The Discrete Calderón Reproducing Formula of Frazier and Jawerth

Transcription

1 The Discrete Calderón Reproducing Formula of Frazier and Jawerth Árpád Bényi and Rodolfo H. Torres Abstract. We present a brief recount of the discrete version of Calderón s reproducing formula as developed by M. Frazier and B. Jawerth, starting from the historical result of Calderón and leading to some of the motivation and applications of the discrete version. Alberto P. Calderón s genius has produced a plethora of deep and highly influential results in analysis [6], and his reproducing formula certainly qualifies as a gem among them. Also commonly referred to as Calderón s resolution of identity, Calderón s Reproducing Formula (CRF) is a strikingly elegant relation, inspired by the simple idea of breaking down a function as a sum of appropriate convolutions or wave like functions (see (1.1) below). Moreover, mathematicians working on wavelets consider nowadays Calderón as one of the forefathers of the theory. The purpose of this expository note is to honor the memory of Björn Jawerth by presenting a brief account of some aspects of one of his major contributions to mathematics: the development in collaboration with Michael Frazier of what they called the φ-transform and how it relates to Calderón s original formula. None of what is presented here is new and we will repeat some arguments commonly found in the literature; but some others that we shall present seem to be part of the folklore of the subject or are hard to locate in the references. We will try to follow a partially historical and partially formal approach to this beautiful formula as we learned it from the horse s mouth, in particular from Björn Jawerth himself, but also from Michael Frazier, Richard Rochberg, Michael (Mitch) Taibleson, and Guido Weiss. 1 The goal is to summarize here in a succinct way some simple motivations while pointing to both classical as well as some not so well-known references, including only 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 uninitiated audience, some of the profound contributions of Jawerth and collaborators, which are sometimes overlooked in the continuing proliferation and rediscovery of results in the area of function space decompositions. We also want to insight the curiosity of those who may have not read numerous original works we shall mention, and motivate them to further explore the reach existing literature Mathematics Subject Classification. Primary: 42B20; Secondary: 42B15, 47G99. 1 In 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 in the 80 s. 1

2 2 Á. BÉNYI AND R. H. TORRES As it is today well-understood, the φ-transform is a way to represent functions and even distributions as linear combinations of translates and dilates of a fixed function. The representation is sometimes also referred to in the literature as the Frazier- Jawerth transform, almost orthogonal decomposition, or non-orthogonal wavelet expansion. In fact, like wavelets, the Frazier-Jawerth decomposition [24, 25] can be 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, but their origins and initial motivations for their developments were somehow different. While Frazier s and Jawerth s were rooted in the analysis of functions spaces, their intrinsic features and atomic decompositions, and the classical operators acting on them, the study of wavelets can be traced back to the rediscovery of CRF in an applied context and the construction of orthonormal bases with certain particular properties. We quote Frazier and Jawerth [25, p. 37]: Wavelets are a collection of functions similar to the representing functions in our decomposition, but which are mutually orthogonal. In fact, wavelets form an unconditional basis for the usual function spaces in harmonic analysis listed above. Thus, unlike our theory, the wavelet theory is immediately connected to the vast literature on the construction of explicit unconditional bases for various function spaces. However, for the applications that we have considered (not related to bases), our more elementary decomposition has been sufficient. Thus, for reasons of simplicity (and perhaps stubbornness) we have presented our results without reference to the beautiful theory of wavelets. However, the reader will readily note that our conclusions generally apply as well to the wavelet decomposition. Likewise, we will not attempt the almost impossible task to provide a comprehensive account of wavelet theory, but just implicitly point out to some common features 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 to the works of Y. Meyer [45], Hernández and Weiss [34], and the references therein.) Nor shall we describe in its fullness the connection of the φ-transform to the very active 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 orthogonality of the analogous wavelet formula for simplicity in the construction. Such construction is astonishingly elementary, and this only adds to its beauty. At a theoretical level, this lack of perfect orthogonality is of no consequence in most uses. In addition, the redundancy and flexibility of frame representations is nowadays preferred in some numerical applications, while in others the perfect orthogonality of wavelets still becomes crucial. The potential of almost orthogonal or quasiorthogonal decompositions in applications has been observed early on in the development of all these and related expansions. For example, I. Daubechies, A. Grosmann, and Y. Meyer stated in [17, p. 1273]: We believe that tight frames and the associated simple (painless!) quasiorthogonal expansions will turn out to be very useful in various questions of signal analysis, and in other domains

3 DISCRETE CALDERÓN S REPRODUCING FORMULA 3 of applied mathematics. Closely related expansions have already been used in the analysis of seismic signals [the authors referred to the works in [27], [30], [31]]. While in their work they were focusing on the construction of frames based on the 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 frames associated with the ax+b group. Tight frames associated with the ax + b group were first introduced in a different context closer to pure mathematics. In Ref. 13(b) [Frazier-Jawerth [24]] one can find a definition of quasiorthogonal families very close to our tight frames, and a short discussion of the similarities between a quasiorthogonal family and an orthonormal basis. For the many miraculous properties of this orthonormal basis, see Ref. 14 [Meyer [42]]. As already mentioned, we will focus mainly on the relation of the φ-transform to the CRF and we want to keep this presentation not too extensive, hence leaving out many wonderful connections to other topics. For more details on the interplay between wavelets, frames, sampling, signal analysis, CRF, and much more, as well as historical accounts and relevant contributions, we refer to the delightful introduction by J. Benedetto to the compendium of articles in [33]. 1. Continuous Calderón s Reproducing Formula The formula in its modern form can be briefly stated as (1.1) u = 0 u φ t φ t dt t, for an appropriate function φ and with convergence also understood in an appropriate sense. Like several other of his profound contributions, the origins of the reproducing identity of Calderón are traced back to his 1964 paper [4] on complex interpolation. The abstractness and the generality of the presentation do not make this 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 reviewer s opinion, to the unfortunate subdivision into two parts, the first giving the definitions and main results, the second giving the detailed proofs; the reader has to spend quite a lot of time just searching for the relevant passage for the proof of each particular statement., whereas C. Fefferman and E. Stein [19] eloquently summarize this masterpiece of Calderón as follows: This important and lengthy paper of Calderón contains significant conceptual insights of broad interest, but at the same time requires a number of ingenious and tricky technical devices for executions. Though highly appreciated as one of the foundation rocks on which modern interpolation theory was built upon, the article indeed contains many other results,

