Ad Honorem Yves Meyer

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1378 Notices of the AMS Volume 65, Number 11 Ronald Coifman, Guest Editor of papers in areas of application ranging from signal processing to medical diagnostics. Modern engineering depends on his methods. Numerical analysis uses his Introduction tools for efficient numerical computation of linear and Yves Meyer was awarded the 2017 . His work nonlinear maps. has impacted mathematics in a broad and profound More recently Meyer has introduced new tools for way. Perhaps even more importantly, he has led a broad, analysis of the Navier-Stokes equations for fluid flow, multifaceted, worldwide network of research collabora- discovering remarkable profound mechanisms relating tions of mathematics with music, chemistry, physics, and oscillation to stability and blowups. signal processing. He has made seminal contributions Some of Meyer’s close collaborators have kindly pro- in a number of fields, from to applied vided some descriptions of his work, with a goal of mathematics. Meyer started his career on the interface between covering a broad panorama of analysis. Stéphane Mallat, Fourier analysis and number theory. Early in his career who formalized with Meyer the orthogonal multiresolu- he introduced the theory of model sets [1], which have tion framework, describes Meyer’s celebrated work on become an important tool in the mathematical study . His student Stéphane Jaffard (1989), who wrote of aperiodic order two years before the discovery of the account [J] of his Abel Prize for the Société Mathé- Penrose pavings by Roger Penrose and ten years before matique de Gazette des Mathématiciens, focuses the discovery of quasi-crystals by Dan Shechtman. on time-frequency analysis. Alexander Olevskii describes Around 1975 he initiated the field of nonlinear Fourier Meyer’s Sets, which modeled quasi-crystals before they analysis as a tool for organizing and analyzing nonlinear were discovered. My own contribution describes Meyer’s functional transformations of mathematics. He devel- work on nonlinear Fourier analysis. His student Albert oped all the fundamental tools and concepts necessary Cohen (1990) describes Meyer’s impact on sparse analysis. to understand the nonlinear dependence of solutions of Meyer’s work is characterized by an extraordinary boundary value problems on the shape of the bound- depth, solving longstanding problems, and starting new ary. In particular, together with Coifman and McIntosh fields of mathematics and applications. These seminal, [4], he solved in 1982 the last outstanding problem of broad contributions have had a profound impact on classical harmonic analysis by proving the continuity of the Cauchy integral operator on Lipschitz curves. It had different areas of science and establish him as a major been the key obstacle to the solvability of boundary figure in mathematics. value problems for Lipshitz domains (e.g. domains with corners). His methodologies prepared the way for Bony’s References para-differential calculus, Wu’s proof of the existence [DS] Bjørn Ian Dundas and Christian Skau, Inter- of water waves in three dimensions, and the proof of view with Abel Laureate Yves Meyer, Newsletter Kato’s conjecture, essentially changing the landscape of Eur. Math. Soc., Sept. 2017, 14–22, www.ems-ph.org analysis. /journals/newsletter/pdf/2017-09-105.pdf#page=16, pp. 14–22; reprinted in the May 2018 Notices https: Around 1984 Meyer [6]–[8] discovered the relation //www.ams.org/journals/notices/201805/rnoti-p520 between the analytic tools used in harmonic analysis .pdf. MR3726777 and various signal processing algorithms used in seismic [J] S. Jaffard, Yves Meyer, Prix Abel 2017 (French) exploration. In his Abel Prize interview [DS], Meyers said, Gaz. Math. No. 153 (2017), 20–26. smf4.emath.fr “Morlet, Grossmann, and Daubechies were in a sense ahead /Publications/Gazette/2017/153/. MR3701591 of me in their work on wavelets. So I was the ‘Quatrieme Mousquetaire.’ They were Les Trois Mousquetaires.” He A Few Representative Works by Meyer recognized their work as a rediscovery of Calderón’s [1] Algebraic Numbers and Harmonic Analysis, North Holland, formulas in harmonic analysis, thereby bridging fifty New York (1972). MR0485769 years of multiscale harmonic analysis with “wavelets.” [2] Wavelets and Operators, Vol. 1 and 2, revised version of 3, This discovery led later to the construction of the Meyer Cambridge Univ. Press (1992, 1997). MR1719426 basis, an orthonormal basis of functions localized [3] Wavelets, paraproducts and Navier–Stokes equations, in in space and frequency. His work inspired Daubechies to Current Developments in Mathematics 1996, MIT Press discover the compactly supported orthonormal wavelet (1997). bases, which profoundly affected the field of engineering, [4] (with R. R. Coifman and A. McIntosh) L’intégrale de Cauchy leading in subsequent work to nonlinear adapted Fourier sur les courbes Lipschitziennes, Annals of Math. 116 (1982) 361–387. MR672839 analysis and signal processing [3]–[5]. [5] (with Marco Cannone) Littlewood–Paley decompositions and The technological impact has been remarkable. For Navier-Stokes equations, J. Methods Appl. Anal. (1995) 307– example, the current JPEG 2000 standard for image 319. MR1362019 compression has evolved from the wavelet tools invented [6] (with A. Grossmann, and I. Daubechies) Painless nonorthog- by Meyer. The field of wavelet analysis has thousands onal expansions. Journal of Mathematical Physics 27, 1271 (1986). MR836025 Ronald Coifman is professor of mathematics at . [7] Wavelets and Applications, Proc. Int. Congr. Math. Kyoto His email address is [email protected]. (1990) 1619–1626. Math. Soc. Japan.

December 2018 Notices of the AMS 1379 [8] (with M. Farge, E. Goirand, F. Pascal, and V. Wickerhauser) Lorenzo Brandolese 2001 Improved predictability of two-dimensional turbulent flows Diego Chamorro 2006 using wavelet packet compression, Fluid Dynamics Research Jérôme Gilles 2006 (1992) 229–250. Xiaolong Li 2006 Photo Credit Ronald Coifman Opening photo courtesy of Stéphane Jaffard. Yves Meyer’s Work on Nonlinear Fourier Analysis: Meyer’s PhD students Beyond Calderón-Zygmund Antoine Ayache It is a privilege to be able to relate ideas, explorations, Jean-Louis Clerc and visions that Yves, his collaborators, and his students Michel Bruneau (codir. P. A. Meyer) developed over the last forty years, for some of which I Sylvia Dobyinski was an active participant and observer. François Gramain We have had a lot of fun and excitement in this adven- Jean-Lin Journé ture, exploring and discovering beauty and structure. Pierre-Gilles Lemarié-Rieusset I will focus my narrative on the simplest illustrations Noel Lohoue and examples of Yves Meyer’s foundational contributions Martin Meyer to nonlinear harmonic analysis, and try to illuminate and Freddy Paiva motivate some key programmatic issues that continue Fabrice Planchon and build beyond the Calderón-Zygmund vision and Philippe Tchamitchian program. It was Zygmund’s view that harmonic analysis Chantale Tran-Oberlé provides the infrastructure linking all areas of analysis, Mohamed El Hodaibi from complex analysis to partial differential equations to Fatma Trigui probability and geometry. Taoufik El Bouayachi In particular he pushed forward the idea that the Oscar Barraza remarkable tools of complex analysis, such as contour Henri Oppenheim integration, conformal mappings, and factorization, used Patrick Andersson to provide miraculous proofs in real analysis should be Guillaume Bernuau deciphered and converted to real variable tools. Together Mehdi Abouda with Calderón, he bucked the trend for abstraction, preva- Frederic Oru lent at the time, and formed a school pushing forward this Ramzi Labidi interplay between real and complex analysis. A principal Soulaymane Korry (codir. Bernard Maurey) bridge was provided by real variable methods, multiscale Ali Haddad analysis, Littlewood Paley theory, and related Calderón Aline Bonami 1970 representation formulas, later rediscovered by Morlet and Jean-Pierre Schreiber 1972 others. They will be discussed here in relation to wavelets. Marc Frisch 1977 Jean-Paul Allouche 1978 Bilinear Convolutions, the Calderón Commutator, Salifou Tembely Complex Analysis, and Paraproducts Qixiang Yang In order to understand some of the basic ideas and Guillaume Bernuau methods introduced by Meyer and to illustrate the scope François Lust-Piquard 1978 (codir. N.Varopoulos) of the program, we start with the basic example of “para- Guy David 1981 calculus” introduced by Calderón as a bilinear operator Michel Zinsmeister 1981 needed to extend smooth pseudo-differential calculus to Alain Yger 1982 rough environments. He managed by an analytical tour Gérard Bourdaud 1983 de force using complex function theory to prove that Michel Emsalem 1987 the Calderón commutator defined below is a bounded Miguel Escobedo Martinez 1988 (codir. H.Brezis) operator on 퐿2. Pascal Auscher 1989 Given a Lipschitz function 퐴 on the real line (so ′ ∞ Stéphane Jaffard 1989 퐴 ∶= 푎 ∈ 퐿 (ℝ)) one formally defines the linear operator Albert Cohen 1990 퐶1(푓) by the formula Sylvain Durand 1993 퐴(푥) − 퐴(푦) (1) 퐶 (푓) = ∫ 푓(푦)푑푦 1 2 Marco Cannone 1994 ℝ (푥 − 푦) Khalid Daoudi 1996 (codir. J.Levy-Véhel, INRIA) = (퐴|푑/푑푥| − |푑/푑푥|퐴)푓 = [퐴, |푑/푑푥|](푓), Abderrafiaa El-Kalay 1996 Hong Xu 1996 where the meaning of the absolute value of the derivative Patrik Andersson 1997 operator is given as (2) |푑/푑푥|푓(푥) = 1/2휋 ∫ exp(푖푥휉)|휉|푓(휉)푑휉̂ ℝ

