54] seems to have been thefirst to rigorously address this issue. For the benefit of the reader, and since thereference [46] is not always readily available, we include here a sketch of Peetre’sresult.

Theorem 2.2. Let φ be an admissible function satisfying (2.3). Then, for eachu ∈ S ′, there exist a sequence of polynomials {PN}N of bounded degree (dependingon u) and another polynomial P such that

u = limN→∞

(N∑ν=0

u ∗ φ2−ν ∗ φ2−ν

)+ limN→∞

( −1∑ν=−N

u ∗ φ2−ν ∗ φ2−ν − PN

)+ P,

where the limits are taken in S ′.

Proof. Similar arguments to those in Proposition 1.3 show that for any g ∈ S∞∑ν=0

φ(2−νξ)2g(ξ)

converges in S, so if 〈·, ·〉 denotes the pairing of distributions with test functions,we have

limN→∞

⟨N∑ν=0

(φ2−ν )2u, g

⟩=

⟨u,

∞∑ν=0

(φ2−ν )2g

⟩.

It follows that

u∞ ≡ limN→∞

(N∑ν=0

u ∗ φ2−ν ∗ φ2−ν

)4Note that the conditions on the support of ψ make φ a smooth function!


exists in S ′.Next, since u ∈ S ′, there exists a continuous seminorm qL,M on S,

qL,M (g) = supξ∈Rn,|α|<M

(1 + |ξ|)L|∂αg(ξ)|,

such that

|〈u, g〉| . qL,M (g)

for all g ∈ S. Since |ξ| ≈ 2ν on the support of φ2−ν (ξ), for any ν < 0 and |γ| = Mwe can write

|〈u, ξγ(φ2−ν )2g〉| . qL,M (ξγ(φ2−ν )2g) . 2ν .

It follows that ∂γ(∑−1

−N u ∗ φ2−ν ∗ φ2−ν

)converges in S ′ for all |γ| = M . But

this is equivalent to the existence of a sequence of polynomials {PN} of degrees no

larger than M − 1, such that(∑−1

ν=−N u ∗ φ2−ν ∗ φ2−ν

)− PN converges in S ′ to

some distribution that we denote by u−∞.Finally, notice that if g ∈ S is supported away from the origin, we have in

particular g ∈ S0 and so

〈u∞ + u−∞, g〉 =

⟨u,

∞∑ν=−∞

(φ2−ν )2g

⟩= 〈u, g〉 ,

where we have used the fact that the formula (2.4) converges in S for g ∈ S0. Itfollows that u − (u∞ + u−∞) is a distribution supported at the origin and henceu− (u∞ + u−∞) is some polynomial P . �

As with the continuous CRF, the representation (2.4) can be obtained and con-vergence can be established in various function spaces under less stringent conditionson φ. In particular, one can use two different functions φ and ψ (in applications, oneof them is usually constructed after the other one is given) with the same propertiesof an admissible function except that they are now jointly normalized to verify

(2.5)∑ν∈Z

φ(2−νξ)ψ(2−νξ) = 1, ξ 6= 0.

Another very convenient version could be obtained with a function φ ∈ S, such

that φ is with the same support and satisfying a non-vanishing Tauberian conditionas before; and another function θ ∈ S, which is itself supported on the unit ballcentered at the origin and has vanishing moments

´xγθ(x) dx = 0 for all |γ| up

to a prescribed order M . Moreover, this can be done so that θ still satisfies theTauberian condition of φ (although obviously not the compact support one), and

(2.6)∑ν∈Z

φ(2−νξ)θ(2−νξ) = 1, ξ 6= 0;

see [24, pp.783-784].

Starting from the first discretization of the CRF, Frazier and Jawerth arrived atthe formula (2.1) through a very clever use of a generalized sampling theorem. Theclassical Shannon sampling theorem (yet another reproducing formula rediscovered


many times in both theoretical and applied mathematics) states that if f ∈ L2(Rn)

and, say, supp f ⊂ {ξ ∈ Rn : |ξ| < π}, then

(2.7) f(x) =∑k∈Zn

f(k)

n∏j=1

sin(π(xj − kj))π(xj − kj)

.

Note that, since f has compact support, we may assume f is continuous; in fact, fagrees almost everywhere with a smooth function of exponential type. The sampling

formula is easy to establish by expanding f in a Fourier series and then using Fourierinversion. It is also easy to see that (2.7) converges in L2 (as well as uniformly). Apoint usually made about (2.7) is its slow rate of convergence given the slow decayat infinity of the function

sinc (x) ≡n∏j=1

sin(πxj)

πxj.

