Fundamental papers in wavelet theory. With a foreword by Ingrid Daubechies. Introduction by John J. Benedetto.

*(English)*Zbl 1113.42001
Princeton, NJ: Princeton University Press (ISBN 978-0-691-11453-8/hbk; 978-0-691-12705-7/pbk). xviii, 878 p. (2006).

There are two reviews of this item, the original review from 2006 by Kai
Schneider and a “looking back” review from 2013 by Joseph Lakey.

First review by Kai Schneider (Marseille) (2006):

The book contains a reprint collection of 37 early papers on wavelets and its precursors which were essential and most significant for the development of modern wavelet theory. Several seminal papers have been translated into English, which were upto now only available in French or German. Precursors in signal processing are presented, like the Laplacian pyramid for image coding by Burt and Adelson (1983) and the application of quadrature mirror filters by Esteban and Galand (1977). The orignal papers on the continuous wavelet transform by Grossmann and Morlet (1984) and papers on the relation of wavelets with coherent states in physics associated to the affine group are also contained.

The early mathematical publications on wavelet bases of Hilbert spaces are reprinted starting with an English translation of the seminal paper by Haar (1910) and the original paper of Franklin (1928). Strömberg’s paper on spline wavelets (1983) can also be found together with the English translation of Meyer’s and Lemarié’s early work (1987, 1986). Papers on atom and frame decompositions starting with Duffin and Schaeffer’s work (1952) defining frames show that the integral representation of the continuous wavelet transform can be replaced by suitable Riemann sums and that wavelets yield a suitable decomposition of the time-frequency plane.

The original papers introducing the concept of multiresolution analysis by Mallat (1989) and on compactly supported wavelets by Daubechies (1988) can be found together with a typed version of Meyers manuscript on wavelets with compact support presented in 1987 at the University of Chicago. Original work on multiresolution analysis in several dimensions and on the construction of the corresponding wavelet bases is encountered in the papers by Meyer and Gröchenig. First constructions of non separable wavelet bases are shown in two papers by Kovačević and Vetterli (1992) and by Gröchenig and Madyck (1992).

Finally, several papers on selected applications of wavelets like for compression of matrices in wavelet bases (Beylkin et al., 1991), the compression of wavelet decompositions (DeVore et al., 1992) and wavelet denoising (Donoho and Johnstone, 1995), are presented.

The book gives an almost complete view on the development of wavelet theory and its history.

This collection is a very useful reference book and the introductory essays of each chapter are a guide for exploring the different historic papers.

Editorial addition from the publisher’s description:

“This book traces the prehistory and initial development of wavelet theory, a discipline that has had a profound impact on mathematics, physics, and engineering. Interchanges between these fields during the last fifteen years have led to a number of advances in applications such as image compression, turbulence, machine vision, radar, and earthquake prediction.

This book contains the seminal papers that presented the ideas from which wavelet theory evolved, as well as those major papers that developed the theory into its current form. These papers originated in a variety of journals from different disciplines, making it difficult for the researcher to obtain a complete view of wavelet theory and its origins. Additionally, some of the most significant papers have heretofore been available only in French or German.

Heil and Walnut bring together these documents in a book that allows researchers a complete view of wavelet theory’s origins and development.

Section I. Precursors in Signal Processing

Introduction: Jelena Kovačević 23

1. Peter J. Burt and Edward H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Commun. 31 (1983), 532-540. 28

2. R. E. Crochiere, S. A. Webber,, and J. L. Flanagan, Digital coding of speech in sub-bands, Bell Syst. Tech. J. 55 (1976), 1069-1085. 37

3. D. Esteban and C. Galand, Application of quadrature-mirror filters to split-band voice coding schemes, ICASSP ’77, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 2, April 1977, 191-195. 54

4. M.J.T. Smith and T. P. Barnwell III, A procedure for designing exact reconstruction filter banks for tree-structured subband coders, ICASSP ’84, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 9, March 1984, 421-424. 59

5. Fred Mintzer, Filters for distortion-free two-band multirate filter banks, IEEE Trans. Acoust., Speech, and Signal Proa, 33 (1985), 626-630. 63

6. Martin Vetterli, Filter banks allowing perfect reconstruction, Signal Processing, 10 (1986), 219-244. 68

7. P. P. Vaidyanathan, Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property, IEEE Trans. Acoust., Speech, and Signal Proa, 35 (1987), 476-492. 94