1380 Notices of the AMS Volume 65, Number 11 and 푓̂ is the Fourier transform of 푓. Applications to Analytic Dependence This is the first commutator of Calderón. The simplest One class of problems in nonlinear Fourier analysis particular case is obtained when 퐴(푥) = 푥 and 퐶1(푓) concerns the nonlinear analytic dependence of linear becomes the classical Hilbert transform. He introduced operators on functional arguments. As we will see, such an auxiliary related operator, linking complex function problems are deeply connected to all aspects of harmonic theory and Fourier analysis as we now describe. analysis [CM, 4]. We shall focus on the seminal example Let 푎 and 푓 be two periodic functions on the circle of the Riemann mapping functional on rectifiable curves of power series type, with 푎 bounded and 푓 in 퐿2. Let ℎ in the complex plane. satisfy ℎ′ = 푎푓′. The fundamental mathematical question is whether Calderón’s theorem is equivalent to the statement that this dependence is analytic on some families of curves. A ℎ is in 퐿2. Written in terms of Fourier coefficients of 푎, 푓, particular example blending all of this is the following: We and ℎ this equation becomes wish to understand the flow lines of water above a riverbed as in Figure 1, and their dependence on the modification ̂ ̂ (3) ℎ푘 = ∑ (푗/푘)푎푘−푗̂ 푓푗. of the shape of the bed. Since the flow lines are the images 0<푗≤푘 of horizontal lines under the Riemann mapping from the ′ Observe that ℎ(휃) = ∫0<푡<휃 푎(푡)푓 (푡)푑푡 and that the deriva- upper half plane to the region above the curve, we need tive is in the sense of distributions as 푓 is only in 퐿2. Meyer to understand the dependence of the Riemann mapping came out with the following—remarkably simple—proof. on the curve. The link between the geometric description Write of the curve and the analytic problem is provided by 푑훾 the Cauchy transform. This operator is at the center of 푗/푘 = 푠 = 1/휋 ∫ 푠푖훾 . 2 ℝ 1 + 훾 complex analysis, and provides a vehicle to understand the infinite-dimensional Banach manifold of rectifiable Subsituting in (3) we get curves satisfying the chord-arc condition. 푑훾 ℎ̂ = 1/휋 ∫ (푗/푘)푖훾푎̂ 푓̂ , 푘 ∑ 푘−푗 푗 2 ℝ 0<푗≤푘 1 + 훾 leading to the representation of ℎ as 푑훾 ℎ = 휋(푎, 푓) = 1/휋 ∫ 푀 (푎푀 푓) , −훾 훾 2 ℝ 1 + 훾 푖훾 ̂ 푖푘휃 2 where 푀훾푓(휃) = ∑0<푘 푘 푓푘푒 is a contraction on 퐿 , proving that in the simpler case when 푎 is a bounded function and 푓 is in 퐿2 that ℎ is square integrable. Together we obtained the full strength of Calderón’s theorem using real variable methods on 푀−훾 as a singular integral operator on 퐻1. An important property hidden in the paraproduct is the weak continuity of this bilinear expression. Observe that the product of functions is not a bilinear operation that is weakly continuous in the arguments: consider sin(푛푥), which converges weakly to 0 while sin(푛푥) ⋅ sin(푛푥) Figure 1. Flowlines above a riverbed. How are they converges weakly to 1/2. On the other hand, since each going to be affected by the red bump? Fourier coefficient of the product of two functions of analytic type is a combination of a finite number of So consider a Jordan curve in terms of its arc length coefficients, it is clear that the paraproduct is weakly parameterization, i.e., continuous in the arguments. The analysis done by Meyer 푠 ∫ 푖훼(푡) and collaborators introduced real variable versions of the 푧(푠) = 푒 푑푡, commutators or bilinear operators described above. These and its corresponding Cauchy integral operator as: are fundamental building blocks for higher-dimensional 푖훼(푡) nonlinear analysis, starting with bilinear pairings. 푓(푡)푒 푑푡 퐶(훼, 푓)(푠) = 푝.푣.(1/2휋푖) ∫ . The simplest real variable version of the Calderón para- ℝ (푧(푠) − 푧(푡)) product is given in terms of wavelets or Haar functions The natural space for which this operator-valued function as of 훼 is analytic is the space of functions of bounded mean ℎ = 휋(푎, 푓) = ∑ 푚퐼(푎) < 푓, ℎ퐼 > ℎ퐼, oscillation (BMO). In fact the first derivative of 퐶(푡훼, 푓) 퐼 at 푡 = 0 is a bounded operator on 퐿2 if and only if 훼 is where ℎ퐼 are the Haar basis functions supported on the in BMO. The condition of 훼(푠) having a small norm in dyadic interval 퐼 and 푚퐼(푎) is the mean value of a, on BMO is equivalent to the geometric chord-arc condition that interval [2]. |푠 − 푡| < (1 + 푐)|푧(푠) − 푧(푡)| for small 푐.

December 2018 Notices of the AMS 1381 We are now going to sketch the relationship of the question of analyticity to complex analysis, more surpris- ingly to operator functional calculus, and to the BMO manifold of chord-arc curves. This whole theory pro- vides remarkable linkages between functional analysis, operator theory, singular integrals, and geometry. Consider a monotone change of variable of the form ℎ(푥) = ∫푥(1 + 푎(푡))푑푡. We conjugate the opera- tor 푑/푑푥 with the change of variable defined by ℎ(푥) −1 to obtain the operator 1/(1 + 푎)푑/푑푥 = 푈ℎ푑/푑푥푈ℎ , −1 where 푈ℎ푓(푥) = 푓(ℎ (푥)). We now consider the operator sgn(푑/푑푥) defined as Figure 2. Coifman and Meyer having fun while sgn(푑/푑푥)푓 = 1/2휋 ∫ 푒푥푝(푖푥휉) sgn 휉푓(휉)푑휉.̂ working on the 푘th commutators. ℝ We let ℎ(푥) = 푥 + 퐴(푥) to conclude that 푓(푡)(1 + 푎(푡))푑푡 This theme of discovery of the natural functional space sgn((1/(1 + 푎)푑/푑푥) = ∫ . appears again in Meyer’s work for the Navier-Stokes equa- 푥 − 푡 + (퐴(푥) − 퐴(푡) ℝ tions and other nonlinear PDE. The multilinear operators We see that the Cauchy integral for the curve above the arising in the power series can be analyzed and decom- graph of 퐴(푥) is given by the same expression with 푎 posed directly, using Fourier or other transforms. This replaced by 푖푎. approach provides insight and could enable efficient More generally if we replace 푎 by 푧푎, the analyticity numerical implementations. The Cauchy integral gener- in 푎 is equivalent to a bound in 푐푘 for the 푘-th Taylor alizes directly to higher dimensions, for example as a coefficient in 푧, known as the 푘-th commutator. We also double layer potential operator, or more generally, as the can show easily that the analytic dependence of the restriction to a submanifold of a Calderón-Zygmund oper- operator sgn((1/(1 + 푎)푑/푑푥) on the coefficient 1/(1 + 푎) ator of the appropriate homogeneity. The long-standing is equivalent to the boundedness of the Cauchy integral problem of short-time existence of water waves in 2 or 3 operator [1]. dimensions was solved by using these higher- Returning to the Riemann mapping, we observe that dimensional extensions. This example is related to a range the Cauchy transform is an oblique projection from 퐿2 of questions on the dependence and properties of func- onto the space of boundary restrictions (to the curve) tions of operators, or more generally on the dependence of holomorphic functions on one side of curve. The of spectral theory or resolvents on the coefficients. corresponding orthogonal projection, called the Szego The example of a variable-coefficient Laplace operator projection, can be easily expressed as a series involving the Δ in divergence form is related to the Kato conjecture, 1/2 Cauchy transform, and also using the Riemann mapping which states that the domain of Δ is the space of 2 itself, thereby leading to an analytic expression and a functions having a gradient in 퐿 . This conjecture in one proof that the Riemann mapping depends analytically variable is equivalent to the boundedness of the Cauchy on the curve, provided that the Cauchy transform does. integral, as proved by Coifman, McIntosh, and Meyer [1], The distance between operators corresponding to two and then extended to the general case close to the identity curves can be shown to be equivalent to the bounded and finally resolved by Pascal Auscher, Steve Hofmann, mean oscillation norm of the difference between their Michael Lacey, Alan McIntosh, and Philippe Tchamitchian arguments, thereby defining the most general geometry in 2002. on the infinite-dimensional manifold of curves for which all of these objects are real analytic. A Few Representative Works by Meyer Several remarkable features appear here. The Cauchy [1] R. R. Coifman, A. McIntosh, and Y. Meyer, L’integral de Cauchy définit un opérateur borné sur le courbes transform on chord-arc curves is an operator-valued Lipschitziennes, Ann of Math., vol. 116, 361–387, 1982. functional of the argument of the curve; it can be expanded MR672839 in a convergent power series of operators on BMO. The [2] R. R. Coifman and Y. Meyer, On commutators of singular norm on BMO is the operator norm defined by the first integrals and bilinear singular integrals, Trans. Amer. Math. linear term. It defines the largest Banach space for which Soc., 315–331, 1975. MR0380244 the nonlinear operator-valued transform is analytic. [3] R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis Specifically for the Riemann map functional from the and analytic dependence. Proc. Symp. Pure Math 43, American curve 푧(푠) = ∫푠 푒푖훼(푡)푑푡 given in terms of the argument Mathematical Society, Providence RI, 1985. MR812284 [4] R. R. Coifman and Y. Meyer, Wavelets: Calderón-Zygmund 훼(푠) to the argument 훽(푠) of the change of variable and multilinear operators, Cambridge Studies in Advanced 푠 훽(푡) ℎ(푠) = ∫ 푒 푑푡 defined by the Riemann map is an Mathematics, 1997. MR1456993 analytic functional, in the BMO topology of both 훼(푠) and [5] Y. Meyer, Wavelets and Operators, vol. 1, Cambridge 훽(푠). University Press, 1997. MR1228209

1382 Notices of the AMS Volume 65, Number 11 Image Credits where wavelet coefficients exhibit the local variations of 푗 푗 All section images courtesy of Ronald Coifman. 푓 at scale 2 in the neighborhood of 푡 = 2 푛:

⟨푓, 휓푗,푛⟩ = ∫ 푓(푡) 휓푗,푛(푡)푑푡 .