Notice, however, that (2.7) can also be seen through the equality f = f ∗ sinc , as

f ∗ sinc (x) =∑k∈Zn

f(k) sinc (x− k).

It is a well-known fact that replacing sinc by a function g ∈ S with supp g ⊂ {ξ ∈Rn : |ξ| < π} and g(ξ) ≡ 1 on the support of f , one can obtain a much fasterconvergent formula

f(x) = (f ∗ g)(x) :=∑k∈Zn

f(k) g(x− k).

Moreover, if one is only interested in discretizing the (continuous version of the)convolution of f and g and not recovering f , something much more general can beachieved. Frazier and Jawerth proved in [24] the following result.

Theorem 2.3. Let f ∈ S ′and g ∈ S be so that supp f , supp g ⊂ {ξ ∈ Rn : |ξ| <π}. Then

(2.8) (f ∗ g)(x) =∑k∈Zn

f(k) g(x− k)

with convergence in S ′. Moreover, if f is actually in S, then the convergence is inS.

Proof. The formula is easily established for sufficiently nice functions as in theclassical sampling theorem, that is, by expanding g in a Fourier series on [−π, π]n,writing

f g(ξ) =∑k∈Zn

(2π)−n

(ˆ[−π,π]n

g(z)eikz dz

)eikξ f(ξ)

and then using Fourier inversion. The details to obtain the convergence in S ′ via astandard regularization process once the formula is established for smooth functionscan be found in [24] and [26], so we will not repeat them here.

We do indicate the computations for the stronger convergence when f ∈ S sincewe will appeal to it later. This is rather simple, since for any multi-index α we candifferentiate term by term

∂α(f ∗ g)(x) = (f ∗ ∂αg)(x) =∑k∈Zn

f(k) (∂αg)(x− k),


and for any M > 0 we can write∣∣∣∣∣∣(1 + |x|2)M∑|k|>N

f(k) (∂αg)(x− k)

∣∣∣∣∣∣ .α,M∑|k|>N

|f(k)| (1 + |x|2)M

(1 + |x− k|2)M

.α,M∑|k|>N

|f(k)| (1 + |k|2)M → 0 as N →∞.

In the estimate above, we have used Peetre’s inequality and the fast decay of thesamples |f(k)| for f ∈ S. �

As with the classical sampling theorem, there is of course a rescaled versionof (2.8), with samples taken according to the Nyquist rate at a distance inverselyproportional to the radius of the spectrum of the functions f and g. More precisely,

if supp f , supp g ⊂ {ξ ∈ Rn : |ξ| < 2νπ}, then

(2.9) (f ∗ g)(x) = 2−νn∑k∈Zn

f(2−νk) g(x− 2−νk).

Using now the formula (2.4), and then for each ν the formula (2.9) with f =u ∗ φ2−ν and g = φ2−ν , Frazier and Jawerth [24] arrived in a truly “painless” wayto a simple yet deep and amazing discrete CRF: their φ-transform.

Theorem 2.4. If φ is admissible, then

(2.10) u(·) =∑ν∈Z

2−νn∑k∈Zn

(u ∗ φ2−ν )(2−νk)φ2−ν (· − 2−νk),

where for u ∈ L2 the convergence is in L2, for u ∈ S0 the convergence is in S, and foru ∈ S ′ the convergence is in S ′ modulo polynomials (in the sense of Theorem 2.2).

The just stated theorem above gives us, at last, a rigorous proof of the formula

(2.11) u =∑ν∈Z

∑k∈Zn

〈u, φνk〉φνk,

with φνk as in (2.2).It is remarkable that the Frazier-Jawerth approach produces so “easily” the

above representation, which looks precisely like an expansion in an orthonormalbasis of wavelets. By contrast, the construction of orthonormal wavelets with agenerating function in S is equally beautiful but a rather intricate task; it was firstdone by Lemarie and Meyer [40] and it is quite remarkable that such a constructionis possible. For an earlier construction of spline wavelets, see the work of Stromberg[53], while for further compactly supported wavelets, see Daubechies [15].