4 4 Á. BÉNYI AND R. H. TORRES including the reproducing formula. Frazier, Jawerth, and Weiss, point in [26] to [4, 34] as the birthplace of Calderón s identity. And it is there, in 14 in fact, a hidden 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, Calderón s idea, reduced to a particular case, is as follows. For a given function ϕ on R n, and t > 0, let us write as usual ϕ t (x) = t n ϕ(t 1 x). τ y ϕ(x) = ϕ(x y) denotes the translation operator, and denotes the operation of convolution of two functions. Letting u L 2 (R), and ϕ C (R) be a spherically symmetric function having vanishing moments up to a prescribed order, Calderón first introduces (in [4, 14]) the L 2 (R)-valued function of t ˆ F (t) = T u = (τ y u)t 1 ϕ(t 1 y) dy, R where the integral is to be interpreted as Riemann vector valued. In other words, F (t)(u) = T u(t) = u ϕ t. Letting now ψ 1, ψ 2 C, spherically symmetric and with compact support, he further defines the operator where S(F, u) = S(T u, u) = u ψ 2 + Cu, Cu = 1 ˆ R τ y F (t 1 )ψ 1 (ty)dy dt. Note now that, by selecting ψ 1 = ϕ, and further writing φ t = ϕ t 1 we have in fact Cu = 1 u φ t φ t dt t. Calderón claims and later proves (in 34) that it is possible to select ψ 1 and ψ 2 to recover u through the operator S: u = S(T u, u). It follows that u = u ψ u φ t φ t dt t, which bears a strong resemblance to what is nowadays commonly referred to as Calderón s reproducing formula. The formula, even in a more modern form, appears also to have been known to Calderón and others around him. But surprisingly, the next published version of a more detailed CRF that we are aware of did not appear until 10 years later in an article by N. J. H. Heideman in 1974, [32]. Therein the author writes in page 68: We first prove a theorem of A.P. Calderón in [3] [here Calderón [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 L 1 (R n ) function so that its Fourier transform µ(ξ) = µ(x)e ix ξ dx verifies that, as a function of t, µ(tξ) is not identically zero for R n ξ 0. Then, there exist a function φ in the Schwartz space S whose Fourier transform is compactly supported outside the origin and another function ψ S so that dt (1.2) u = u µ t φ t t, and (1.3) u = u ψ + 0 ˆ 1 0 u µ t φ t dt t,

5 DISCRETE CALDERÓN S REPRODUCING FORMULA 5 for each tempered distribution u S. Actually Heideman stated the theorem more generally, the dilations by 1/t replaced by a certain one parameter group of operators T t and µ a Borel measure with an appropriate Tauberian condition replacing the non-vanishing condition in the statement above. Heideman implicitly uses in his proof the existence of the improper integral in (1.2) as a distribution, allowing him to freely interchange the integration with the action of the Fourier transform. That this was the first explicit version in print of the formula in a form more similar to how it is used today seems to be corroborated by the comments in the 1981 article by S. Janson and M. Taibleson [37]. In fact, the authors state on page 29: Il teorema di rappresentazione é stato formulato per la prima volta nel corso di un seminario alla University of Chicago tenuto da Alberto Calderón intorno al 1960 ed é stato usato (implicitamente) da Calderón in [4]. In forma esplicita compare nell articolo di N. Heideman [2] [here [32]] e in molti altri successive. Like those around Calderón in Chicago learning about his formula in seminars in the 60 s, those around Washington University in the 80 s (such as the second named author of this article) learned about Janson and Taibleson s paper directly from Mitch. The authors in [37] made explicit the implicit weak convergence argument in Heideman s paper looking at the limit of truncated integrals ˆ R ɛ u µ t φ t dt t in the sense of distributions. They rigorously showed, in particular, that given the function µ one can always construct the associated φ so that for all ξ 0, and ψ S, 0 ψ(ξ) = µ(tξ) φ(tξ) dt t = 1 1 µ(tξ) φ(tξ) dt t for ξ 0 and ψ(ξ) 1 in a neighborhood of the origin, such that (1.3) holds for u 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 and Taibleson 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 distributions whose Fourier transforms are supported at the origin. Such distributions are the inverse Fourier transforms of finite linear combinations of the delta function and its derivatives at the origin, hence they are polynomials. Janson and Taibleson established then the validity of the homogeneous formula (1.2) for tempered distributions but with weak convergence modulo polynomials. An earlier study of this modulo polynomial convergence, but in a simpler discrete version of the CRF, is in the book by Peetre [47] (we will come back to this convergence in the next section). Presumably, some of the several subsequent works referred to by Janson and Taibleson in the above quote from their work implicitly include Calderón s three