Section II. Precursors in Physics: Affine Coherent States

Introduction: Jean-Pierre Antoine 113

1. Erik W. Aslaksen and John R. Klauder, Continuous representation theory using the affine group, J. Math. Phys. 10 (1969), 2267-2275. 117

2. A. Grossmann, and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723-736. 126

3. A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations I, J. Math. Physics, 26 (1985), 2473-2479. 140

Section III. Precursors in Mathematics: Early Wavelet Bases

Introduction: Hans G. Feichtinger 149

1. Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme [On the theory of orthogonal function systems], Math. Ann. 69 (1910), 331-371. Translated by Georg Zimmermann. 155

2. Philip Franklin, A set of continuous orthogonal functions, Mathematische Annalen, 100 (1928), 522-529. 189

3. Jan-Olov Strömberg, A modified Franklin system and higher-order spline systems on \(\mathbb R^n\) as unconditional bases for Hardy spaces, Conf. on Harmonic Analysis in Honor of A. Zygmund, Vol. II, W. Beckner et al., eds., Wadsworth (Belmont, CA), (1983), 475-494. 197

4. Yves Meyer, Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs [Uncertainty principle, Hilbert bases, and algebras of operators], Séminaire Bourbaki, 1985/86. Astérisque No. 145-146 (1987), 209-223. Translated by John Horvath. 216

5. P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes [Wavelets and Hilbert bases], Rev. Mat. Iberoam. 2 (1986), 1-18. Translated by John Horvath. 229

6. Guy Battle, A block spin construction of ondelettes. I, Commun. Math. Phys. 110 (1987), 601-615. 245

Section IV. Precursors and Development in Mathematics: Atom and Frame Decompositions

Introduction: Yves Meyer 263

1, R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72 (1952), 341-365. 269 [Zbl 0049.32401]

2. Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Am. Math. Soc. 83 (1977), 569-645. 295 [Zbl 0358.30023]

3. Ingrid Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. 372 [Zbl 0608.46014]

4. Michael Frazier and Björn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. 385 [Zbl 0551.46018]

5. Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86 (1989), 307-340. 408 [Zbl 0691.46011]

6. Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 39 (1990), 961-1005. 442

Section V. Multiresolution Analysis

Introduction: Guido Weiss 489

1. Stephane G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11 (1989), 674-693. 494

2. Yves Meyer, Wavelets with compact support, Zygmund Lectures, U. Chicago (1987). 514

3. Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases for \(L^2(\mathbb R)\), Trans. Am. Math. Soc, 315 (1989), 69-87. 524

4. A. Cohen, Ondelettes, analysis multiresolutions et filtres miroirs en quadrature. [Wavelets, multiresolution analysis, and quadrature mirror filters], Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990), 439-459. Translated by Robert D. Ryan. 543

5. Wayne M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), 1898-1901. 560

6. Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41 (1988), 909-996. 564

Section VI. Multidimensional Wavelets

Introduction: Guido Weiss 655

1. Yves Meyer, Ondelettes, fonctions splines et analyses graduées [Wavelets, spline functions, and multiresolution analysis], Rend. Sem. Mat. Univ. Politec. Torino, 45 (1987), 1-42. Translated by John Horvath. 659

2. Karlheinz Gröchenig, Analyse multi-echelle et bases d’ondelettes [Multiscale analyses and wavelet bases], C. R. Acad. Sci. Paris Série I, 305 (1987), 13-17. Translated by Robert D. Ryan. 690

3. Jelena Kovačević and Martin Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for \(\mathbb R^n\), IEEE Trans. Inform. Theory 38 (1992), 533-555. 694

4. K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases and self-similar tilings of \(\mathbb R^n\) , IEEE Trans. Inform. Theory, 38 (1992), 556-568. 717

Section VII. Selected Applications

Introduction: Mladen Victor Wickerhauser 733

1. G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun. Pure Appl. Math., 44 (1991), 141-183. 741 [Zbl 0722.65022 ]

2. Ronald A. DeVore, Bjorn Jawerth, Vasil Popov, Compression of wavelet decompositions, Am. J. Math., 114 (1992), 737-785. 784 [Zbl 0764.41024]

3. David L. Donoho and Iain M. Johnstone, Adapting to unknown smoothness by wavelet shrinkage, J. Am. Stat. Assoc., 90 (1995), 1200-1224. 833

4. Stéphane Jaffard, Exposants de Hölder en des points donnés et coefficients d’ondelettes [Ho\"lder exponents at given points and wavelet coefficients], C. R. Acad. Sci. Paris Série I, 308 (1989), 79-81. Translated by Robert D. Ryan. 858 [Zbl 0665.42012]

5. Jerome M. Shapiro, Embedded image coding using zero trees of wavelet coefficients, IEEE Trans. Signal Process. 41 (1993), 3445-3462. 861.”