The method of Frazier and Jawerth, however, does not produce an orthonor-mal basis5 since not all the functions φνk are orthogonal to each other. Moreover,it does not even produce a basis but only a frame in L2 when φ ∈ S. Nonethe-less, as mentioned earlier, this is of no technical consequence in many uses of therepresentation.

5It is actually possible to construct smooth orthonormal wavelets (Shannon wavelets) from

the sampling theorem approach if one starts with φ a characteristic function of an appropriate

annulus. Moreover such approach can also be carried out in other groups beyond Rn; see [22].However this gives wavelets that, for example in dimension one, only decay like |x|−1 at infinity.


Once again, the method allows for a lot of flexibility and one can use twodifferent functions φ and ψ satisfying (2.5) instead of (2.3). This leads to theexpansion

(2.12) u(·) =∑ν∈Z

2−νn∑k∈Zn

(u ∗ φ2−ν )(2−νk)ψ2−ν (· − 2−νk),

or

(2.13) u =∑ν∈Z

∑k∈Zn

〈u, φνk〉ψνk,

with ψνk defined in same way as φνk. Moreover, there is also an inhomogeneousversion (as with discrete wavelets) in which all the low frequency supported functionsφ2−ν and ψ2−ν are replaced by two single functions Φ and Ψ, so that

(2.14) Φ(ξ)Ψ(ξ) +∑ν≥1

φ(2−νξ)ψ(2−νξ) = 1

for all ξ ∈ Rn. This in turn leads to

(2.15) u =∑k∈Zn

〈u,Φ0k〉Ψ0k +∑ν≥1

∑k∈Zn

〈u, φνk〉ψνk,

where Φ0k = Φ(· − k) and, similarly, Ψ0k = Ψ(· − k); see [25]. An advantage ofthis formula is that it now “honestly” converges in S and S ′ and it is best suited tostudy inhomogeneous Besov and Triebel-Lizorkin spaces.

3. Characterization of function spaces, atoms, and molecules

The fact that the Fourier transform is no longer an isometry in Lp for p 6= 2is a great example of how unfortunate constraints could sometimes be turned intofantastic opportunities (at least in the right hands of virtuous mathematicians).Trying to overcome the impossibility of measuring Lp properties of functions di-rectly from the Fourier transform promoted the creation of powerful mathematicaltools, which have had and continue to have impressive efficacy in harmonic analysis,operator theory and partial differential equations. Littlewood-Paley theory breaksthe Fourier transform of a function into mildly interfering frequency bands thatallow to recover information related to function spaces beyond the Hilbert spacecontext. Owing its origin to Littlewood and Paley in the 30’s, the theory took itsfull force in Rn with Stein’s introduction of his g-function in the late 50’s. ChapterIV of his book [51] has a marvelous and not only authoritative but also still oneof the most didactic presentations of the topic. We quote from Stein’s presentation[51, p. 81] of the g-function, of which he says that

“aside from its applications, it illustrates the principle that oftenthe most fruitful way of characterizing various analytic situations(such as finiteness of Lp norms, existence of limits almost every-where, etc.) is in terms of appropriate quadratic expressions.”

Calderon’s reproducing formula is strongly connected to Littlewood-Paley the-ory. This is nicely illustrated in [26]. Let P denote the Poisson kernel for the upperhalf-space,

P (x) = cn(1 + |x|2)−(n+1)/2,


U(x, t) = (Pt ∗ u)(x), and consider a particular version of the g-function,

g(u)(x) =

(ˆ ∞0

|t∂tU(x, t)|2 dtt

)1/2

=

(ˆ ∞0

|φt ∗ u(x)|2 dtt

)1/2

,

where φ(ξ) = −|ξ|P (ξ) = −|ξ|e−|ξ|. Littlewood-Paley theory says that, for 1 < p <∞, we have

‖u‖Lp ≈ ‖g(u)‖Lp ,while by the CRF,

u(x) =

ˆ ∞0

φt ∗ φt ∗ u(x)dt

t.

That is, we can viewu→ φt ∗ u

as a bounded map from Lp(Rn) into Lp(Rn, L2((0,∞), dtdxt )) which is inverted bythe CRF. Likewise, there is a discrete version of this fact: if φ is an admissiblefunction, then

‖u‖Lp ≈

∥∥∥∥∥∥(∑ν∈Z|φ2−ν ∗ u|2

)1/2∥∥∥∥∥∥Lp

,

while other quadratic (or q-power) functions characterize numerous function spaces.It is perhaps not surprising then that the formulas (2.11), (2.13) or (2.15) could berelated through Littlewood-Paley theory to the characterization of such spaces.It is quite incredible, however, that it is just the size of the coefficients in thoseexpansions what gives the characterization.