6 6 Á. BÉNYI AND R. H. TORRES articles on parabolic H p spaces, two in collaboration with A. Torchinsky, [7], [8] and [5]. These works were published in the period , hence before [37]. In particular, it is indicated in [8, pp ] that the formula converges pointwise for smooth functions u with compact support, and in [5, pp ] that it converges when paired against test functions if u H p. The 80 s saw a proliferation of very important results in harmonic analysis that were proved taking advantage of the CRF. Some relevant appearances of the formula that should be mentioned are in the works of Chang-Fefferman [9] (where the formula is stablished by formally exchanging Fourier transform and integration), and Uchiyama [60] and Wilson [62], where the CRF plays a central role in constructing atomic decompositions. During this time, the formula started to be presented more frequently as dt u = u φ t φ t t, 0 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 T 1-theorem of David and Journé in [18]. In that work, the authors also prove the convergence in L 1 of the truncated integrals in the CRF for functions in L 1 with mean zero. In their own proof of the T 1-theorem, Coifman and Meyer [12] made a very nice use of the so-called P t Q t formula (which goes back to their work with McIntosh [11]) where one can recognize traces of Calderón s own formula, except this time used to decompose an operator instead of a function. In an application to study the minimality of the Besov space Ḃ0,1 1, Meyer [44] proved the convergence of the formula in such space norm. In the mid 80 s the CRF was rediscovered within the mathematical physics community; it is studied in the article of Grossman and Morlet [30] and other related works which helped jump start much of the wavelets revolution. In fact, the CRF can be written as (1.4) u = 0 ˆ R n u, φ t,y φ t,y (x) dydt t n+1, where now φ t,y (x) = t n/2 φ((x y)/t) and the last expression is nowadays referred to as the continuous wavelet representation of u. It turns out that the CRF converges in a very strong sense, namely in L 2. This has been folklore in the subject based on a formal computation using the Fourier transform, but we could not find a published explicit rigorous argument for this convergence until the book by Frazier, Jawerth and Weiss [26]. This work was published in 1991 but was based on lectures at a CBMS conference in Alabama in Within the wavelet literature, the L 2 convergence can be traced to the book of I. Daubechies [16, pp ], where pointwise convergence for some continuous function is also presented. Though not commonly found in the literature, the CRF converges almost everywhere for arbitrary functions in L 2, as stated in the following theorem. Theorem 1.2. Assume that φ S(R n ) is real valued, radial, with Fourier transform compactly supported away from the origin, and such that (1.5) 0 φ(tξ) 2 dt t = 1

7 DISCRETE CALDERÓN S REPRODUCING FORMULA 7 for all ξ 0. Then, for all u L 2 (R n ), we have (1.6) u = where the convergence u = 0 lim u ɛ,r = ɛ 0 + R + u φ t φ t dt t, lim ɛ 0 + R + ˆ R ɛ u φ t φ t dt t holds both in the L 2 and the pointwise almost everywhere sense. Moreover, the convergence also holds for u L p for 1 < p <. Proof. As mentioned, the statement for L 2 appeared in this form in [26, 1], from where we repeat the proof. It is clearly not a challenge to construct a function in S, real valued, radial, and with Fourier transform compactly supported away from the origin. For any such function, it is easy to see that the integral in the righthand side of (1.5) is (a real) constant for any ξ 0. Hence, normalizing any such 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 φ t (ξ) = φ(tξ), using the integral condition on φ, the Fourier transform of the right hand side simplifies to û. The rigorous 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 L 2 L 1 goes as follows. One observes that in this case u φ t φ t L 1 (R n ) for all t > 0; in fact, u φ t φ t L 1 u L 1 φ 2 L 1. A similar reasoning shows that, for all 0 < ɛ < R, we have u ɛ,r L 1 ln(r/ɛ) u L 1 φ 2 L 1, thus u ɛ,r L 1 (R n ). Now, using Fubini s theorem we get û ɛ,r (ξ) = û(ξ) ˆ R ɛ φ(tξ) 2 dt t. Therefore, combining Plancherel and Lebesgue s dominated convergence theorems leads to u ɛ,r u 2 L = 2 (2π) n û ɛ,r û 2 L ˆ 2 ( ˆ R = (2π) n û(ξ) 2 1 φ(tξ) 2 dt ) 2 dξ 0 R n ɛ t as ɛ 0 and R. Next, consider the case of a general function u L 2 (R n ) and let u j L 2 L 1 be such that u j u L 2 0 as j. Then (1.7) u ɛ,r u L 2 u j u L 2 + (u j ) ɛ,r u j L 2 ˆ R dt + (u j u) φ t φ t t. L 2 ɛ Let τ > 0 be arbitrary. As explained in the L 2 L 1 case above, we immediately see now that (u j ) ɛ,r u j 2 L = 2 (2π) n (u j ) ɛ,r û j 2 L 2

8 8 Á. BÉNYI AND R. H. TORRES ˆ ( û j û 2 L + û 2 j (ξ) 2 1 R n u j u 2 L + τ, 2 ˆ R ɛ φ(tξ) 2 dt ) 2 dξ t for ɛ sufficiently small and R sufficiently large, since the L 2 norms of the functions u j are uniformly bounded. Also, using Minkowksi s inequality, we can bound the third term on the right hand side of (1.7) by ˆ R dt (u j u) φ t φ t L 2 t ɛ ˆ R ɛ u j u L 2 φ t 2 L 1 dt t ln(r/ɛ) u j u L 2. All in all, we have obtained that for ɛ and 1/R sufficiently small depending on τ but fixed, u ɛ,r u L 2 u j u L 2(1 + ln(r/ɛ)) + τ. Finally, for j large enough, we can then write u ɛ,r u L 2 τ, thus proving Calderón s reproducing formula in this case as well. The pointwise almost everywhere convergence can be established in the following way. Following [18, p. 376], we observe that if φ S is as above, then there exists a function η S such that η(0) = 1 and (1.8) t d dt (u η t) = u φ t φ t. We note that (1.8) can be viewed as an identity relating the families of operators Q t (u) = u φ t and P t (u) = u η t, namely Q 2 t = t d dt P t. To verify the conditions on η simply set ˆ 1 η(ξ) = 1 φ(tξ) 2 dt 0 t. We note immediately that η(0) = 1. Then, since φ S, for all multi-indices α and all N N, we can write for ξ 0 ξ N α η(ξ) 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) = φ(sξ) 2 ds t s, which implies d dt η(tξ) = 1 t ( φ t ) 2 (ξ). The last equality gives the identity (1.8). If u L 2 (R n ), we simply need to recall that for such η S(R n ) with η(0) = 1, P t forms an approximation to the identity. That is, we have the almost everywhere convergence of u η t to u as t 0 +. Note also that (1.9) u η t (x) u L 2 η t L 2 t n/2 u L 2, since η t is L 1 -normalized, and hence u η t 0 as t. Therefore, using now (1.8), we see that ˆ R lim u φ t φ t (x) dt ˆ R ɛ 0 + ɛ t = lim d ɛ 0 + ɛ dt (u η t)(x)dt = u(x) R + R +