Looking-back review by Joseph Lakey (Las Cruces) (2013):

Well into the 21st century, to most researchers familiar with wavelets and in most contexts, Wavelet Theory refers to the principles underlying the generation of wavelet or multi-wavelet bases, usually for \(L^2(\mathbb{R})\) or sometimes \(L^2(\mathbb{R}^n)\), with a number of variations that depend on desired properties including regularity, optimal spatial or frequency localization, symmetry, adaptation to differential operators, and so on. In one variable the design issues are well understood. In multiple variables, they are not. Practitioners are satisfied with separable wavelets for some multivariate applications, but better adapted tools are often sought, including curvelets or shearlets—conceptual descendants of wavelets—the effective design and implementation of which rely on understanding the broad perspectives laid out in this volume. Continuous wavelet transforms (CWTs) remain an important tool for detection problems and new CWT methods are still being developed, such as Daubechies, Lu and Wu’s synchrosqueezing transform, [I. Daubechies, J. Lu and H.-T. Wu, Appl. Comput. Harmon. Anal. 30, No. 2, 243–261 (2011; Zbl 1234.94018)] but, for the most part, wavelet theory refers to wavelet bases.

In contrast, in the late 1980’s and early 1990’s, when speaking of wavelets one had to supply context: otherwise one might mean almost any complete family of oscillating functions corresponding to the orbit, or discretized orbits, of a single function or finite number of functions, under a group acting on a class of functions defined on a homogeneous space, typical examples being the affine or Weyl-Heisenberg group acting on \(\mathbb{R}^n\). One of the fundamental questions in this broader context asked when a discrete family of functions thus generated might form a complete set, a frame, or another type of synthesizing family for which a Banach norm could be encoded with an equivalent coefficient space norm, or a basis. Mathematicians working in this area were continuously being challenged by engineers and scientists in parallel disciplines to develop tools that could lead to practical and useful implementations.

This collection chronicles—through thirty-seven fundamental contributions divided into seven categories—the early development of wavelet theory, primarily in the 1980s and early 1990’s, but dating all the way back to Haar’s 1910 thesis, written under Hilbert’s direction and appearing here in English translation for the first time. Quoting the editors’ preface, “This volume traces the development of modern wavelet theory by collecting into one place many of the fundamental original papers in signal processing, physics, and mathematics that stimulated the rise of wavelet theory, along with many major papers in the early development of the subject.”

The first section, introduced by Jelena Kovačević, documents the development of subband coding by engineers working primarily in speech and image processing, leading to the concept of a quadrature mirror filter. The second and shortest section, introduced by Jean-Pierre Antoine, documents the development of the concept of an affine coherent state. The third paper in this section, Grossmann, Morlet and Paul’s Transforms associated to square integrable group representations I provides a mathematical physicist’s view of the context in which the continuous wavelet transform is a simple but useful example. The third section, introduced by Hans G. Feichtinger, documents through examples of early wavelet bases, including the Haar and Franklin systems, Meyer’s construction of Schwartz class wavelets, and Battle’s block spin construction, various early wavelet constructions. Reading Sects. I and III one sees what the engineers had that the mathematicians did not, and vice-versa. In particular one sees the divergent perspectives that prevented an earlier formulation of the multiresolution analysis structure developed in Sect. V. It might have been tempting for the editors to place the materials of Sects. I, III and V adjacently, but doing so might have given an inaccurate impression of the mathematical context in which discrete wavelets developed. Orthonormal wavelet bases seemed a miracle to mathematicians who were accustomed to thinking in terms of frames and, in particular, of discrete wavelet and Gabor transforms as discretizations of their continuous parameter counterparts. In the case of Weyl-Heisenberg discretizations, bases with good time-frequency localization were known to be impossible. Thus Sect. IV, which develops atom and frame decompositions in the context of harmonic analysis, is placed where it is. This section is introduced by Yves Meyer. Both Sects. V and VI—the latter one addressing wavelet bases for \(L^2(\mathbb{R}^n)\), \(n>1\)—are introduced by Guido Weiss. The final section, introduced by Mladen Victor Wickerhauser, develops some of the early applications of orthonormal wavelets, particularly numerical algorithms, compression and denoising, and image coding. Jaffard’s short paper Hölder exponents at given points and wavelet coefficients is also found in English translation in this section. Besides Haar’s thesis and Jaffard’s note, six additional works appear here in English translation from French: Yves Meyer’s Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs; Lemarié and Meyer’s Ondelettes et bases hilbertiennes; Meyer’s 1987 Zygmund Lectures, titled Wavelets with compact support, which provides an alternative proof of existence of compactly supported regular wavelets—the lectures were given while Daubechies’ work was in press (and were distributed as handwritten notes); A. Cohen’s Ondelettes, analysis multirésolutions et filtres mirroirs en quadrature; Meyer’s Ondelettes, fonctions splines et analyses graduées; and Gröchenig’s Analyse multi-échelle et bases d’ondelettes.