Indeed, Frazier and Jawerth showed in [24] and [25] that any space X in theBesov or Triebel-Lizorkin scales can be characterized by a corresponding space ofcoefficients S(X)6. These discrete spaces are defined for sequences {sQ} indexed

by the dyadic cubes and with norms in terms of {|Q|c(X)|sQ|}, where the powerc(X) depends on the function space X. For some L2-based spaces X, the spacesS(X) are simply weighted l2 spaces. For example, for X = L2, the associated space

is S(L2) = l2, and for L2α (= Bα,22 = Fα,22 ), the homogeneous Sobolev spaces of

functions with their “derivatives of order α” in L2, one has the weighted l2 norm

‖{sQνk}‖S(L2α) =

∑ν,k

(|Qνk|−α/n|sQνk |

)2

1/2

.

For most spaces X based on Lp with p 6= 2, S(X) is more complicated than aweighted lp space, but it is still a Banach or quasi-Banach lattice with norm onlydepending on |sQ|; see [25].

In proving the characterizations

‖u‖X ≈ ‖{|〈u, φνk〉|}‖S(X),

Frazier and Jawerth [24, 25] developed also smooth atomic and molecular decom-position of the function spaces considered. There is a rich history in harmonicanalysis related to atomic decompositions going back to the pioneering work ofCoifman [10], but the atomic decomposition in [24, 25] seems most influenced byUchiyama’s one for BMO, [60]. Frazier and Jaweth also mention “an alternate

6Frazier and Jawerth denote the sequence spaces associated with Bα,qp , respectively Fα,qp , by

bα,qp , respectively fα,qp .


direction” to decompositions (though some how similar spirit) in the articles byCoifman and Rochberg [13] and Ricci and Taibleson [48].

A very interesting fact is the universality of the construction in [24, 25] ofsmooth atomic decompositions for Lp, Hardy, and Sobolev spaces, and more gener-ally the whole scales of Besov or Triebel-Lizorkin spaces.

Starting with the decomposition

u =∑ν∈Z

θ2−ν ∗ φ2−ν ∗ u,

where θ is selected as in (2.5), having now compact support and a prescribed numberof vanishing moments (which could be arbitrary large but finite), one can rewrite afunction u ∈ X as

(3.1) u =∑ν∈Z

∑k∈Zn

sQaQ.

Here Q = Qνk is a dyadic cube in Rn,

sQ = sQνk = |Q|1/2| supy∈Q|φ2−ν ∗ u(y)|

and

aQ = aQνk =1

sQνk

ˆQνk

θ2−ν (x− y)φ2−ν ∗ u(y) dy.

The family {aQ} of smooth atoms, as defined in [24, 25], satisfies some desirableproperties: the functions aQ are C∞, compactly supported on a dilate of Q, andhave as many vanishing moments as θ. They are also normalized7 as the functionsφνk and so

|∂γaQ(x)| .γ |Q|−1/2−|γ|/n.

Moreover, if aQ have enough vanishing moments depending on the space X, it canbe proved that

‖{sQ}‖S(X) . ‖u‖X .Likewise, Frazier and Jawerth showed that whenever

u =∑ν∈Z

∑k∈Zn

sQφQ,

with φQ = φν,k as before, then also ‖{sQ}‖S(X) . ‖u‖X .The converse inequality needed for the characterization of the function spaces

is actually established for a more general class of building blocks: a family {mQ}of smooth molecules. The molecules, for a given space X, are a rougher and lessoscillating version of the wavelets φνk. They also have some limited decay awayfrom the cube Q = Qνk and are normalized like the functions φνk. More precisely,they satisfy

(3.2) |∂γmQ(x)| . |Q|−1/2−|γ|/n

(1 +|x−xQ|l(Q) )N

=2ν(n/2+|γ|)

(1 + |2νx− k|)N

for all |γ| ≤ M and with N and M dependending on the function space X consid-ered. In addition, the molecules have a number of vanishing moments also dependingon X. The family of atoms and the family of functions φνk are hence molecules as

7We note that actually the normalization in [24] is different from the one in [25], but consis-tent with appropriate modifications in the definitions of the spaces S(X) in those works.


well. It is important to note that, although the mQ satisfy estimates which behaveas if the molecules were the translations and dilations of a single function m, theydo not need to be so. Frazier and Jawerth showed that whenever

f =∑ν∈Z

∑k∈Zn

sQmQ

(most generally in S ′ modulo polynomials), then f ∈ X and

‖f‖X . ‖{sQ}‖S(X),

completing the characterization of X in terms of the coefficients in the expansion(2.11). There are also inhomogeneous versions of all of the above characterizations.