ABOUT THE AUTHOR In 1909, Alfréd Haar constructed the first wavelet basis, generated by a piece-wise constant wavelet Ronald (Raphy) Coifman has been −1/2 a close friend and collaborator 휓 = 2 (1[0,1/2] − 1[1/2,1]). of Yves Meyer for the last forty It is supported on [0, 1] and discontinuous. years, jointly encouraging a global It was also known that an orthonormal wavelet basis network of harmonic analysts. could be generated by the Shannon wavelet 휓 whose Fourier transform 휓(휔)̂ is compactly supported

Ronald Coifman 휓̂ = 1[−2휋,−휋] + 1[휋,2휋]. Stéphane Mallat The Shannon wavelet 휓(푡) is C∞ but has a slow decay because its Fourier transform is discontinuous. The fact Orthogonal Wavelets: Bridging Beyond Mathe- that it defines a wavelet orthonormal basis is closely matics related to the Shannon sampling theorem. Yves Meyer is a pure . This seemed obvious Yves knew about wavelets through his encounter with to us in the amphitheatre of 400 students at École the physicist Alex Grossmann and the geophysicist Jean Polytechnique in , where Yves was enthusiastically Morlet, who had introduced redundant families of regular 2 describing the beauty of harmonic analysis as “a Mozart wavelets with fast decay that spanned the space L (ℝ) symphony.” It was 1984, and Meyer already had the aura without being orthogonal. But is it possible to find a regular wavelet 휓 having a fast decay, which generates an of an exceptional mathematician, with spectacular results 2 in number theory and harmonic analysis. We thought orthonormal basis of L (ℝ)? In 1986, this seemed highly that he was living in the limbo of abstractions, but we unlikely otherwise it would certainly have been discovered were wrong. I later realized that his intellectual home is before. The Balian-Low theorem proved that orthogonal bases composed of localized complex exponentials could transient, a nomad he sometimes calls himself, which is not be constructed with regular functions having fast a root of his profound originality. decay. To verify a similar property for wavelets, Yves In 1986 it was under the umbrella of the Bourbaki tried to build an orthogonal basis with a regular and well- seminar that Yves published his discovery of wavelet localized wavelet, hoping to find the reason why it did orthonormal bases, with C∞ functions having a fast not exist. To his surprise, he discovered wavelets 휓 that decay. This was not a 100-year-old conjecture, perhaps are C∞ with a fast decay and that generate orthonormal not looking spectacular for such a mathematical artist, bases. Meyer wavelets resemble Shannon wavelets, with a but nonetheless opening a door to a world of science and Fourier transform that has a compact support but that is mathematics. It led to more than a hundred thousand C∞. As a result 휓(푡) is also C∞ with a fast decay: the best papers in many fields of science and mathematics, books, of both worlds, illustrated in Figure 1. Yves also extended patents, industrial applications, and start-ups. He did this construction to define wavelet bases of functions of not just open a door, but also penetrated the world several variables, which led to important applications in of applications and led the way for a whole generation image processing. of . Another spectacular outcome was to These wavelet constructions could have been anec- bridge pure and applied mathematics in France, the nation dotal if Yves had not realized that these bases were of Bourbaki. unconditional bases of most classical functional spaces, But let me return to 1986. It was known that one could from Lp(ℝ) for 푝 < 1 < ∞ to Sobolev or Hölder spaces. 2 build an orthonormal basis of L (ℝ) with functions that Hence one can read the regularity of a function 푓 from the 푗 푗 are dilations by 2 and translations by 2 푛 of a single amplitude of its wavelet coefficients |⟨푓, 휓 ⟩| and their 2 푗,푛 real-valued function 휓(푡) for (푗, 푛) ∈ ℤ : decay as the scale 2푗 goes to 0. At fine scales, wavelet −푗/2 −푗 푗 coefficients become very small in domains where 푓 is {휓푗,푛(푡) = 2 휓(2 (푡 − 2 푛))} . (푗,푛)∈ℤ2 regular. If 푓 has few singularities, then at fine scales 2푗 it As a result, any function 푓 ∈ L2(ℝ) can be expanded as has few large wavelet coefficients ⟨푓, 휓푗,푛⟩. One can thus approximate a piecewise regular function 푓 by keeping 푓 = ∑ ⟨푓, 휓푗,푛⟩휓푗,푛 , relatively few large coefficients ⟨푓, 휓푗,푛⟩ in its wavelet (푗,푛)∈ℤ2 decomposition. These properties opened the possibility to transport Stéphane Mallat is professor of applied mathematics at the pure harmonic analysis to applications. It provided a Collège de France. His email address is stephane.mallat@college mathematical framework to understand a wide range of -de-france.fr. scientific and industrial applications. But Yves did not

December 2018 Notices of the AMS 1383 Finding conditions on ℎ and 푔 to recover 푥 from 푥1 and 푥2 ψ(t) became a major research issue in signal processing. Nec- essary and sufficient “conjugate mirror filter” conditions were established by Smith and Barnwell on the Fourier transforms ℎ(휔)̂ and 푔(휔)̂ of the two filters: 1 |ℎ(휔)|̂ 2 + |ℎ(휔̂ + 휋)|2 = 2 and 푔(휔)̂ = 푒−푖휔 ℎ̂∗(휔 + 휋) . Although seemingly unrelated to wavelets, these filters 0.5 then appeared to be at the core of wavelet orthonormal bases. In computer vision, similar questions appeared in the 1980s, but for other reasons. To recognize objects in 0 images, researchers proposed to detect edges, which appear at different scales. Reducing the resolution of images is important to eliminate fine details when they 0.5 are not needed. In 1983, Peter Burt developed a pyramidal algorithm that iteratively averages the image with a filter ℎ, while detecting multiscale edges with another filter 푔 1 derived from ℎ. 5 0 5 In 1986, I had left École Polytechnique to do a PhD in image processing at the University of Pennsylvania. I ∞ Figure 1. Graph of a Meyer wavelet. It is a C was learning image pyramids and conjugate mirror fil- function with a fast decay, whose translations and ters when a friend gave me Meyer’s paper on orthogonal 2 dilations define an orthonormal basis of L (ℝ). wavelets. The connections became apparent. Multiresolu- tion image pyramids could be formalized as projections in embedded subspaces of L2(ℝ), providing progressively stay in the cocoon of his native mathematics commu- finer approximations of functions as the scale decreases. nity, and crossed the door with enthusiasm. Similarly Moreover, connection across scales appeared to be gov- to Calderón and Zygmund in harmonic analysis, he be- erned by the conjugate mirror filters ℎ and 푔 studied in came the concert master of a wide research program, signal processing, which also insure orthogonality proper- providing a vision and connections between mathematics ties. As a result, wavelets generating an orthonormal basis 2 and multiple fields of science, and providing space for L (ℝ) have a Fourier transform obtained by cascading the many researchers to contribute. This generosity drew a Fourier transform of these filters: whole generation of young mathematicians along this ∞ ̂ −1/2 −1 ̂ −푝 path, without frontiers between pure and applied mathe- 휓(휔) = 2 푔(2̂ 휔) ∏ ℎ(2 휔) . 푝=2 matics, science, and engineering applications. His student Pierre-Gilles Lemarié together with Guy Battle began by This also implies that the wavelet coefficients ⟨푓, 휓푗,푛⟩ showing that other wavelet bases could be constructed of a discretized function 푓 can be computed with a fast with polynomial splines, having exponential decay. algorithm, by cascading convolutions and subsampling But this is not with ℎ and 푔, as in filter bank algorithms used by signal a story where the processing engineers. For appropriate filters, its requires a story where world was illumi- fewer operations than the Fast Fourier Transform. mathematics and nated by mathematics, I sent my manuscript to Yves, who enthusiastically but where mathemat- brought me to Chicago where he was working in the other sciences ics and other sciences office of Zygmund. In three days, he solved all remaining grew together. Engi- mathematical problems. Yves had previously realized grew together neers had not been that there was an embedding-space structure underlying wavelets. We gave a sufficient condition to guarantee waiting arms crossed. 2 Applications were too convergence in L (ℝ) of the infinite filter product and important. For telecommunications, you need to multi- obtain a wavelet that generates an orthonormal basis. plex signals—to aggregate multiple signals into a single This meant that new wavelet orthonormal bases could be constructed by defining filters that satisfy the conjugate one for transmission and then be able to separate them. In mirror conditions discovered in signal processing. A 1977, two engineers, Esteban and Galand, had introduced necessary and sufficient condition was then found by a procedure to split 푁 coefficients of a sequence 푥[푛] for Albert Cohen, a student of Yves, to relate wavelets to 0 ≤ 푛 < 푁 into two sequences of 푁/2 coefficients each, these filters. computed with a convolution of 푥 with two “filters” ℎ and At this point, many signal processing engineers looked 푔 while keeping one sample out of two: at wavelets as a useless mathematical abstraction. What 2 푥1[푛] = 푥 ⋆ ℎ[2푛] and 푥2[푛] = 푥 ⋆ 푔[2푛] . should they care about functions in L (ℝ) when all

1384 Notices of the AMS Volume 65, Number 11 computations are performed over finite sequences, with filters they had discovered before? A first answer was given by the remarkable work of . Ingrid showed that more conditions had to be imposed on conjugate mirror filters to obtain regular wavelets of compact support. The Daubechies filters reduced computations, and the regularity of the resulting wavelets appeared to be important to avoid introducing visible artifacts when compressing images. In collaboration with Albert Cohen and Christian Feauveau, she designed the wavelet filters that have been adopted in the image compression standard JPEG-2000. This com- pression algorithm decomposes images in a wavelet basis and represents the non-zero coefficients with an efficient coding scheme. It can code images with 20 times fewer bits, with barely visible artifacts, because few wavelet coefficients of large amplitude need to be coded. The position of these large coefficients are located in near edges, as shown in Figure 2. A second reason why mathematics became useful came from relations with Hölder regularity exponents, proved by Stéphane Jaffard, another student of Yves. He found necessary conditions and sufficient conditions to characterize the regularity of a function 푓 at any point 푡, from the decay of wavelet coefficients in a neighborhood of this point. It gave a mathematical framework to understand relations between wavelet coefficients and signal properties and to analyze very irregular functions such as multifractals. A surprising connection appeared with statistics. David Donoho and Iain Johnstone realized that the uncondi- tional basis properties proved by Meyer were exactly what was needed to suppress additive noise from signals having sparse wavelet coefficients. They showed that a simple thresholding, which set to zero the smallest wavelet coefficients of a noisy signal, is a nearly opti- mal non-linear estimator over large classes of signals. It opened a new field in non-linear statistics, which is still alive today. Applications in numerical analysis came from the work of Gregory Beylkin, Raphy Coifman, and Vladimir Rocklin, who proved that many singular operators are represented by sparse matrices in a wavelet orthonormal basis. Matrix multiplications can then be computed with far fewer Figure 2. Top: original image 푓. It is an array of pixels operations, for applications to calculations of solutions that encode the image intensity. Bottom: each small of partial differential equations, integral equations, and image displays as dark points the wavelet variational problems. coefficients that have a large amplitude, at a Wavelet orthonormal bases have found a multitude particular scale and orientation. Large coefficients are of applications, in chemistry, physics, many branches of located where the image intensity has a sharp information processing, and mathematics. They provided transition, near contours. Most coefficients are nearly new tools in statistics and approximation theory, to zero (white), which is why they are compressed with specify multiscale properties of random processes or to much fewer bits than the original image pixels. develop fast numerical analysis algorithms. This is how the free nomadism of a curious and incredibly talented mathematician has had such high impact and created unexpected openings between many fields within and outside mathematics.