The contributions within each section are ordered so as to give a sense of how the concepts developed over time. Consequently, the ordering is mostly, but not entirely, chronological. Burt and Adelson’s paper The Laplacian pyramid as a compact image code, which appears first in Sect. I, is one exception. As Kovačević explains in her section introduction, the remaining papers in Sect. I did develop a theory conceptually in sequential order, whereas the relevance of Burt and Adelson’s paper to the acoustics and speech signal processing community was not understood until later. Another exception occurs in Sect. V. After Meyer’s remarkable but relatively elementary 1985 construction of an orthonormal basis of wavelets in the Schwartz class (found in Sect. III), the subject developed rapidly. Mallat’s IEEE paper A theory for multiresolution signal decomposition: The wavelet representation—which first quantified the mutiresolution structure of wavelet bases and appears first in Sect. V—was not published until July 1989, two years after it was submitted. By that time it was firmly established that regular orthonormal wavelets can always be associated with a multiresolution analysis, but a lot of fundamental questions remained about the relationship between the two. The papers by Cohen and Lawton in Sect. III went a long way toward clarifying the relationship, though the basic question of when a function \(\psi\) is an orthonormal wavelet for \(L^2(\mathbb{R})\) was not settled until the mid-90’s, as Guido Weiss explains in the introduction to Sect. III. Daubechies’ seminal 1988 Orthonormal bases of compactly supported wavelets, which not only first constructed continuous orthonormal wavelets having compact support, but also quantified how pointwise regularity depends on the subband filters and addressed implementation issues, is placed at the end of this section. Placed here, the broad perspective that Daubechies drew from contributions in earlier sections of this collection becomes evident, and questions about the relationship between MRAs and wavelets are less distracting. The introductions to the sections set the context for the contributions themselves, in most cases providing first hand accounts of the contexts in which those contributions arose, and filling in the picture regarding the roles played by further fundamental contributions that were not included in this volume. Meyer’s introduction to Sect. IV stands out among these. Meyer was front and center of the wavelet revolution in the 1980s and his introduction transcends the section, creating a broad but personal account of the collisions of streams of concepts from geoexploration, group representation theory, and continuous and discrete mathematical representations found in Sects. II–IV, that contributed to his own fundamental discoveries and the fundamental contributions of others, especially those of Mallat and Daubechies in Sect. V. While Meyer’s introduction does not say so much about some of the specific contributions in Sect. IV, Benedetto’s introduction to the full volume does specifically address, or at least touch on, the contributions of Duffin and Schaefer, of Frazier and Jawerth, and of Feichtinger and Gröchenig found in Sect. IV. Both Daubechies’ foreword and Benedetto’s introduction to the full volume also shed a lot of light on the crucial roles played by diverse perspectives in the early development of wavelets.

While each section is laid out so as to expose how key concepts evolved, the ordering of sections within the volume serves to build a picture of what Daubechies called the “cross-fertilization between the signal analysis applications and the purely mathematical aspects …” (see p. 448). Presenting the buildup to quadrature mirror filters in Sect. I shows that the signals community basically had it all except for some obscure but crucial details about orthogonality and regularity of iterated function systems. Setting contributions on affine coherent states in Sect. II shows that (continuous) wavelets are but one example of reproducing systems associated with square-integrable group representations. Recalling early mathematical constructions (Sect. III) indicates that mathematicians had all the tools but lacked intuition as to how they might be used. The atoms and frame decompositions of Sect. IV remind us that, over time, mathematicians developed uses for the tools they forgot they already had, and so on.