Incidentally, the characterization of function spaces in terms of the coefficientsproves also the convergence of the representation formula in the norm of such spaces.Regarding pointwise convergence, various results can be obtained. We point to thework on wavelets by Kelly, Kon, and Raphael [38, 39] where rates of convergenceare also given, and to the article by Tao [55], where various summation methodsfor pointwise convergence are considered.

Molecular characterizations of functions spaces have a long history as well; see,for example, the works of Taibleson and Weiss [54] and Coifman and Weiss [14]. Itturns out that, as with other types of molecules in the literature, singular integralsand other important multiplier operators map smooth atoms into molecules (butnot atoms into atoms). We will show a very simple application of this in the nextsection.

What we have just briefly summarized about the characterization of functionsspaces takes actually a tremendous arsenal of harmonic analysis tools to be rig-orously achieved, as well as a wise use of them [24, 25]. Such tools include, inparticular, the Plancherel-Polya inequality for functions of exponential type, thePetree maximal function, and the Fefferman-Stein vector valued maximal function,which should warn the reader of the deep technical aspects of the proofs we haveomitted. Another crucial notion in this development is that of almost diagonaloperator which we will now succinctly describe.

4. Almost orthogonality and almost diagonal operators

Let us recall in an informal way the philosophy behind the diagonalization ofoperators by a family of generating functions (essentially, linear algebra!). Let T besome linear operator acting on one of the spaces X. Starting with the φ-transformfor u ∈ X,

u =∑Q

〈u, φQ〉φQ,

and applying T (while ignoring for the moment convergence issues) one gets

T (u) =∑Q

〈u, φQ〉T (φQ).

Expanding now T (φQ), we obtain

T (u) =∑Q

〈u, φQ〉∑P

〈T (φQ), φP 〉φP =∑P

∑Q

〈T (φQ), φP 〉〈u, φQ〉φP .


It follows from the characterization of X in terms of the discrete space of coefficientsS(X) that, to study the boundedness properties of T on X, it would be enough tostudy the boundedness properties of an infinite matrix

AT = {AT (P,Q)}P,Q = {〈T (φQ), φP 〉}P,Qindexed by the dyadic cubes and acting on S(X).

Ideally, if the family {φQ} are a basis of eigenvectors for T , then AT is just adiagonal matrix which is trivial to study. However, this requires constructing a gen-erating family of functions for each operator T to be studied; something extremelyunlikely to be done given all the constrains the functions φQ must satisfy to ex-pand the space X. The notion of almost diagonalitzation consists then in relaxingthe perfect diagonalization of each operator by looking instead at whole familiesof operators which come close to be diagonalized by the φ′Qs. That is, instead ofrequiring

T (φQ) = cQφQ,

one is content with

T (φQ) ≈ φQin some appropriate sense.

Still wishfully thinking along these lines, one would then have

〈T (φQ), φP 〉 ≈ 〈φQ, φP 〉,and for the matrix of AT ,

(4.1) AT (P,Q) → 0

as the entry (P,Q) “moves away” from the diagonal of the matrix. Moreover, recallthat the spaces S(X) are defined only in terms of the size of the coefficients |sQ|,so it may be possible to study the positive matrix |AT | instead of AT . Studyingpositive matrices is often a lot easier; in particular, there is the well-known criterionto study them given by Schur’s test.

Frazier and Jawerth found a very precise quantification of (4.1), which impliesthe boundedness of a matrix on S(X) for different spaces X. For example, ifQ = Qνk and P = Pµl, then a matrix A = {A(P,Q)}P,Q satisfying the almostdiagonal condition

(4.2) |A(P,Q)| . 2−(µ−ν)(ε+n/2)

(1 + 2ν |2−νk − 2−µl|)n+ε

for some ε > 0 and ν ≤ µ, and symmetric estimates if ν < µ, is bounded on S(Lp),1 < p <∞. Notice that

1 +diam(Q ∪ P )

l(Q)≈ 1 +

|2−νk −2−µl|2−ν

so we can also rewrite (4.2) as

|A(P,Q)| .(l(P )

l(Q)

)ε+n/21

(1 + l(Q)−1diam(Q ∪ P ))n+ε.