December 2018 Notices of the AMS 1385 Image Credits All section images courtesy of Stéphane Mallat.

ABOUT THE AUTHOR Stéphane Mallat works in math- ematics applied to data analysis and machine learning, with ap- plications to image and audio recognition, and learning models of physical systems.

Stéphane Mallat Stéphane Jaffard

Time-Frequency Analysis, Chirps, and Local Regularity By the middle of the twentieth century the limitations of classical Fourier analysis for signal processing were patent. For example, musical recordings, which are a succession of notes of limited duration, clearly call for a localized Fourier analysis, as do chirps, a generic denomination which covers signals 푓 that locally look like a pure frequency, evolving slowly and smoothly with time. These defining conditions for chirps can be formalized as Figure 1. The gravitational wave as recorded by LIGO on September 14, 2015, after denoising; and the 푖휑(푡) (4) 푓(푡) = 푅푒 (푎(푡)푒 ), corresponding chirp as predicted by general where the modulation factor 푎 and the instantaneous relativity. frequency 휑 satisfy 푎′(푡) window 휑 is both smooth and well localized then, for any | | << 휑′(푡) and |휑″(푡)| << (휑′(푡))2. 푎(푡) choice of 푎 and 푏, a system of the form The most important examples of chirps are supplied by 휑(푡 − 푎푘) 푒푖 푏푛 푡 푘, 푛 ∈ ℤ gravitational waves (Figure 1): The very first one detected, is either incomplete or over-complete. Another Nobel in September 2015, was of the form |푡 − 푡 |−1/4 cos(휔|푡 − 0 laureate, K. Wilson, suggested a way to overcome this 푡 |5/8 + 휂). 0 obstruction, by allowing for a double localization around Analysis of such data should involve time-frequency opposite frequencies. Such bases, where the complex analysis. The signal is first localized by multiplying it exponentials are simply replaced by sines and cosines, with “windows.” Gaussians are a natural choice because were constructed by Daubechies, Jaffard, and Journé and of their optimal localization in space and frequency. Then supply orthonormal bases of 퐿2(ℝ) of the form: a Fourier analysis of the localized signal is performed. This idea was introduced in the 1940s by Nobel laureate 휑0,푘(푡) = 휑(푡 − 푘) 푘 ∈ ℤ, D. Gabor and leads to the short-time Fourier transform of 푘 a function 푓, defined as ⎪⎧ √2휑 (푡 − ) cos(2휋푛푡) if 푘 + 푛 ∈ 2ℤ, ⎪ 2 −2푖휋푡휉 휑푛,푘(푡) = 퐺푓(푥, 휉) = ∫ 푓(푡)휑(푡 − 푥) 푒 푑푡, ⎪⎨ ℝ ⎪ 푘 ⎪ √2휑 (푡 − ) sin(2휋푛푡) if 푘 + 푛 ∈ 2ℤ + 1. where 휑 is the window. A continuous transform is ⎩ 2 computationally inefficient, and this raises the question of S. Klimenko, who was the designer of Coherent Waveburst, the existence of appropriate orthogonal decompositions, the algorithm used in the signal processing part of the and of the corresponding fast transforms. Orthonormal gravitational wave detection, chose this basis because it bases cannot follow directly by sampling 퐺푓 because of meets the following requirements: the Balian-Low theorem (1981), which states that if a • The window 휑 can be Meyer’s [LM] scaling function (Figure 2). This choice is motivated by the fact that Stéphane Jaffard is professor of mathematics at Université Paris a window with compact support in the Fourier Est, France. His email address is [email protected]. domain allows elimination of noise components

1386 Notices of the AMS Volume 65, Number 11 where the windows 휑푛,푘 start at 푎푘 and have arbitrary lengths 푙푘. These bases found a remarkable application in speech segmentation: E. Wesfreid and V. Wickerhauser devised an entropy minimization criterion that allows the lengths of the windows to adapt to the changes in the signal and thus performs automatic segmentation. A different point of view can be developed for modeling chirps, where the instantaneous frequency 휑(푡) in (4) diverges at a point 푥0, leading to functions that have a singularity at 푡0. This leads to pointwise singularities that are typically of the form 1 풞 (푡) = |푡 − 푡 |ℎ sin ( ). ℎ,훽 0 훽 |푡 − 푡0| Based on the heuristic supplied by such toy-examples, Meyer [JM1] developed a general framework for such behaviors, where the sine function is replaced by a fairly arbitrary oscillating function, and showed that they are characterized by precise estimates on the wavelet coefficients. A remarkable application is supplied by the analysis of Riemann’s function ∞ sin(휋푛2푡) ℛ(푡) = , ∑ 2 푛=1 푛 which was proposed by Riemann as a candidate for a continuous nowhere differentiable function. It took a century to disprove Riemann’s intuition and show that ℛ is differentiable at rational points of the form (2푝 + 1)/(2푞+1) (J. Gerver, 1970). Meyer considerably improved Figure 2. Meyer’s scaling function and an element of our comprehension of the behavior of Riemann’s function the corresponding Wilson basis. Meyer’s scaling in the neighbourhood of these points by exhibiting a function stands at the junction between two major complete chirp expansion about 푡 = 1: extensions of Fourier analysis at the end of the 0 twentieth century, time-scale and time frequency 푡 푘+1/2 1 (−푘) ℛ(1 + 푡) ∼ − + ∑ |푡| 푔푘 ( ) where 푔푘 ∼ ℛ analysis. It is the building stone that allows 2 푘≥1 푡 generation of orthonormal bases in both settings. (see Figure 3; 푔푘 is essentially a primitive of order 푘 of the Riemann function itself!). that lie away from the main Fourier area of These explorations were the first stones that paved the interest. way towards a classification of the pointwise singularities • Fast decomposition algorithms. of everywhere irregular functions, now referred to as • Fast translation algorithms: one needs to compare multifractal functions. Meyer [M1], [M2] introduced a new the two signals recorded in the two LIGO detectors, regularity exponent, the weak scaling exponent, which has which do not arrive exactly at the same time, and the remarkable property of being covariant with respect therefore need to be shifted. to fractional primitives or derivatives. He revisited Wilson • Gravitational waves are sparse in Wilson bases. bases, constructed new bridges with wavelet decompo- This procedure is now referred to as the Wilson- sitions, and built a whole variety of bases that can be Daubechies-Meyer transform. tailored to particular chirp behaviors. With Jaffard [JM2] Variants of Wilson bases were independently con- he uncovered an unexpected relationship between spar- structed by H. Malvar. They also are time-frequency sity in a wavelet basis and pointwise regularity: They orthonormal bases of 퐿2(ℝ) of the form showed that generically, in the sense of Baire category, 1 functions that have sparse wavelet expansions (the de- 휑 (푡) = 휑(푡 − 푘) cos [휋 (푛 + ) (푡 − 푘)] 푘, 푛 ∈ ℤ. creasing rearrangement of the sequence of their wavelet 푛,푘 2 coefficients has fast decay) are multifractal, and they de- The corresponding decomposition, called MDCT (Modified termined their multifractal spectrum (i.e. the Hausdorff Discrete Cosine Transform), is currently used in audio dimensions of their pointwise Hölder singularities); Jaf- compression formats, e.g. MP3 or MPEG2 AAC. Malvar fard, Meyer, and Ryan [JMR] provide a user-friendly review bases of adaptive lengths were introduced by R. Coifman on all these topics. and Y. Meyer [CM]: They are of the form Meyer has been at the center of the effervescent and 휋 1 prolific multidisciplinary network that made the success 휑푛,푘(푡) = 휑푘(푡) cos [ (푛 + ) (푡 − 푎푘)] , of wavelets. He is a living proof that the limits between 푙푘 2

December 2018 Notices of the AMS 1387 [M2] Y. Meyer, Oscillating patterns in image processing and non- linear evolution equations, University Lecture Series, Vol. 22, AMS (2000). MR1852741

Image Credits Figure 1 dx.doi.org/10.7935/K5MW2F23. Figures 2, 3, and author photo courtesy of Stéphane Jaffard.

ABOUT THE AUTHOR Stéphane Jaffard did his PhD un- der the supervision of Yves Meyer at the very beginning of wavelet theory. His main subject of re- search is harmonic and multifrac- tal analysis and their applications in signal and image processing. Stéphane Jaffard

Alexander Olevskii

Meyer’s Sets and Related Problems Many years ago Yves Meyer introduced remarkable con- cepts and constructions, which allowed him to solve Figure 3. The Riemann function, and a zoom around some important problems in harmonic analysis. After many years these concepts and ideas remain important the chirp at 푡0 = 1. Meyer exhibited a complete chirp expansion near the rare differentiable points. and are applicable to new problems. Below I am going to focus on some aspects of this new development, in which Meyer is one of the main actors. pure and applied science do not exist, and he repeatedly has shown that deep mathematical concepts can be the Poisson Summation Formula key to spectacular applications. He is famous for being We start with the classical Poisson formula: a passionate lecturer. Extremely generous, he always pushed his many students and collaborators to the front; 1 ∗ ∑ 푓(휆)̂ = ∑ 푓(휆 ). each of them can testify to the importance he laid to 휆∈Λ |Λ| 휆∗∈Λ∗ the transmission of science and the values of intellectual Here Λ is a lattice in ℝ푛, |Λ| is the volume of its fundamen- integrity, humanism, and tolerance. In a time driven by ∗ material values and short range-profit, Meyer stands as tal parallelepiped, Λ is the dual lattice, 푓 is any function ̂ an example for younger generations of scientists. in the Schwartz class 푆(ℝ), and 푓 its Fourier transform. Equivalently, if 휇 is the sum of unit masses on a lattice Cited Papers of Meyer (a Dirac comb), then its Fourier transform ̂휇 (in sense of [CM] R. Coifman and Y. Meyer, Remarques sur l’analyse de distributions) is again a Dirac comb. The spectrum of 휇 Fourier à fenêtre, C. R. Acad Sci. Paris, Sér I, Math. Vol. 312, (:= the support of ̂휇) in this case is the dual lattice. pp. 259–261(1991). MR1089710 The Poisson formula has many applications. In partic- [JM1] S. Jaffard and Y. Meyer, Wavelet methods for pointwise ular, it plays a seminal role in the 푋-ray diffraction. regularity and local oscillations of functions, Memoirs of the The following problem is important: do there exist some AMS, Vol. 123, no. 587 (1996). MR1342019 other discrete measures 휇 such that the distributional [JM2] S. Jaffard and Y. Meyer, On the pointwise regularity of Fourier transform 휇 is also a discrete measure? Here functions in critical Besov spaces, Vol. 175, N. 2, pp. 415–434 (2000). MR1780484 one looks for measures 휇 that are not finite sums of [JMR] S. Jaffard, Y. Meyer and R. Ryan, Wavelets, tools for Dirac combs, translated and modulated. This problem Science and Technology, SIAM (2001). MR1827998 was studied in the 1950s by J.-P. Kahane - S. Mandelbrot [LM] P.-G. Lemarié and Y. Meyer, Ondelettes en bases hilberti- and by J.-P. Guinand. ennes, Revista Matematica Iberoamericana Vol. 2, pp. 1–18 (1986). MR864650 [M1] Y. Meyer, Wavelets, Vibrations and Scalings, CRM Mono- Alexander Olevskii is professor of mathematics at Tel Aviv Uni- graph Series, Vol. 9, AMS (1998). MR1483896 versity. His email address is [email protected].