9 DISCRETE CALDERÓN S REPRODUCING FORMULA 9 for a.e x. The method used for the pointwise convergence and again a simple limiting argument give also the convergence in L p for 1 < p <. Indeed, adapting the arguments from [58], we first note that for u L 2 L p, u η t converges to u in L p as t 0 + and u η t (x) Mu(x), where M is the Hardy-Littlewood maximal function. By dominated convergence, u η t converges to 0 in L p as t and hence u ɛ,r converges to u. Now, for a general u L p, we can approximate it by functions u j L 2 L p. Observe also that, by Minkowksi s inequality, u ɛ,r makes sense in L p and while u ɛ,r (u j ) ɛ,r L p ɛ,r M(u u j ) L p ɛ,r u u j L p, (u η ɛ u η R ) (u j η ɛ u j η R ) L p M(u u j ) L p u u j L p. It follows that we also have, for arbitrary u L p and any 0 < ɛ < R <, u ɛ,r = u η ɛ u η R. Finally, notice that in place of (1.9) we have u η R (x) u L p η R L p R n/p u L p, where 1/p + 1/p = 1, and so the same arguments used in the case p = 2 give the convergence for general p. We want to remark that the conditions we have imposed on the function φ are to 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 is possible 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 conditions on φ for the formula to hold and also about the converge of the CRF in many other senses and various function spaces. For example, S. Saeki [50] proved that for u L p (R n ), 1 < p <, the integrals y u φ t φ t (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 L p (w), where w is an A p weight, allowing also more general truncations of the improper integral in the formula. K. Li and W. Sun [41] established then pointwise almost everywhere convergence in L p (w). Wilson also obtained in [64] convergence in H 1 and weak in BMO (as the dual of H 1 ). More recently, the second named author of this article and E. Ward proved in the already cited article [58] the convergence of a natural analog of the CRF in mixed Lebesgue spaces L p L q (R n R). There is also a rich theory with versions of the CRF in more abstract group theoretic setting. One of the first works in this direction is the article by H. Feichtinger and K. Gröchenig [20] involving square integrable representations. More references to the literature dealing with wavelets generated through the action of other groups are given in the survey by E. Wilson and G. Weiss [61]. One may speculate that the continuous CRF converges in norm as a limit of the functions u ɛ,r in any (homogeneous) space which admits a Littlewood-Paley

10 10 Á. BÉNYI AND R. H. TORRES decomposition; in particular, in all the homogeneous Besov and Triebel-Lizorkin spaces Ḃα,q p and F p α,q. 2 However, we have not been able to locate an explicit reference for this fact in the literature. Such a general result does exists for the discrete formula in the next section, as proved by Frazier and Jawerth [24, 25] (see also [44] and the references therein for discrete orthonormal wavelets). Nonetheless, in many applications, the following very strong convergence in the appropriate common dense subspace for most Ḃα,q p and F p α,q spaces suffices to justify many uses of the CRF. Proposition 1.3. Let ˆ S 0 = {u S : u(x)x α dx = 0 for all multi-indices α}. Then, if u S 0, lim ɛ 0 u ɛ,1/ɛ = u in the topology of S. Proof. Note that since φ has compact support away from the origin, for each ɛ > 0 we have ( ˆ ) 1/ɛ û(ξ) 1 φ(tξ) 2 dt = 0 t unless ξ ɛ or ξ 1/ɛ. Also, for all α, M, we have ɛ α û(ξ) α,m (1 + ξ ) M because û S. In addition, for all multi-indices α, β, α û(ξ) α,β ξ β because α û(0) = 0 for all α. With these estimates, it is easy to see that (1 + ξ ) M α (û(ξ) û ɛ,1/ɛ (ξ)) α,m ɛ, which proves the required convergence. Naturally, a simpler version of the reproducing formula also holds: supposing that ψ is such that ψ(tξ) dt t = 1, then 0 u = 0 u ψ t dt t. But the advantage of having the double convolution φ t φ t in Calderón s formula is that both convolution factors u φ t and φ t are with compact support on the Fourier side, while in the simpler version above only one of the factors, ψ t, has this property. As it will become clear, having the double convolution proves to be a crucial ingredient in the discrete version of the formula by Frazier and Jawerth. 2 We 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 is consistent with the convergence modulo polynomials of the CRF for arbitrary distributions.