We clearly see in this particular case that the entries of the matrix are small if theEuclidean distance between the cubes is large or if their sizes are very different.Moreover, Frazier and Jawerth showed that such matrices are essentially those ofoperators which map atoms into molecules or molecules into molecules. Instead of


presenting the technical details of such results, we look at a very intuitive reasoningof why it may be possible to obtain something like (4.2).

Assume that T (φQ) = mQ, where mQ is a molecule, and consider the matrix

AT = {〈T (φQ), φP 〉}P,Q = {〈mQ, φP 〉}P,Q.

We want to see that (4.1) holds true. Again, let Q = Qνk and P = Pµl. First, sincethe functions mQ and φP are respectively localized around Q and P and decay awayfrom them (cf. (3.2)), each function “is very small where the other one is not” so

〈mQ, φP 〉 =

ˆRnmQ(x)φP (x) dx→ 0

as |xQ − xP | → ∞. Note that no cancelation of the wavelike functions is neededhere.

On the other hand, if xQ ≈ xP but, say, ν << µ, then the almost orthogonalityis a manifestation of the physical fact that “waves with very different wavelengthsare invisible to each other”. In fact, as illustrated in the figure below, mQ lives at amuch larger scale than φP , which highly oscillates where mQ is essentially constant.

Therefore, in this case,

〈mQ, φP 〉 =

ˆRnmQ(x)φP (x) dx ≈ 0,

since we are integrating a function with mean zero against a function which is es-sentially constant. The further vanishing moments of φP can be combined with thesmoothness of mQ to quantify this. It follows that only the entries with ν ≈ µ andk ≈ l are significant in the matrix.

Calderon-Zygmund operators are one example of operators whose matrices arealmost diagonal. This was proved in [21], [23], and [57] by showing that, undersuitable cancelations conditions as in the reduced form of the T1-Theorem of Davidand Journe [18], they map atoms into molecules. We will not define these operatorshere but we note that at the function level it makes no sense to take the absolutevalue of a Calderon-Zygmund operator. Taking the absolute value of its kernel killscrucial cancelations needed to be able to even define the operator. However, atthe sequence space level we can still treat the corresponding matrices as positive.Somehow, mysteriously or not, the sizes of the pairings 〈T (φQ), φP 〉 manage toencode all the needed information.

We want to conclude this section with a simple application to pseudo- differen-tial operators, which are much more easily handled than general Calderon-Zygmundoperators.


Let Ta be a pseudo differential operator with symbol a ∈ S01,1; that is,

Ta(f)(x) =

ˆRneixξa(x, ξ)f(ξ) dξ,

where the symbol a(x, ξ) is a C∞(Rn × Rn) function satisfying

|∂βξ ∂γxa(x, ξ)| .β,γ (1 + |ξ|)|γ|−|β|,

for all β, γ. Such operators are easily seen to be at least bounded from S to S ′, whichwill suffice for our purposes. The class of symbols S0

1,1 is sometimes referred to as“exotic”, and they are notorious for producing operators with Calderon-Zygmundkernels (we will not need this property) but for failing to be bounded on L2; see,for example, [35, 36]. They are, however, bounded on some spaces of smoothfunctions as shown in the works of Meyer [43], Runst [49], Bourdaud [3], and Stein[52] among others. The Frazier-Jawerth machinery can be used to obtain similarresults in a very simple way. In fact the φ-transform encodes so much informationthat the proof reduces to a straighforward computation and an integration by parts.We sketch the arguments from [56].

Let f ∈ S. By the convergence of the inhomogeneous formula (2.15) and thecontinuity of Ta on S,

Ta(f) =∑k∈Zn

〈f,Φ0k〉Ta(Ψ0k) +∑ν≥1

∑k∈Zn

〈f, φνk〉Ta(ψνk).

We will show that Ta(Ψ0k) and Ta(ψνk) are molecules associated to the cubes Q0k

and Qνk for the Sobolev spaces L2α with α > 0. For such spaces all that we need to

verify are the conditions (3.2) for some N > n and all |γ| ≤ [α] + 1; no vanishingmoments are required on these molecules [25]. If we verify such estimates, we willthen be able to conclude that

‖T (f)‖L2α. ‖f‖L2

α

for all f ∈ S, and by density for all f ∈ L2α.