1388 Notices of the AMS Volume 65, Number 11 Model Sets and Reconstruction of Signals Here I discuss a new application of the model sets as sampling sets. For a bounded set 푆 ⊂ ℝ, denote by 푃푊푆 (the Paley– Wiener space with spectrum 푆) the space of functions (signals) 푓 ∈ 퐿2(ℝ) whose Fourier transform is supported on 푆. Given a uniformly discrete set Λ, one wishes to recover every 푓 ∈ 푃푊푆 from its values (samples) on Λ. The following inequality would allow one to solve this problem in a stable way: 2 2 (6) ‖푓‖ ≤ 퐶(푆, Λ) ∑ |푓(휆)| , ∀푓 ∈ 푃푊푆. 휆∈Λ Given 푆, for which sets Λ does this condition hold? When 푆 is an interval, A. Beurling proved that the validity of (2) is essentially equivalent to the condition that Λ Figure 1. The Voronoi cells of a model set provide a must be “dense” in a certain sense. H. Landau extended nonperiodic tiling, such as this Penrose tiling in the necessity of the density condition to the case of Keskuskatu Square, Helsinki. disconnected spectra. However, in general no reasonable sufficient condition can be expressed in terms of some density of Λ. The geometric structure of 푆 plays a crucial Meyer’s Model Sets role. Hence, a special construction of a “sampling set” Λ Yves Meyer [1] discovered a family of non-periodic discrete is needed in the general case. Is it possible to find Λ so that (2) holds for every measures with point spectrum. spectrum 푆 of given measure, independently of its struc- The construction is based on a concept of model sets, ture and localization? This problem was put forward by obtained from lattices by the cut-and-project procedure. A. Olevskii and A. Ulanovskii, who proved in 2006 that Here is the definition in the simplest case. In the plane for compact spectra such “universal sampling sets” do (푥, 푦), take a lattice Γ in the general position. Fix a set 푄 (a exist. Such sets were constructed as certain perturbations “window”) that is a finite union of intervals on the 푦-axis. of lattices. A new remarkable proof of this result was Consider the set of lattice points that lie in ℝ × 푄, and found in 2008 by B. Matey and Y. Meyer, who showed that project them onto the 푥-axis. This way one gets a Meyer appropriate model sets satisfy the same property. For model set 푀 in ℝ. A similar construction can be done in more about the subject see Olevskii and Ulanovskii [2]. ℝ푛, starting with a lattice Γ in ℝ푛 × ℝ푚. Here are some properties of these model sets 푀: Quasicrystals - Every 푀 is a nonperiodic uniformly discrete set. It is surprising that Meyer’s concept of model sets turned Uniform discreteness means that the pairwise distances out to be a recognized mathematical model for a physical between the different elements are separated from zero. phenomenon discovered much later. In the early 1980s - The “Voronoi cells” corresponding to 푀 provide a 푛 Dan Shechtman and his colleagues found aperiodic atomic tiling of ℝ by non-overlapping translates of a finite family structures whose diffraction patterns consist of isolated of polyhedra. A well-known example of such a tiling is due spots. For this discovery he received the 2011 Nobel Prize. to Penrose (Figure 1). De Bruijn proved that the Penrose Such a phenomenon has been perceived as a contradiction tiling can be obtained by the cut-and-project procedure to classical crystallography. The structures got the name from a 5-D lattice. “quasicrystals.” - Every model set 푀 supports a measure 휇 with point A parallel with Meyer’s model sets is clear: each 푀 spectrum. is an aperiodic uniformly discrete set and it supports Here is a proof of the last property in the simplest a measure 휇 with point spectrum. Since 푓̂ in (1) is a case: let 푀 be a 1-D model obtained from a lattice Γ and Schwartz function, one can prove that the “visible” part window 퐼. Take a function 푓 ∈ 푆(ℝ) supported on 퐼. Set of the spectrum, which is the set of spectral atoms whose size exceeds a fixed small number, is uniformly discrete 휇 ∶= 푓(푦)훿 . ∑ 푥 (consists of spots!). 푥,푦∈Γ Similarly, discrete aperiodic measures with pure point Then 휇 is supported on 푀. Using the Poisson formula in spectrum are often called “Fourier quasicrystals.” Meyer ℝ2 one can verify that sets provide important examples of such sets. On the other hand, if both the support and the spectrum ̂ (5) ̂휇= ∑ 푓(푣)훿푢. of a measure 휇 in ℝ푛 are uniformly discrete, then it is a (푢,푣)∈Γ∗ finite sum of translates and modulates of a Dirac comb, So, ̂휇 is a discrete measure with a dense support 푆. as proved by N. Lev and A. Olevskii [4]. Their result shows

December 2018 Notices of the AMS 1389 Cited Papers of Meyer [1] Y. Meyer, Algebraic numbers and harmonic analysis, North– Holland, Amsterdam, 1972. MR0485769 [2] A. Olevskii and A. Ulanovskii, Functions with Disconnected Spectrum: Sampling, Interpolation, Translates, University Lecture Series, AMS, 2016. MR3468930 [3] J. C. Lagarias, Mathematical quasicrystals and the prob- lem of diffraction. Directions in mathematical quasicrystals, In: CRM Monograph Series, vol. 13, pp. 61–93, AMS, 2000. MR1798989 [4] N. Lev and A. Olevskii, Quasicrystals and Poisson’s sum- mation formula, Invent. Math. 200 (2015), no. 2, 585–606. MR3338010 [5] Y. Meyer, Measures with locally finite support and spec- trum, Proc. Natl. Acad. Sci. USA 113 (12) (2016) 3152–3158. MR3482845

Photo Credits Figure 1 courtesy of YIT Corporation. Photo of Dan Shechtman by Technion–Israel Institute of Tech- nology (File:Shechtman_(2).jpg) [CC BY-SA 3.0 (https: //creativecommons.org/licenses/by-sa/3.0)], via Wiki- media Commons. Photo of Alexander Olevskii used with his permission.

Dan Shechtman, Nobel Laureate for the discovery of quasicrystals, for which Meyer’s model sets serve as mathematical models. ABOUT THE AUTHOR Alexander Olevskii received his ScD degree in 1966 at Moscow that in some natural situations a quasicrystal can not be State University. His research has observed. been concentrated on problems For 푛 = 1 the result is true in full generality, in of classical harmonic analysis and higher dimensions for positive measures. In particular, related areas. the support of such measures is periodic. This answered a problem raised by J. Lagarias [3]. Alexander Olevskii Nonclassical Poisson Formulas The characterization of measures as the sums of Dirac Albert Cohen combs remains true even if the support of 휇 is just locally finite (whilst the spectrum is still uniformly discrete). The Emergence of Sparse Analysis through the On the other hand, there are measures such that both Works of Yves Meyer the support and the spectrum are locally finite but not Since the turn of the century, the concept of sparsity has periodic [4]. Further examples of this kind were found by become prominent in many areas of applied mathematics, M. Kolountzakis and Y. Meyer. at the crossroad with other disciplines such as signal Each such measure generates a Poisson-type summa- and image processing. Loosely speaking, sparse approx- tion formula. A spectacular one, due to Meyer [5], has the imation refers to the possibility of representing certain form complex objects by very few numbers up to a controlled ̂휇= 푐휇, loss of accuracy. This task is usually made possible once a proper mathematical representation of these objects has where 휇 is a purely atomic measure on ℝ supported at been provided, for example through a change of basis. the points ±|푘 + 푎|, where 푎 ∈ ℝ3∖ ℤ3, 푘 runs over ℤ3, Such economical representations play an obvious role and the masses of atoms are effectively defined. in data compression. They also found powerful applica- Recently Meyer has moved forward substantially this tions in statistical estimation and inverse problems. The line of research. He found a series of non-classic Poisson spectacular development of compressed sensing led to formulas with special arithmetics of the nodes. a deeper mathematical understanding of how sparsity Note that all examples known so far, in one way or another, are based on the classic Poisson formula. It Albert Cohen is professor of mathematics at Laboratoire Jacques- would be interesting to know whether there are examples Louis Lions, Sorbonne Université, Paris, France. His email address of a different nature. is [email protected].