While Mallat’s A theory for multiresolution signal decomposition… and Daubechies’ Orthonormal bases of compactly supported wavelets and several other papers included here are cornerstones of the subject, other watershed moments in the development of wavelets are less obvious. Quoting from the editors’ preface: “We tried to choose papers that represented a significant leap forward or breakthrough that led to a significant development in the theory, and did not necessarily choose papers in which a concept first appeared.” An example of this is Coifman and Weiss’s 1977 work, Extensions of Hardy spaces and their use in analysis in Sect. IV on atom and frame decompositions. The notion of a Hardy space atom appeared earlier in separate of works of R.R. Coifman [Stud. Math. 51, 269–274 (1974; Zbl 0289.46037)] and R.H. Latter [Stud. Math. 62, 93–101 (1978; Zbl 0398.42017)] and of C. Herz in the martingale case [Trans. Am. Math. Soc. 193, 199–215 (1974; Zbl 0321.60041)]. But Coifman and Weiss’s 1977 work illustrated the role that the atomic characterization could play in extending Hardy space theory to broader settings and later would play in the operator theory of those spaces. A lot of other fundamental subtopics had to be omitted entirely due to lack of space. A partial list can be found in Daubechies’ foreword, which suggests a “missing chapter” on key concepts relating subdivision and refinement to computer graphics.

While wavelets are not the subject of intense focus they once were, it is generally agreed that wavelets were at the center of an early renaissance in applied and computational harmonic analysis that brought about sustained interaction between pure mathematics and concrete applications. Niels Abel famously said, “It appears to me that if one wishes to make progress in mathematics, one should study the masters …” Wavelet theory has an extraordinary number of monographical accounts both by undisputed masters including [I. Daubechies, Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics (1992; Zbl 0776.42018)]; [S. Mallat, A wavelet tour of signal processing. The sparse way. 3rd ed. Amsterdam: Elsevier/Academic Press (2009; Zbl 1170.94003)]; [Y. Meyer, Ondelettes et opérateurs I: Ondelettes. Actualités Mathématiques. Paris: Hermann, Éditeurs des Sciences et des Arts (1990; Zbl 0694.41037)] and [E. Hernández and G. Weiss, A first course on wavelets. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press (1996; Zbl 0885.42018)] among others, and by pupils too many to mention. But the collection under review contains the original works, unfiltered except for translations, guided by first hand accounts from several of the masters themselves. Even if one is not interested in proving new results about wavelets, this volume is valuable in a sense that amounts to an extrapolation of Abel’s maxim to the context of interactions between mathematics and applications: if one wants to learn how to do applied harmonic analysis well, one should study this volume.

First review by Kai Schneider (Marseille) (2006):

The book contains a reprint collection of 37 early papers on wavelets and its precursors which were essential and most significant for the development of modern wavelet theory. Several seminal papers have been translated into English, which were upto now only available in French or German. Precursors in signal processing are presented, like the Laplacian pyramid for image coding by Burt and Adelson (1983) and the application of quadrature mirror filters by Esteban and Galand (1977). The orignal papers on the continuous wavelet transform by Grossmann and Morlet (1984) and papers on the relation of wavelets with coherent states in physics associated to the affine group are also contained.

The early mathematical publications on wavelet bases of Hilbert spaces are reprinted starting with an English translation of the seminal paper by Haar (1910) and the original paper of Franklin (1928). Strömberg’s paper on spline wavelets (1983) can also be found together with the English translation of Meyer’s and Lemarié’s early work (1987, 1986). Papers on atom and frame decompositions starting with Duffin and Schaeffer’s work (1952) defining frames show that the integral representation of the continuous wavelet transform can be replaced by suitable Riemann sums and that wavelets yield a suitable decomposition of the time-frequency plane.

The original papers introducing the concept of multiresolution analysis by Mallat (1989) and on compactly supported wavelets by Daubechies (1988) can be found together with a typed version of Meyers manuscript on wavelets with compact support presented in 1987 at the University of Chicago. Original work on multiresolution analysis in several dimensions and on the construction of the corresponding wavelet bases is encountered in the papers by Meyer and Gröchenig. First constructions of non separable wavelet bases are shown in two papers by Kovačević and Vetterli (1992) and by Gröchenig and Madyck (1992).