Fix k ∈ Zn and ν ≥ 1 (the arguments for ν = 0 are similar but easier) and letψQ = ψνk. We can compute

T (ψQ)(x) =

êixξa(x, ξ)ψQ(ξ)dξ = 2νn/2TQ(ψ)(2νx− k),

where

TQ(f)(x) =

êixξa(2−ν(x+ k), 2νξ)f(ξ)dξ.

Furthermore, letting N > n/2, taking γ derivatives in x inside the integral sign,and then integrating by parts in ξ, we also compute

(∂γTQ(ψ))(x) =

ˆRneixξ

(I −∆ξ)N

(1 + |x|2)N

∑β

Cβ(iξ)β∂γ−βx (a(2−ν(x+ k), 2νξ)ψ(ξ))

dξ,(4.3)

where the sum runs over all multi-indices β such that βj ≤ γj for all j = 1, . . . , n.Note that when ρ of the (I − ∆ξ)

N derivatives fall on the symbol a, we canestimate

|∂ρξ∂γ−βx (a(2−ν(x+ k), 2νξ)| . 2ν|ρ|2−ν(|γ|−|β|)(1 + |2νξ|)|γ|−|β|−|ρ| . 1,


because the integral takes place on the support of ψ where |ξ| ≈ 1. By the same

reasoning, any derivatives falling on (iξ)βψ(ξ) are bounded by a constant. It followsthat

|∂γTQ(ψ)(x)| . (1 + |x|2)−N ,

and

|∂γT (ψQ)(x)| = |∂γ(2νn/2TQ(ψ))(2νx− k)| . 2νn/22ν|γ|

(1 + |2νx− k|2)N,

which is precisely (3.2).For L2 or L2

α with α < 0, the molecules Ta(ψQ) must have vanishing moments.This can be achieved if one imposes some cancelations on the transpose operatorT ∗a of the form T ∗a (xγ) = 0; see [56] for details. The approach also works for all theBesov and Triebel-Lizorkin spaces. Other pseudodifferential operators with homo-geneous symbols (and of arbitrary order) were treated with the same arguments in[28]. In a multilinear setting, almost diagonal matrices (tensors) and applicationsto multilinear operators were investigated in this fashion in [29], [1], and [2].

5. Concluding remarks

There are many other applications of the reproducing formula and functionspace characterizations presented in [25]. Among others, there are results about in-terpolation, pointwise and Fourier multipliers, and traces involving several Triebel-Lizorkin spaces. It is impressive to see the outreach of the theory to many previouslyexisting topics but also to an abundance of new results. Moreover, we believe thatmany new applications of these powerful techniques are yet to come. Frazier andJawerth modestly state in [25, p. 36]:

“In any case, we want to make clear that virtually all of ourtechniques already exist in some antecedent form. Nevertheless,their particular combination here leads to new conclusions andto sharpened versions of known results. Moreover, our presenta-tion reveals an elementary discrete structure underlying a diverserange of topics in harmonic analysis.”

Wavelets have become a great mathematical success story based in part onthe discrete underlying structures referred to above. Their applications in imageprocessing, signal analysis, compression of information, and so on, have impactedtechnology and our everyday life. It is stimulating to think that part of the genesisof this important mathematical contribution is in an obscure formula in an inter-polation paper by Calderon. Bjorn Jawerth’s original mathematical training was infunction spaces and interpolation, areas in which he has made many other importantcontributions. But he also embraced the applications of wavelets theory and, duringhis late years, he did a lot of translational research, software and even hard-weardevelopment, and founded a couple of companies. Yet many of the “miracles” ofwavelets can only be fully explained and understood through the detailed harmonicanalysis that preceded the applications and the arsenal of functional analytic toolsof which Bjorn was so fond of. The works [24, 25] are great contributions in thisanalysis. Personally, we will always be thankful to Bjorn and Michael for their workon discrete decompositions, for the mathematics they have shared with us, and forthe influential role their ground breaking research has had in our own professionalcareers.


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Department of Mathematics, Western Washington University, 516 High Street,

Bellingham, WA 98225, USAE-mail address: [email protected]

Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Lawrence,Kansas 66045-7523

E-mail address: [email protected]

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