1390 Notices of the AMS Volume 65, Number 11 could be exploited in many such applications. While Yves of orthonormal wavelet bases that belong to the Schwartz Meyer’s work was not primarily focused on such areas, class his contributions played a key role in the emergence and (3) spreading of this concept and in the development of its 풮(ℝ) ∶= {푓 ∈ 퐶∞(ℝ) ∶ sup |푥|푘|푓(푙)(푥)| < ∞, 푘, 푙 ≥ 0}, mathematical foundations. 푥∈ℝ The present paper includes parts of a more detailed and are therefore well localized both in time and frequency. survey on the research works of Meyer published in Another orthonormal wavelet basis with smoothness and the Abel volume [2], which can be consulted for further localization properties had been obtained earlier in the references. work of Jan-Olov Strömberg. By its elegant simplicity, Meyer’s construction was celebrated as a milestone. A Golden Decade A major turning point occurred in 1986 when Stéphane Mallat [5] introduced the natural framework that was the The process of analyzing and representing an arbitrary key to the general construction of wavelets, as well as function 푓 by means of more elementary functions has to fast decomposition and reconstruction algorithms. A been at the heart of fundamental and applied advances multiresolution approximation is a dense nested sequence in science and technology for several centuries. In more of approximation spaces recent decades, implementation of this process on com- 2 puters by fast algorithms has become of ubiquitous use in (4) {0} ⋯ ⊂ 푉푗−1 ⊂ 푉푗 ⊂ 푉푗+1 ⊂ ⋯ 퐿 (ℝ), scientific computing. In the foundational example of the generated by a so-called scaling function 휑 in the sense 푗/2 푗 univariate Fourier expansions these elementary building that (2 휑(2 ̇−푘))푘∈ℤ is a Riesz basis of 푉푗. A countable blocks are the 1-periodic complex exponential functions family (푒푘)푘∈ℱ in a Hilbert space 푉 is called a Riesz basis defined by if it is complete and there exists constants 0 < 푐 ≤ 퐶 < ∞ 푖2휋푛푡 such that (1) 푒푛(푡) = 푒 , 푛 ∈ ℤ, (5) 푐 |푥 |2 ≤ ‖ 푐 푒 ‖2 ≤ 퐶 |푥 |2 2 ∑ 푘 ∑ 푘 푘 푉 ∑ 푘 and they form an orthonormal basis of 퐿 (]0, 1[). 푘∈ℱ 푘∈ℱ 푘∈ℱ The functions 푒푛 are perfectly localized in frequency holds for any finitely supported coefficient sequence but have no localization in time, since their modulus is (푥푘)푘∈ℱ, and therefore by density for any sequence in equal to 1 independently of 푡. This property constitutes ℓ2(ℱ). a major defect when trying to efficiently detect the local In this framework, the generating wavelet 휓 is then frequency content of functions by means of Fourier anal- constructed so that the functions (2푗/2휓(2푗 ̇−푘)) con- ysis. It also makes Fourier representations numerically 푘∈ℤ stitute a Riesz basis for a complement 푊푗 of 푉푗 into ineffective for functions that are not smooth everywhere. 푉푗+1. This approach allowed in particular the construc- For example, the Fourier coefficients 푐푛(푓) of a 1-periodic tion of compactly supported orthonormal wavelets by piecewise smooth function 푓 with a jump discontinuity at Ingrid Daubechies [4]. −1 a single point 푡0 ∈ [0, 1] decay like |푛| , which affects The multiresolution analysis framework was immedi- the convergence of the Fourier series on the whole of ℝ. ately extended by Stéphane Mallat and Yves Meyer to Wavelet bases are orthonormal bases of 퐿2(ℝ) with the multivariate functions, by tensorizing the spaces 푉푗 in general form the different variables. This leads to multivariate wavelet 푗/2 푗 bases of the form (2) 휓푗,푘(푥) = 2 휓(2 푥 − 푘), 푗 ∈ ℤ, 푘 ∈ ℤ, 휀 푑푗/2 푗 푑 (6) 휓휀,푗,푘 = 2 휓휀(2 ⋅ −푘), 푗 ∈ ℤ, 푘 ∈ ℤ , where 휓 is a function such that ∫ℝ 휓 = 0. Due to their 푑 increased resolution as the scale level 푗 tends to +∞, for 휀 = (휀1, … , 휀푑) ∈ {0, 1} ∖ {(0, … , 0)}, where they are better adapted than Fourier bases for capturing (7) local phenomena. The most basic example of the Haar 휓휀(푥1, … , 푥푑) ∶= 휓휀1 (푥1) ⋯ 휓휀푑 (푥푥), 휓0 ∶= 휑, 휓1 ∶= 휓. system, which corresponds to 휓 = Χ[0,1/2[ − Χ[1/2,1[, has Adaptation of these bases to more general bounded do- been known since 1911. In this example, the function 휓 mains of ℝ푑 as well as to various types of manifolds came suffers from a lack of smoothness, also reflected by the in the following years, again based on the multiresolution slow decay at infinity of its Fourier transform. concept. The construction of modern wavelet theory took place All these developments are well documented in the during the decade of 1980-1990. It benefited greatly classical textbooks [4],[8]. A major stimulus was the from ideas coming from various (and sometimes com- vision of powerful applications in areas as diverse as pletely disjoint) sources: theoretical harmonic analysis, signal and image processing, statistics, and fast numer- approximation theory, computer vision and image anal- ical simulation. This perspective was confirmed in the ysis, computer aided geometric design, digital signal following decades. Meyer played a key role in identifying processing. One of the fundamental contributions of the mathematical properties that are of critical use in Yves Meyer was to recognize and organize these separate such applications, in particular the ability of wavelets to developments into a unified and elegant theory. characterize a large variety of function spaces. As we next After some attempts to disprove their existence, Meyer discuss, these properties naturally led to the concept of turned the table in 1985 and gave a beautiful construction sparse approximation. He was also one of the first to point

December 2018 Notices of the AMS 1391 푝 out some intrinsic limitations of wavelets and promote having distributional derivatives up to order 푚 in 퐿loc: alternative strategies. apart from the Hilbertian case 푝 = 2, for which one has 푚,2 2푚 2 (10) 푓 ∈ 푊per (]0, 1[) ⟺ ∑ (1 + |푛| )|푐푛(푓)| < ∞, 푛∈ℤ no such characterization is available when 푝 ≠ 2. Meyer showed that, in contrast to the trigonometric system, wavelet bases are unconditional bases for most classical function spaces that are known to possess one. The case of 퐿푝 spaces for 1 < 푝 < ∞ is treated by the following observation: if the general wavelet 휓 has 퐶1 smoothness, the multiplier operator (9) by a bounded sequence belongs to a classical class of integral operators introduced by Calderón and Zygmund, which are proved to act boundedly in 퐿푝(ℝ푑). Conversely, Meyer showed that Calderón–Zygmund operators are “almost diagonalized” by wavelet bases in the sense that the resulting matrices have fast off-diagonal decay. This property plays a key role in the numerical treatment of partial differential and integral equations by wavelet methods. Yves Meyer receives Abel Prize from King Harald of The characterization of more general function spaces Norway. by the size properties of wavelet coefficients is particularly simple for an important class of smoothness spaces introduced by Oleg Besov. There exist several equivalent Function Spaces and Unconditional Bases definitions of Besov spaces. The original one uses the 푝 When expanding a function 푓 into a given basis (푒푛)푛≥0, 푚-th order 퐿 -modulus of smoothness a desirable feature is that the resulting decomposition 푚 (11) 휔푚(푓, 푡)푝 ∶= sup ‖Δℎ 푓‖퐿푝 , 푓 = ∑푛≥0 푥푛푒푛 is numerically stable: operations such as |ℎ|≤푡 perturbations, thresholding or truncation of the coeffi- 푚 where Δℎ is the 푚-th power of the finite difference cients 푥푛 should effect the norm of 푓 in a well-controlled operator Δℎ ∶ 푓 ↦ 푓(⋅ + ℎ) − 푓. For 푠 > 0, any integer manner. Such prescriptions can be encapsulated in the 푚 > 푠, and 0 < 푝, 푞 < ∞, a function 푓 ∈ 퐿푝(ℝ푑) belongs 푠,푝 푑 following classical property. to the space 퐵푞 (ℝ ) if and only if the function 푔 ∶ 푡 → −푠 푞 A sequence (푒푛)푛≥0 in a separable Banach space 푋 is 푑푡 푡 휔푚(푓, 푡)푝 belongs to 퐿 ([0, ∞[, 푡 ). One may use an unconditional basis if the following properties hold: (12)

‖푓‖ 푠,푝 ∶= ‖푓‖ 푝 + |푓| 푠,푝 , with |푓| 푠,푝 ∶= ‖푔‖ 푞 푑푡 , (i) It is a Schauder basis: every 푓 ∈ 푋 admits a unique 퐵푞 퐿 퐵푞 퐵푞 퐿 ([0,∞[, 푡 ) expansion ∑푛≥0 푥푛푒푛 that converges towards 푓 in as a norm for such spaces, also sometimes denoted 푋. 푠 푝 푑 푠,푝 푑 by 퐵푞(퐿 (ℝ )). Roughly speaking, functions in 퐵푞 (ℝ ) (ii) There exists a finite constant 퐶 ≥ 1 such that for have up to 푠 (integer or not) derivatives 퐿푝. The third any finite set 퐹 ⊂ ℕ, index 푞 may be viewed as a fine tuning parameter, which (8) appears naturally when viewing Besov spaces as real |푥푛| ≤ |푦푛|, 푛 ∈ 퐹 ⟹ ‖ ∑ 푥푛푒푛‖ ≤ 퐶‖ ∑ 푦푛푒푛‖ . interpolation spaces between Sobolev space: for example, 푋 푋 푛∈퐹 푛∈퐹 with 0 < 푠 < 푚, 푠 푝 푝 푚,푝 The property (8) means that membership of 푓 in 푋 only (13) 퐵푞(퐿 ) = [퐿 , 푊 ]휃,푞, 푠 = 휃푚. depends on the moduli of its coordinates |푥푛|. In other 푠,∞ 푠 Particular instances are the Hölder spaces 퐵∞ = 퐶 and words, multiplier operators of the form 푠,푝 푠,푝 Sobolev spaces 퐵푝 = 푊 , when 푠 is not an integer or (9) 푇 ∶ ∑ 푥푛푒푛 → ∑ 푐푛푥푛푒푛 when 푝 = 2 for all values of 푠. 푛≥0 푛≥0 Let (휓휆) denote a multivariate wavelet basis of the type (6), where for simplicity 휆 denotes the three indices should act boundedly in 푋 if (푐푛)푛≥0 is a bounded sequence. Orthonormal and Riesz bases are obvious (푒, 푗, 푘) in (6). Denoting by |휆| ∶= 푗 = 푗(휆) the scale level of 휆 = (푒, 푗, 푘), we consider the expansion examples of unconditional bases in Hilbert spaces. While the trigonometric system (1) is a Schauder basis (14) 푓 = ∑ 푑휆휓휆, in 퐿푝(]0, 1[) when 1 < 푝 < ∞, it does not constitute an |휆|≥0 unconditional basis when 푝 ≠ 2, and it is thus not possible where the coarsest scale level |휆| = 0 also includes the 푝 to characterize the space 퐿 through a property of the translated scaling functions that span 푉0. 푠,푝 푑 moduli of the Fourier coefficients. The same situation is The characterization of 퐵푞 (ℝ ) established by Meyer met for classical smoothness spaces, such as the Sobolev for such expansions requires some minimal prescriptions: 푚,푝 spaces 푊per (]0, 1[) that consist of 1-periodic functions one assumes that for an integer 푟 > 푠 the univariate