11 DISCRETE CALDERÓN S REPRODUCING FORMULA Discrete Calderón s Reproducing Formula A full discretization of Calderón s identity through appropriate Riemann sums may be used to motivate the Frazier-Jawerth φ-transform or discrete wavelet decompositions. We first work at a formal level. Suppose that φ is as in the statement of Theorem 1.2. For ν Z, k Z n, let Q = {Q νk } ν,k be the collection of dyadic cubes, where Q νk has side length l(q) = 2 ν and lower left corner x Q = 2 ν k. By making the change of variable t = 2 µ in (1.6), we see that, modulo a multiplicative constant, u(x) = ˆ ν+1 ˆ u φ 2 µ(y)φ 2 µ(x y)dy dµ. ν Z ν R n The integrand in the last expression depends smoothly on µ so we can roughly approximate the integration in µ by the value of the integrand at say µ = ν. Hence, braking the integral in R n into dyadic cubes of side length 2 ν, we get u(x) ˆ u φ 2 ν (y)φ 2 ν (x y)dy ν Z Q νk k Z n Note now that the Fourier transforms of both functions u φ 2 ν and φ 2 ν are supported around frequency of the order 2 ν, and so the functions themselves cannot oscillate too much on a cube of side length 2 ν. We can then attempt to write the Riemann sum approximation u(x) u φ 2 ν (2 ν k)φ 2 ν (x 2 ν k)2 νn. ν Z k Z n Of course, we have ignored issues about convergence and have not precisely quantified any of the approximation, but this simple reasoning clearly suggests the possibility of a discrete φ-transform or wavelet expansion (2.1) u = ν,k u, φ νk φ νk, where the functions 3 (2.2) φ νk (x) = 2 νn/2 φ(2 ν x k) = 2 νn/2 φ 2 ν (x 2 ν k) 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 Calderón s identity, has been extensively used in the context of function spaces on R n and their Littlewood- Paley characterizations; see the books by Stein [51], Peetre [46], Triebel [59], and the 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 away from zero on a smaller annulus π/4 + ɛ ξ π ɛ, and (2.3) φ(2 ν ξ) 2 = 1, ξ 0. ν Z 3 For simplicity, we are assuming φ to be radial.

12 12 Á. BÉNYI AND R. H. TORRES Such a function φ is sometimes referred to as admissible. The existence of a function ψ which satisfies all the above properties except possibly (2.3) is of no concern. Noticing that any such function satisfies for some 0 < c < C < c < ψ(2 ν ξ) 2 < C, ν Z one can define φ as desired by letting ψ(ξ) φ(ξ) = ( ψ(2 ) 1/2. 4 ν Z ν ξ) 2 Then again, in various appropriate senses, (2.4) u = ν Z u φ 2 ν φ 2 ν. For example, we have the following result. Proposition 2.1. If u L 2, then the convergence of (2.4) holds in L 2 sense, while, if u S 0, then the convergence is in S. In fact, the proof of the convergence in L 2 is rather trivial while the one for functions in S 0 can be obtained in a similar fashion to Proposition 1.3. Also, although the righthand side of the above formula makes sense for an arbitrary distribution u S, there are issues with the convergence since, yet again, the functions φ 2 ν are completely blind to distributions supported at the frequency point ξ = 0. As mentioned earlier, Peetre [46, pp.52-54] seems to have been the first to rigorously address this issue. For the benefit of the reader, and since the reference [46] is not always readily available, we include here a sketch of Peetre s result. Theorem 2.2. Let φ be an admissible function satisfying (2.3). Then, for each u S, there exist a sequence of polynomials {P N } N of bounded degree (depending on u) and another polynomial P such that u = lim N ( N ) u φ 2 ν φ 2 ν ν=0 where the limits are taken in S. + lim N ( 1 ν= N u φ 2 ν φ 2 ν P N ) + P, Proof. Similar arguments to those in Proposition 1.3 show that for any g S φ(2 ν ξ) 2 g(ξ) ν=0 converges in S, so if, denotes the pairing of distributions with test functions, we have N ( φ 2 ν ) 2 û, g = û, ( φ 2 ν ) 2 g. It follows that lim N ν=0 ν=0 ( N ) u lim u φ 2 ν φ 2 ν N ν=0 4 Note that the conditions on the support of ψ make φ a smooth function!

13 DISCRETE CALDERÓN S REPRODUCING FORMULA 13 exists in S. Next, since u S, there exists a continuous seminorm q L,M on S, such that q L,M (g) = sup (1 + ξ ) L α g(ξ), ξ R n, α <M û, g q L,M (g) for all g S. Since ξ 2 ν on the support of φ 2 ν (ξ), for any ν < 0 and γ = M we can write û, ξ γ ( φ 2 ν ) 2 g q L,M (ξ γ ( φ 2 ν ) 2 g) 2 ν. ( 1 ) It follows that γ N u φ 2 ν φ 2 ν converges in S for all γ = M. But this is equivalent to the existence ( of a sequence of polynomials {P N } of degrees no 1 ) larger than M 1, such that ν= N u φ 2 ν φ 2 ν P N 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 S 0 and so û + û, g = û, ( φ 2 ν ) 2 g = û, g, ν= where we have used the fact that the formula (2.4) converges in S for g S 0. It follows that û (û + û ) is a distribution supported at the origin and hence u (u + u ) is some polynomial P. As with the continuous CRF, the representation (2.4) can be obtained and convergence can be established in various function spaces under less stringent conditions on φ. In particular, one can use two different functions φ and ψ (in applications, one of them is usually constructed after the other one is given) with the same properties of an admissible function except that they are now jointly normalized to verify (2.5) φ(2 ν ξ) ψ(2 ν ξ) = 1, ξ 0. ν Z 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 condition as before; and another function θ S, which is itself supported on the unit ball centered at the origin and has vanishing moments x γ θ(x) dx = 0 for all γ up to a prescribed order M. Moreover, this can be done so that θ still satisfies the Tauberian condition of φ (although obviously not the compact support one), and (2.6) φ(2 ν ξ) θ(2 ν ξ) = 1, ξ 0; see [24, pp ]. ν Z Starting from the first discretization of the CRF, Frazier and Jawerth arrived at the formula (2.1) through a very clever use of a generalized sampling theorem. The classical Shannon sampling theorem (yet another reproducing formula rediscovered

14 14 Á. BÉNYI AND R. H. TORRES many times in both theoretical and applied mathematics) states that if f L 2 (R n ) and, say, supp f {ξ R n : ξ < π}, then (2.7) f(x) = n sin(π(x j k j )) f(k). π(x k Z n j=1 j k j ) Note that, since f has compact support, we may assume f is continuous; in fact, f agrees 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 Fourier inversion. It is also easy to see that (2.7) converges in L 2 (as well as uniformly). A point usually made about (2.7) is its slow rate of convergence given the slow decay at infinity of the function sinc (x) n j=1 sin(πx j ) πx j. Notice, however, that (2.7) can also be seen through the equality f = f sinc, as f sinc (x) = k Z n f(k) sinc (x k). It is a well-known fact that replacing sinc by a function g S with supp ĝ {ξ R n : ξ < π} and ĝ(ξ) 1 on the support of f, one can obtain a much faster convergent formula f(x) = (f g)(x) := k Z n 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 be achieved. Frazier and Jawerth proved in [24] the following result. Theorem 2.3. Let f S and g S be so that supp f, supp ĝ {ξ R n : ξ < π}. Then (2.8) (f g)(x) = k Z n f(k) g(x k) with convergence in S. Moreover, if f is actually in S, then the convergence is in S. Proof. The formula is easily established for sufficiently nice functions as in the classical sampling theorem, that is, by expanding ĝ in a Fourier series on [ π, π] n, writing f ĝ(ξ) = (ˆ ) (2π) n g(z)e ikz dz e ikξ f(ξ) k Z n [ π,π] n and then using Fourier inversion. The details to obtain the convergence in S via a standard regularization process once the formula is established for smooth functions can 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 since we will appeal to it later. This is rather simple, since for any multi-index α we can differentiate term by term α (f g)(x) = (f α g)(x) = f(k) ( α g)(x k), k Z n

15 DISCRETE CALDERÓN S REPRODUCING FORMULA 15 and for any M > 0 we can write (1 + x 2 ) M f(k) ( α g)(x k) α,m k >N α,m k >N k >N f(k) f(k) (1 + k 2 ) M 0 as N. (1 + x 2 ) M (1 + x k 2 ) M In the estimate above, we have used Peetre s inequality and the fast decay of the samples f(k) for f S. As with the classical sampling theorem, there is of course a rescaled version of (2.8), with samples taken according to the Nyquist rate at a distance inversely proportional to the radius of the spectrum of the functions f and g. More precisely, if supp f, supp ĝ {ξ R n : ξ < 2 ν π}, then (2.9) (f g)(x) = 2 νn k Z n f(2 ν k) g(x 2 ν k). Using now the formula (2.4), and then for each ν the formula (2.9) with f = u φ 2 ν and g = φ 2 ν, Frazier and Jawerth [24] arrived in a truly painless way to a simple yet deep and amazing discrete CRF: their φ-transform. Theorem 2.4. If φ is admissible, then (2.10) u( ) = 2 νn (u φ 2 ν )(2 ν k) φ 2 ν ( 2 ν k), ν Z k Z n where for u L 2 the convergence is in L 2, for u S 0 the convergence is in S, and for u 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 = u, φ νk φ νk, ν Z k Z n with φ νk 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 orthonormal basis of wavelets. By contrast, the construction of orthonormal wavelets with a generating function in S is equally beautiful but a rather intricate task; it was first done by Lemarié and Meyer [40] and it is quite remarkable that such a construction is possible. For an earlier construction of spline wavelets, see the work of Strömberg [53], while for further compactly supported wavelets, see Daubechies [15]. The method of Frazier and Jawerth, however, does not produce an orthonormal basis 5 since not all the functions φ νk are orthogonal to each other. Moreover, it does not even produce a basis but only a frame in L 2 when φ S. Nonetheless, as mentioned earlier, this is of no technical consequence in many uses of the representation. 5 It 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 R n ; see [22]. However this gives wavelets that, for example in dimension one, only decay like x 1 at infinity.

16 16 Á. BÉNYI AND R. H. TORRES Once again, the method allows for a lot of flexibility and one can use two different functions φ and ψ satisfying (2.5) instead of (2.3). This leads to the expansion (2.12) u( ) = 2 νn (u φ 2 ν )(2 ν k) ψ 2 ν ( 2 ν k), ν Z k Z n or (2.13) u = u, φ νk ψ νk, ν Z k Z n with ψ νk defined in same way as φ νk. Moreover, there is also an inhomogeneous version (as with discrete wavelets) in which all the low frequency supported functions φ 2 ν and ψ 2 ν are replaced by two single functions Φ and Ψ, so that (2.14) Φ(ξ) Ψ(ξ) + φ(2 ν ξ) ψ(2 ν ξ) = 1 ν 1 for all ξ R n. This in turn leads to (2.15) u = u, Φ 0k Ψ 0k + u, φ νk ψ νk, k Z n ν 1 k Z n where Φ 0k = Φ( k) and, similarly, Ψ 0k = Ψ( k); see [25]. An advantage of this formula is that it now honestly converges in S and S and it is best suited to study 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 L p for p 2 is a great example of how unfortunate constraints could sometimes be turned into fantastic opportunities (at least in the right hands of virtuous mathematicians). Trying to overcome the impossibility of measuring L p properties of functions directly from the Fourier transform promoted the creation of powerful mathematical tools, which have had and continue to have impressive efficacy in harmonic analysis, operator theory and partial differential equations. Littlewood-Paley theory breaks the Fourier transform of a function into mildly interfering frequency bands that allow to recover information related to function spaces beyond the Hilbert space context. Owing its origin to Littlewood and Paley in the 30 s, the theory took its full force in R n with Stein s introduction of his g-function in the late 50 s. Chapter IV of his book [51] has a marvelous and not only authoritative but also still one of 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 often the most fruitful way of characterizing various analytic situations (such as finiteness of L p norms, existence of limits almost everywhere, etc.) is in terms of appropriate quadratic expressions. Calderón s reproducing formula is strongly connected to Littlewood-Paley theory. This is nicely illustrated in [26]. Let P denote the Poisson kernel for the upper half-space, P (x) = c n (1 + x 2 ) (n+1)/2,