1392 Notices of the AMS Volume 65, Number 11 generating wavelet 휓 and scaling functions 휑 that defines (6) have derivatives up to order 푟 that decay sufficiently fast at infinity, for instance faster than any polynomial +∞ 푘 rate, and that ∫−∞ 푡 휓(푡)푑푡 = 0 for all 푘 = 0, 1, … , 푟 − 1. Then, one has the norm equivalence

푠,푝 푞 (15) ‖푓‖퐵푞 ∼ ‖휀‖ℓ , where the sequence 휀 = (휀푗)푗≥0 is defined by

푑 푑 푠푗 ( − 푝 )푗 (16) 휀푗 ∶= 2 2 2 ‖(푑휆)|휆|=푗‖ℓ푝 . A closely related characterization of Besov spaces uses the Littlewood-Paley decomposition

(17) 푓 = 푆0푓 + ∑ Δ푗푓, Δ푗푓 ∶= 푆푗+1푓 − 푆푗푓, 푗≥0

−푗 where 푆̂푗푓(휔) ∶= Θ(2 휔)푓(휔)̂ with Θ a smooth com- pactly support function with value 1 for |휔| ≤ 1. It has the same form as above, with now 휀푗 ∶= ‖Δ푗푓‖퐿푝 . In the wavelet characterization the dyadic blocks are further dis- cretized into the local components 푑휆휓휆. Similar results have been obtained for Besov spaces defined on general bounded Lipschitz domains Ω ⊂ ℝ푑 with wavelet bases adapted to such domains. The norm equivalence (15) shows that membership of 푓 in Besov spaces is characterized by simple weighted summability properties of its wavelet coefficients. In the particular case 푞 = 푝, this equivalence takes the very simple form Yves Meyer at his Abel Prize Ceremony. (푠+ 푑 − 푑 )|휆| 푠,푝 2 푝 푝 (18) ‖푓‖퐵푝 ∼ ‖(2 푑휆)‖ℓ . As an immediate consequence, classical results such as 푠,푝 2 푑 푑 or dictionary expansion. Sparse approximation in uncon- the critical Sobolev embedding 퐵푝 ⊂ 퐿 for 푠 = 푝 − 2 ditional bases was identified by David Donoho as a key take the trivial form of the embedding ℓ푝 ⊂ ℓ2 for 푝 < 2. ingredient for powerful applications in data compression While this embedding is not compact, an interesting and statistical estimation, in particular through thresh- approximation property holds: when retaining only the 푛 olding algorithms that he developed jointly with Iain largest coefficients in the wavelet decomposition of 푓, the Johnstone, Gérard Kerkyacharian, and Dominique Picard. resulting approximation 푓푛 satisfies Pushed into the forefront by the work of Meyer, Donoho, −푟 푠 (19) ‖푓 − 푓 ‖ 2 ≤ 퐶푛 ‖푓‖ 푠,푝 , 푟 ∶= . and DeVore, sparse approximation became within a few 푛 퐿 퐵푝 푑 years a prominent concept in signal processing and This follows immediately from the fact that, for 푝 < 2, scientific computing. the decreasing rearrangement of (푑푘)푘≥1 of a sequence 푝 (푑휆) ∈ ℓ satisfies the tail bound Taking Off from the Wavelet World 1/2 1 1 2 2 − 푝 −푟 (20) ( ∑ 푑푘 ) ≤ 푛 ‖(푑휆)‖ℓ푝 . The estimate (20) shows that the rate 푛 of best 푛-term 푘≥푛 approximation of a function, using an orthonormal or 푝 푝 This last estimate shows that ℓ summability governs the Riesz basis (휓휆), is implied by the ℓ summability of is 1 1 compressibility of a sequence, in the sense of how fast it coefficient sequence (푑휆) with 푝 = 푟 + 2 . A more refined can be approximated by 푛-sparse sequences. The theory analysis shows that this rate is exactly equivalent to the 푝 of best 푛-term wavelet approximation, generalizing the slightly weaker property that (푑휆) belongs to 푤ℓ , which above remarks, has been developed by Ronald DeVore and means that its decreasing rearrangement has the decay his collaborators, in close relation with other nonlinear property approximation procedures such as free knot splines or (21) 푑 ≤ 퐶푘−1/푝. rational approximation. 푘 A particularly useful feature of nonlinear wavelet ap- The spaces ℓ푝 and 푤ℓ푝 are thus natural ways of quan- proximation is that piecewise smooth signals, such as tifying sparsity of a function when decomposed in an images, can be efficiently captured since the large coeffi- arbitrary orthonormal or Riesz basis (휓휆) of a Hilbert cients are only those of the few wavelets whose supports space. contain the singularities. This is an instance of sparse ap- In the case of wavelet bases, these summability prop- proximation which aims to accurately capture functions erties are equivalent to Besov smoothness. From an by a small number of well chosen coefficients in a basis applicative point of view, a more natural question is:

December 2018 Notices of the AMS 1393 given a class of functions 풦 in a Hilbert space, which representation methods have since then been proposed basis should be used in order to obtain the sparsest and studied for better capturing geometry: contourlets, possible representations of the element of this class? In shearlets, bandlets, anisotropic finite elements. view of the previous observations, this basis should be Returning to univariate signals, one object of long- picked so that the coefficient sequence of any element of term interest to Meyer is signals whose “instantaneous 푝 풦 belongs to 푤ℓ , for the smallest possible value of 푝. frequency” evolves with time in some controlled manner. One class of particular interest for modeling real images Such signals are called chirps and take the general form is the space BV(푄) of bounded variation functions on the unit cube 푄 ∶= [0, 1]2 that consists of functions (26) 푓(푡) = 푅푒 (푎(푡)푒푖휑(푡)), 푓 ∈ 퐿1(푄) such that ∇푓 is a finite measure. In particular, if Ω ⊂ 푄 is a set of finite perimeter, the characteristic ′ where | 푎 (푡) | << |휑′(푡)| and |휑″(푡)| << |휑′(푡)|2. function ΧΩ belongs to BV(푄). More generally, piecewise 푎(푡) smooth images with edge discontinuities across curves Typical examples of chirp are ultrasounds emitted by of finite length have bounded variation. While the space bats and recordings of voice signals, but the most famous BV(푄) admits no unconditional basis, we showed together one is the gravitational wave signal first detected in 2015, with DeVore, Pencho Petrushev, and Hong Xu that it can be which has for a large part the behaviour “almost” characterized by its decomposition in a bivariate −1/4 5/8 wavelet basis (휓휆) in the following sense: if 푓 = ∑ 푑휆휓휆, (27) 푓(푡) ∼ |푡 − 푡0| cos(|푡 − 푡0| + 휑0). one has Wavelets are not the right tool for sparse representation (22) (푑 ) ∈ ℓ1 ⟹ 푓 ∈ BV(푄) ⟹ (푑 ) ∈ 푤ℓ1. 휆 휆 of chirps. Time-frequency analysis such as the short- In view of the previous remarks this shows that for time Fourier transform provides more natural tools, once general images of bounded variation, the rate of best proper orthonormal bases have been provided. −1/2 푛-term approximation in wavelet bases is 푛 . It can The first example of such a basis was originally sug- be shown that this rate is also the best that can be gested by Kenneth Wilson and formalized by Ingrid achieved by any basis. In particular, no polynomial rate Daubechies, Stéphane Jaffard, and Jean-Lin Journé: an can be achieved when using the Fourier basis. In this orthonormal basis of 퐿2(ℝ) is constructed by taking for sense wavelets appear as the optimal tool for piecewise all 푛 ∈ ℤ the functions 휑 (푡) = 휑(푡 − 푛) and smooth images with edges of finite length. 0,푛 The situation becomes quite different if one considers (28) 푛 images with edges enjoying some geometric smoothness ⎪⎧√2휑(푡 − ) cos(2휋푙푡) 푙≥0, 푙+푛∈2ℤ, ⎪ 2 in addition to finite length. The simplest model consists 휑 (푡) = 푙,푛 ⎨ of piecewise constant images with straight edges. For ⎪ 푛 ⎪√2휑(푡 − ) sin(2휋푙푡) 푙>0, 푙+푛∈2ℤ+1. such images, Meyer [9] noticed that the decreasingly ⎩ 2 rearranged Fourier coefficients decay at rate −1 The generating function 휑 should satisfy certain symme- (23) 푐푘 ≤ 퐶푘 log(푘), try properties. One possible choice is the scaling function therefore comparable to wavelet coefficients up to the associated with the orthonormal wavelet basis of Meyer, logarithmic factor. When going to a higher dimensional which is defined by 휑̂ = √휅, where 휅 is the symmetric and cube 푄 = [0, 1]푑, this rate persists for Fourier coefficients smooth cut-off function. A variant of this system, where while wavelet representations become less effective. the family is made redundant by additional dilations, was A more elaborate model consists of the functions which are piecewise 퐶푚 with edge discontinuities having proposed in the papers of Sergei Klimenko and his col- 퐶푛 geometric smoothness. For such classes 풦(푛, 푚), both laborators for the sparse representation of gravitational wavelet and Fourier decompositions can be outperformed waves and used for their detection. by more sophisticated representations into functions that In recent years, sparse approximation has also been in- combine local support with directional selectivity. One tensively exploited for the treatment of high-dimensional representative example are the curvelets, introduced by approximation. Problems that involve functions of a very Emmanuel Candès and Donoho, which have the form large number of variables are challenged by the so-called 3푗/2 푗 푙 푗 “curse of dimensionality”: the complexity of standard (24) 휓휆 = 2 휓(퐷 푅푗 ⋅ −푘), 푘 ∈ ℤ, 푙 = 0, … , 2 − 1, discretization methods blows up exponentially as the 4 0 where 퐷 is the anisotropic dilation matrix (0 2) and 푅푗 number of variables grows. Such problems arise naturally −푗−1 the rotation of angle 2 휋. The anisotropic scaling and in learning theory, partial differential equations, and nu- angular selectivity allow to better capture the geometry merical models depending on parametric or stochastic of edges, leading to improved sparsity: for example, it is variables. Wavelet representations are not well suited for known that extracting sparsity in such high-dimensional applications. 푝 (25) 푓 ∈ 풦(2, 2) ⟹ (푑휆) ∈ ℓ , 푝 > 2/3, This motivated the development of better adapted tools, where 푑휆 are the coefficients of 푓 in the curvelet ex- such as sparse grids, sparse polynomials, and sparse pansion. The value 2/3 is optimal for this class. Other tensor formats.