17 DISCRETE CALDERÓN S REPRODUCING FORMULA 17 U(x, t) = (P t u)(x), and consider a particular version of the g-function, ( g(u)(x) = t t U(x, t) 2 dt ) 1/2 ( = φ t u(x) 2 dt ) 1/2, t t 0 where φ(ξ) = ξ P (ξ) = ξ e ξ. Littlewood-Paley theory says that, for 1 < p <, we have u L p g(u) L p, while by the CRF, u(x) = φ t φ t u(x) dt 0 t. That is, we can view u φ t u as a bounded map from L p (R n ) into L p (R n, L 2 ((0, ), dtdx t )) which is inverted by the CRF. Likewise, there is a discrete version of this fact: if φ is an admissible function, then ( ) 1/2 u L p φ 2 ν u 2, ν Z 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 be related 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 those expansions what gives the characterization. Indeed, Frazier and Jawerth showed in [24] and [25] that any space X in the Besov or Triebel-Lizorkin scales can be characterized by a corresponding space of coefficients S(X) 6. These discrete spaces are defined for sequences {s Q } indexed by the dyadic cubes and with norms in terms of { Q c(x) s Q }, where the power c(x) depends on the function space X. For some L 2 -based spaces X, the spaces S(X) are simply weighted l 2 spaces. For example, for X = L 2, the associated space is S(L 2 ) = l 2, and for L 2 α (= Ḃα,2 2 = F α,2 2 ), the homogeneous Sobolev spaces of functions with their derivatives of order α in L 2, one has the weighted l 2 norm {s Qνk } S( L2 α ) = ν,k 0 L p ( Q νk α/n s Qνk ) 2 For most spaces X based on L p with p 2, S(X) is more complicated than a weighted l p space, but it is still a Banach or quasi-banach lattice with norm only depending on s Q ; see [25]. In proving the characterizations u X { u, φ νk } S(X), Frazier and Jawerth [24, 25] developed also smooth atomic and molecular decomposition of the function spaces considered. There is a rich history in harmonic analysis related to atomic decompositions going back to the pioneering work of Coifman [10], but the atomic decomposition in [24, 25] seems most influenced by Uchiyama s one for BM O, [60]. Frazier and Jaweth also mention an alternate ḃ α,q p 6 Frazier and Jawerth denote the sequence spaces associated with Ḃ α,q p, respectively f α,q p. 1/2., respectively F p α,q, by

18 18 Á. BÉNYI AND R. H. TORRES direction to decompositions (though some how similar spirit) in the articles by Coifman and Rochberg [13] and Ricci and Taibleson [48]. A very interesting fact is the universality of the construction in [24, 25] of smooth atomic decompositions for L p, Hardy, and Sobolev spaces, and more generally the whole scales of Besov or Triebel-Lizorkin spaces. Starting with the decomposition u = ν Z θ 2 ν φ 2 ν u, where θ is selected as in (2.5), having now compact support and a prescribed number of vanishing moments (which could be arbitrary large but finite), one can rewrite a function u X as (3.1) u = s Q a Q. ν Z k Z n Here Q = Q νk is a dyadic cube in R n, s Q = s Qνk = Q 1/2 sup φ 2 ν u(y) y Q and a Q = a Qνk = 1 ˆ θ s 2 ν (x y) φ 2 ν u(y) dy. Qνk Q νk The family {a Q } of smooth atoms, as defined in [24, 25], satisfies some desirable properties: the functions a Q are C, compactly supported on a dilate of Q, and have as many vanishing moments as θ. They are also normalized 7 as the functions φ νk and so γ a Q (x) γ Q 1/2 γ /n. Moreover, if a Q have enough vanishing moments depending on the space X, it can be proved that {s Q } S(X) u X. Likewise, Frazier and Jawerth showed that whenever u = s Q φ Q, ν Z k Z n with φ Q = φ ν,k as before, then also {s Q } 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 {m Q } of smooth molecules. The molecules, for a given space X, are a rougher and less oscillating version of the wavelets φ νk. They also have some limited decay away from the cube Q = Q νk and are normalized like the functions φ νk. More precisely, they satisfy (3.2) γ m Q (x) Q 1/2 γ /n (1 + x x Q l(q) ) = 2ν(n/2+ γ ) N (1 + 2 ν x k ) N for all γ M and with N and M dependending on the function space X considered. In addition, the molecules have a number of vanishing moments also depending on X. The family of atoms and the family of functions φ νk are hence molecules as 7 We note that actually the normalization in [24] is different from the one in [25], but consistent with appropriate modifications in the definitions of the spaces S(X) in those works.

19 DISCRETE CALDERÓN S REPRODUCING FORMULA 19 well. It is important to note that, although the m Q satisfy estimates which behave as if the molecules were the translations and dilations of a single function m, they do not need to be so. Frazier and Jawerth showed that whenever f = s Q m Q ν Z k Z n (most generally in S modulo polynomials), then f X and f X {s Q } 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 coefficients proves also the convergence of the representation formula in the norm of such spaces. Regarding pointwise convergence, various results can be obtained. We point to the work on wavelets by Kelly, Kon, and Raphael [38, 39] where rates of convergence are also given, and to the article by Tao [55], where various summation methods for 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]. It turns out that, as with other types of molecules in the literature, singular integrals and other important multiplier operators map smooth atoms into molecules (but not atoms into atoms). We will show a very simple application of this in the next section. What we have just briefly summarized about the characterization of functions spaces takes actually a tremendous arsenal of harmonic analysis tools to be rigorously achieved, as well as a wise use of them [24, 25]. Such tools include, in particular, the Plancherel-Polya inequality for functions of exponential type, the Petree maximal function, and the Fefferman-Stein vector valued maximal function, which should warn the reader of the deep technical aspects of the proofs we have omitted. Another crucial notion in this development is that of almost diagonal operator 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 of operators by a family of generating functions (essentially, linear algebra!). Let T be some linear operator acting on one of the spaces X. Starting with the φ-transform for u X, u = Q u, φ Q φ Q, and applying T (while ignoring for the moment convergence issues) one gets T (u) = Q u, φ Q T (φ Q ). Expanding now T (φ Q ), we obtain T (u) = u, φ Q T (φ Q ), φ P φ P = Q P P T (φ Q ), φ P u, φ Q φ P. Q

20 20 Á. BÉNYI AND R. H. TORRES It follows from the characterization of X in terms of the discrete space of coefficients S(X) that, to study the boundedness properties of T on X, it would be enough to study the boundedness properties of an infinite matrix A T = {A T (P, Q)} P,Q = { T (φ Q ), φ P } P,Q indexed by the dyadic cubes and acting on S(X). Ideally, if the family {φ Q } are a basis of eigenvectors for T, then A T is just a diagonal matrix which is trivial to study. However, this requires constructing a generating family of functions for each operator T to be studied; something extremely unlikely to be done given all the constrains the functions φ Q must satisfy to expand the space X. The notion of almost diagonalitzation consists then in relaxing the perfect diagonalization of each operator by looking instead at whole families of operators which come close to be diagonalized by the φ Qs. That is, instead of requiring T (φ Q ) = c Q φ Q, one is content with T (φ Q ) φ Q in some appropriate sense. Still wishfully thinking along these lines, one would then have and for the matrix of A T, T (φ Q ), φ P φ Q, φ P, (4.1) A T (P, Q) 0 as the entry (P, Q) moves away from the diagonal of the matrix. Moreover, recall that the spaces S(X) are defined only in terms of the size of the coefficients s Q, so it may be possible to study the positive matrix A T instead of A T. Studying positive matrices is often a lot easier; in particular, there is the well-known criterion to study them given by Schur s test. Frazier and Jawerth found a very precise quantification of (4.1), which implies the boundedness of a matrix on S(X) for different spaces X. For example, if Q = Q νk and P = P µl, then a matrix A = {A(P, Q)} P,Q satisfying the almost diagonal condition (4.2) A(P, Q) 2 (µ ν)(ε+n/2) (1 + 2 ν 2 ν k 2 µ l ) n+ɛ for some ɛ > 0 and ν µ, and symmetric estimates if ν < µ, is bounded on S(L p ), 1 < p <. Notice that 1 + diam(q P ) l(q) ν k 2 µ l 2 ν so we can also rewrite (4.2) as ( ) ε+n/2 l(p ) 1 A(P, Q) l(q) (1 + l(q) 1 diam(q P )) n+ε. We clearly see in this particular case that the entries of the matrix are small if the Euclidean distance between the cubes is large or if their sizes are very different. Moreover, Frazier and Jawerth showed that such matrices are essentially those of operators which map atoms into molecules or molecules into molecules. Instead of

21 DISCRETE CALDERÓN S REPRODUCING FORMULA 21 presenting the technical details of such results, we look at a very intuitive reasoning of why it may be possible to obtain something like (4.2). Assume that T (φ Q ) = m Q, where m Q is a molecule, and consider the matrix A T = { T (φ Q ), φ P } P,Q = { m Q, φ P } P,Q. We want to see that (4.1) holds true. Again, let Q = Q νk and P = P µl. First, since the functions m Q and φ P are respectively localized around Q and P and decay away from them (cf. (3.2)), each function is very small where the other one is not so ˆ m Q, φ P = m Q (x)φ P (x) dx 0 R n as x Q x P. Note that no cancelation of the wavelike functions is needed here. On the other hand, if x Q x P but, say, ν << µ, then the almost orthogonality is a manifestation of the physical fact that waves with very different wavelengths are invisible to each other. In fact, as illustrated in the figure below, m Q lives at a much larger scale than φ P, which highly oscillates where m Q is essentially constant. Therefore, in this case, ˆ m Q, φ P = m Q (x)φ P (x) dx 0, R n since we are integrating a function with mean zero against a function which is essentially constant. The further vanishing moments of φ P can be combined with the smoothness of m Q to quantify this. It follows that only the entries with ν µ and k l are significant in the matrix. Calderón-Zygmund operators are one example of operators whose matrices are almost diagonal. This was proved in [21], [23], and [57] by showing that, under suitable cancelations conditions as in the reduced form of the T 1-Theorem of David and Journé [18], they map atoms into molecules. We will not define these operators here but we note that at the function level it makes no sense to take the absolute value of a Calderón-Zygmund operator. Taking the absolute value of its kernel kills crucial cancelations needed to be able to even define the operator. However, at the sequence space level we can still treat the corresponding matrices as positive. Somehow, mysteriously or not, the sizes of the pairings T (φ Q ), φ P manage to encode all the needed information. We want to conclude this section with a simple application to pseudo- differential operators, which are much more easily handled than general Calderón-Zygmund operators.