1394 Notices of the AMS Volume 65, Number 11 Compressed Sensing and Quasi-Crystals matrices satisfying RIP of order 푘 require the non-optimal The most usual approach for obtaining a sparse ap- regime 푚 ∼ 푘2 up to logarithmic factors. proximation of a discrete signal represented by a vector In recent years, Meyer studied the problem of sampling 푥 ∈ ℝ푁 is to choose an appropriate basis, compute the continuous bandlimited signals with unknown Fourier coefficients of 푥 in this basis, and then retain only the 푛 support, which may be viewed as an analog counterpart largest of these, with 푛 << 푁. to the above compressed sensing problem. For any set 푑 This approximation process is adaptive since the in- 퐸 ⊂ ℝ , we denote by ℱ퐸 the Paley-Wiener space of dices of the retained coefficients vary from one signal to functions 푓 ∈ 퐿2(ℝ푑) such that their Fourier transform another. The view expressed by Candès, Justin Romberg, ̂ and [1] and Donoho is that since only a few (31) 푓(휔) = ∫ 푓(푥) exp(−푖2휋휔 ⋅ 푥)푑푥, ℝ푑 of these coefficients are needed in the end, it should be possible to compute only a few non-adaptive linear mea- is supported on 퐸. Sampling theory for such functions has surements in the first place and still retain the information been motivated since the 1960s by the development of needed in order to build a compressed representation. discrete telecommunications. It is well known, since the These ideas have led since the turn of the century to the foundational work of and Harry Nyquist, very active area compressed sensing. that regular grids, which are full-rank lattices If 푚 is the number of linear measurements, the (32) 퐿 = 퐵ℤ푑, observed data has the form where 퐵 is a 푑×푑 invertible matrix, are particularly suitable (29) 푦 = Φ푥, for the sampling of certain band-limited functions. This where Φ is an 푚 × 푁 matrix. Any 푛-sparse vector 푥 is may be seen as a direct consequence of the Poisson uniquely characterized by its measurement if and only summation formula, which says that for any sufficiently if no 2푛-sparse vector lies in the kernel of Φ. In other nice function 푓, 푖2휋⟨휆,휔 ∗ words, any submatrix of Φ푇 obtained by retaining a (33) |퐿| ∑ 푓(휆)푒 = ∑ 푓(휔̂ + 휆 ). set 푇 ⊂ {1, … , 푁} of columns with #(푇) = 2푛 should 휆∈퐿 휆∗∈퐿∗ be injective. It is easily seen that a generic 푚 × 푁 Here |퐿| ∶= | det(퐵)| is the measure of the fundamental matrix satisfies this property provided that 푚 ≥ 2푛, and volume of 퐿, and 퐿∗ = (퐵푡)−1ℤ푑 its dual lattice. This therefore 푚 = 2푛 linear measurements are in principle formula shows that, if 퐸 ⊂ ℝ푑 is a compact set with sufficient to be able to reconstruct 푛-sparse vectors. ∗ translates (퐸 + 휆 )휆∗∈퐿∗ having intersections of null However, the reconstruction from 2푛 measurements will measure, functions with Fourier transform supported in generally be computationally untractable when 푁 is large 퐸 are then stably determined by their sampling on 퐿. Such ∗ and unstable due to the fact that Φ푇 Φ푇, even if invertible, sets 퐸 should in particular have measure smaller than the can be very ill-conditioned. density of 퐿, that is, Stability and computational feasibility can be recov- ∗ −1 ered at the expense of a stronger condition introduced by (34) |퐸| ≤ dens(퐿) ∶= |퐿 | = |퐿| . Candès, Romberg, and Tao: the matrix Φ satisfies the re- One elementary example, for which equality holds in the stricted isometry property (RIP) of order 푘 with parameter above, is the fundamental volume of the lattice 퐿∗, that 0 < 훿 < 1 if and only if is, 2 2 2 푘 ∗ −1 푑 (30) (1 − 훿)‖푧‖2 ≤ ‖Φ푇푧‖2 ≤ (1 + 훿)‖푧‖2, 푧 ∈ ℝ , (35) 퐸퐿∗ = (퐵 ) ([0, 1] ), for all set 푇 ⊂ {1, … , 푁} such that #(푇) = 푘. Under such or any of its translates. a property with 푘 = 2푛 and 훿 < 1/3, it was shown that A theory of stable sampling on more general discrete an 푛-sparse vector can be stably reconstructed from its sets was developed in the 1960s by Henry Landau and Arne linear measurements by a convex optimization algorithm, Beurling. The possibility of reconstructing any 푓 ∈ ℱ퐸 which consists in searching for the solution of (29) with from its samples over a discrete set Λ ⊂ ℝ푑 is described minimal ℓ1-norm. by the property of stable sampling: there exists a constant Measurement matrices Φ of size 푚 × 푁 that sat- 퐶 such that isfy RIP to order 푘 are known to exist in the regime (36) ‖푓‖2 ≤ 퐶 |푓(휆)|2, 푓 ∈ ℱ . 푚 ∼ 푘 log(푁/푘). Therefore, with 푘 = 2푛 the measure- 퐿2 ∑ 퐸 휆∈Λ ment budget 푚 is linear in 푛 up to logarithmic factors. Landau proved that a necessary condition for such a However the constructions of such matrices rely upon property to hold is that probabilistic arguments: they are realizations of random matrices for which it is proved that RIP of order 2푛 holds (37) dens(Λ) ≥ |퐸|, with high probability under this type of regime. Two where notable examples are the matrix with entries consisting #(Λ ∩ 퐵(푥, 푅)) of independent centered Gaussian variables of variance (38) dens(Λ) = lim inf푅→∞ inf 1/푚 and the matrix obtained by picking at random 푚 푥∈ℝ푑 |퐵(푥, 푅)| rows from the 푁 × 푁 discrete Fourier transform matrix. is the lower density of Λ, which is the usual density The currently available deterministic constructions of dens(Λ) when the standard limit exists.

December 2018 Notices of the AMS 1395 set Λ = Λ(퐿, 퐾) ∈ ℝ푑 is defined by

(40) Λ ∶= {푝1(푥) ∶ 푥 ∈ 퐿, 푝2(푥) ∈ 퐾}. The density of a model set Λ = Λ(퐿, 퐾) is uniform and given by |퐾| (41) dens(Λ) = . |퐿| Basarab Matei and Meyer [6] showed that the stable sam- pling property holds for any 퐸 under the condition (39) for model sets Λ ∶= Λ(퐿, 퐾) ⊂ ℝ푑 such that 퐾 is a univariate interval. Such sets are called simple quasicrystals. They are universal sampling sets and may therefore be used for the reconstruction of 푠-sparse signals with 2푠 < dens(Λ). A remarkable fact is that, in contrast to compressed sensing matrices, their construction does not rely on any probabilistic argument.

Yves Meyer giving an exuberant interview post Abel Cited Papers of Meyer Prize Ceremony. [1] E. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59, 1207–1223, 2005. MR2230846 [2] A. Cohen, A journey through the mathematics of Yves Meyer, A continuous signal is 푠-sparse in the Fourier domain The Abel Volume, 2018. if it belongs to ℱ퐸 for some set of Lebesgue measure [3] I. Daubechies, Orthonormal bases of compactly supported |퐸| ≤ 푠. Stable reconstruction of any 푠-sparse signal from wavelets, Comm. Pure and Appl. Math. 41, 909–996, 1988. its sampling on a discrete set Λ requires that this set has MR951745 the property of stable sampling for all sets of measure [4] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, |퐸| ≤ 푟 ∶= 2푠. Such sets Λ are called universal sampling 1992. MR1162107 sets. Obviously, they should have density larger than 푟, but [5] S. Mallat, Multiresolution approximation and wavelet or- 2 this condition cannot be sufficient. The case of a regular thonormal bases of 퐿 (ℝ), Trans. Amer. Math. Soc. 315, 69–88, 1989. MR1008470 lattice 퐿 is instructive: on the one hand, the set 퐸퐿∗ has −1 [6] B. Matei and Y. Meyer, Simple quasicrystals are sets of measure |퐸퐿∗ | = |퐿| = dens(퐿) and satisfies the stable stable sampling, Journal of Complex Variables and Elliptic sampling property in view of (33). On the other hand, other Equations 55, 947–964, 2010. MR2674875 sets 퐸 with the same or even smaller measure could have [7] Y. Meyer, Algebraic numbers and harmonic analysis, North- their translates by Λ∗ overlapping with nonzero measure, Holland, New York, 1972. MR0485769 which is a principle obstruction to these properties. This [8] Y. Meyer, Wavelets and Operators, vol. 1, Cambridge phenomenon is well known in electrical engineering as University Press, 1997. MR1228209 aliasing. This shows that universal sampling sets cannot [9] Y. Meyer, Oscillating patterns in image processing and in some nonlinear evolution equations, Jacqueline Lewis be regular lattices. Memorial Lectures, AMS, 2001. MR1852741 Alexander Olevskii and Alexander Ulanovskii gave the first construction of a set Λ of uniform density that has Photo Credits the stable sampling property for any set 퐸 such that Photos of Yves Meyer courtesy of Stéphane Jaffard. (39) |퐸| < dens(Λ). Author photo courtesy of Albert Cohen. Meyer had the intuition that the mathematical models of quasicrystals that emerged from his early work on harmonic analysis and number theory [7] could provide a ABOUT THE AUTHOR natural alternative solution to this problem. Albert Cohen’s current research One such model is obtained by the following cut interests include nonlinear and and project scheme that was implicit in earlier work on high-dimensional approximation algebraic number theory: the set of interest is obtained by theory, statistics, signal-image- projecting a “slice” cut from a higher-dimensional lattice data processing, numerical analy- in general position. More precisely, if 퐿 is a full rank lattice 푑+푚 푑 sis, and inverse problems. of ℝ for some 푑, 푚 > 0, we denote by 푝1(푥) ∈ ℝ 푚 푑+푚 and 푝2(푥) ∈ ℝ the components of 푥 ∈ ℝ such Albert Cohen that 푥 = (푝1(푥), 푝2(푥)) and assume that 푝1 is a bijection between 퐿 and 푝1(퐿) with dense image. A similar property 푚 is assumed for 푝2. Let 퐾 ⊂ ℝ be a Riemann integrable compact set of positive measure. The associated model

1396 Notices of the AMS Volume 65, Number 11