Finally, several papers on selected applications of wavelets like for compression of matrices in wavelet bases (Beylkin et al., 1991), the compression of wavelet decompositions (DeVore et al., 1992) and wavelet denoising (Donoho and Johnstone, 1995), are presented.

The book gives an almost complete view on the development of wavelet theory and its history.

This collection is a very useful reference book and the introductory essays of each chapter are a guide for exploring the different historic papers.

Editorial addition from the publisher’s description:

“This book traces the prehistory and initial development of wavelet theory, a discipline that has had a profound impact on mathematics, physics, and engineering. Interchanges between these fields during the last fifteen years have led to a number of advances in applications such as image compression, turbulence, machine vision, radar, and earthquake prediction.

This book contains the seminal papers that presented the ideas from which wavelet theory evolved, as well as those major papers that developed the theory into its current form. These papers originated in a variety of journals from different disciplines, making it difficult for the researcher to obtain a complete view of wavelet theory and its origins. Additionally, some of the most significant papers have heretofore been available only in French or German.

Heil and Walnut bring together these documents in a book that allows researchers a complete view of wavelet theory’s origins and development.

Section I. Precursors in Signal Processing

Introduction: Jelena Kovačević 23

1. Peter J. Burt and Edward H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Commun. 31 (1983), 532-540. 28

2. R. E. Crochiere, S. A. Webber,, and J. L. Flanagan, Digital coding of speech in sub-bands, Bell Syst. Tech. J. 55 (1976), 1069-1085. 37

3. D. Esteban and C. Galand, Application of quadrature-mirror filters to split-band voice coding schemes, ICASSP ’77, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 2, April 1977, 191-195. 54

4. M.J.T. Smith and T. P. Barnwell III, A procedure for designing exact reconstruction filter banks for tree-structured subband coders, ICASSP ’84, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 9, March 1984, 421-424. 59

5. Fred Mintzer, Filters for distortion-free two-band multirate filter banks, IEEE Trans. Acoust., Speech, and Signal Proa, 33 (1985), 626-630. 63

6. Martin Vetterli, Filter banks allowing perfect reconstruction, Signal Processing, 10 (1986), 219-244. 68

7. P. P. Vaidyanathan, Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property, IEEE Trans. Acoust., Speech, and Signal Proa, 35 (1987), 476-492. 94

Section II. Precursors in Physics: Affine Coherent States

Introduction: Jean-Pierre Antoine 113

1. Erik W. Aslaksen and John R. Klauder, Continuous representation theory using the affine group, J. Math. Phys. 10 (1969), 2267-2275. 117

2. A. Grossmann, and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723-736. 126

3. A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations I, J. Math. Physics, 26 (1985), 2473-2479. 140

Section III. Precursors in Mathematics: Early Wavelet Bases

Introduction: Hans G. Feichtinger 149

1. Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme [On the theory of orthogonal function systems], Math. Ann. 69 (1910), 331-371. Translated by Georg Zimmermann. 155

2. Philip Franklin, A set of continuous orthogonal functions, Mathematische Annalen, 100 (1928), 522-529. 189

3. Jan-Olov Strömberg, A modified Franklin system and higher-order spline systems on \(\mathbb R^n\) as unconditional bases for Hardy spaces, Conf. on Harmonic Analysis in Honor of A. Zygmund, Vol. II, W. Beckner et al., eds., Wadsworth (Belmont, CA), (1983), 475-494. 197

4. Yves Meyer, Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs [Uncertainty principle, Hilbert bases, and algebras of operators], Séminaire Bourbaki, 1985/86. Astérisque No. 145-146 (1987), 209-223. Translated by John Horvath. 216

5. P. G. Lemarié and Y. Meyer, Ondelettes et bases hilbertiennes [Wavelets and Hilbert bases], Rev. Mat. Iberoam. 2 (1986), 1-18. Translated by John Horvath. 229

6. Guy Battle, A block spin construction of ondelettes. I, Commun. Math. Phys. 110 (1987), 601-615. 245

Section IV. Precursors and Development in Mathematics: Atom and Frame Decompositions

Introduction: Yves Meyer 263

1, R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72 (1952), 341-365. 269 [Zbl 0049.32401]

2. Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Am. Math. Soc. 83 (1977), 569-645. 295 [Zbl 0358.30023]

3. Ingrid Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. 372 [Zbl 0608.46014]

4. Michael Frazier and Björn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. 385 [Zbl 0551.46018]

5. Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86 (1989), 307-340. 408 [Zbl 0691.46011]

6. Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 39 (1990), 961-1005. 442

Section V. Multiresolution Analysis

Introduction: Guido Weiss 489

1. Stephane G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11 (1989), 674-693. 494

2. Yves Meyer, Wavelets with compact support, Zygmund Lectures, U. Chicago (1987). 514

3. Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases for \(L^2(\mathbb R)\), Trans. Am. Math. Soc, 315 (1989), 69-87. 524

4. A. Cohen, Ondelettes, analysis multiresolutions et filtres miroirs en quadrature. [Wavelets, multiresolution analysis, and quadrature mirror filters], Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990), 439-459. Translated by Robert D. Ryan. 543

5. Wayne M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), 1898-1901. 560

6. Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41 (1988), 909-996. 564

Section VI. Multidimensional Wavelets

Introduction: Guido Weiss 655

1. Yves Meyer, Ondelettes, fonctions splines et analyses graduées [Wavelets, spline functions, and multiresolution analysis], Rend. Sem. Mat. Univ. Politec. Torino, 45 (1987), 1-42. Translated by John Horvath. 659

2. Karlheinz Gröchenig, Analyse multi-echelle et bases d’ondelettes [Multiscale analyses and wavelet bases], C. R. Acad. Sci. Paris Série I, 305 (1987), 13-17. Translated by Robert D. Ryan. 690

3. Jelena Kovačević and Martin Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for \(\mathbb R^n\), IEEE Trans. Inform. Theory 38 (1992), 533-555. 694

4. K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases and self-similar tilings of \(\mathbb R^n\) , IEEE Trans. Inform. Theory, 38 (1992), 556-568. 717

Section VII. Selected Applications

Introduction: Mladen Victor Wickerhauser 733

1. G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun. Pure Appl. Math., 44 (1991), 141-183. 741 [Zbl 0722.65022 ]

2. Ronald A. DeVore, Bjorn Jawerth, Vasil Popov, Compression of wavelet decompositions, Am. J. Math., 114 (1992), 737-785. 784 [Zbl 0764.41024]

3. David L. Donoho and Iain M. Johnstone, Adapting to unknown smoothness by wavelet shrinkage, J. Am. Stat. Assoc., 90 (1995), 1200-1224. 833

4. Stéphane Jaffard, Exposants de Hölder en des points donnés et coefficients d’ondelettes [Ho\"lder exponents at given points and wavelet coefficients], C. R. Acad. Sci. Paris Série I, 308 (1989), 79-81. Translated by Robert D. Ryan. 858 [Zbl 0665.42012]

5. Jerome M. Shapiro, Embedded image coding using zero trees of wavelet coefficients, IEEE Trans. Signal Process. 41 (1993), 3445-3462. 861.”

Looking-back review by Joseph Lakey (Las Cruces) (2013):

Well into the 21st century, to most researchers familiar with wavelets and in most contexts, Wavelet Theory refers to the principles underlying the generation of wavelet or multi-wavelet bases, usually for \(L^2(\mathbb{R})\) or sometimes \(L^2(\mathbb{R}^n)\), with a number of variations that depend on desired properties including regularity, optimal spatial or frequency localization, symmetry, adaptation to differential operators, and so on. In one variable the design issues are well understood. In multiple variables, they are not. Practitioners are satisfied with separable wavelets for some multivariate applications, but better adapted tools are often sought, including curvelets or shearlets—conceptual descendants of wavelets—the effective design and implementation of which rely on understanding the broad perspectives laid out in this volume. Continuous wavelet transforms (CWTs) remain an important tool for detection problems and new CWT methods are still being developed, such as Daubechies, Lu and Wu’s synchrosqueezing transform, [I. Daubechies, J. Lu and H.-T. Wu, Appl. Comput. Harmon. Anal. 30, No. 2, 243–261 (2011; Zbl 1234.94018)] but, for the most part, wavelet theory refers to wavelet bases.

In contrast, in the late 1980’s and early 1990’s, when speaking of wavelets one had to supply context: otherwise one might mean almost any complete family of oscillating functions corresponding to the orbit, or discretized orbits, of a single function or finite number of functions, under a group acting on a class of functions defined on a homogeneous space, typical examples being the affine or Weyl-Heisenberg group acting on \(\mathbb{R}^n\). One of the fundamental questions in this broader context asked when a discrete family of functions thus generated might form a complete set, a frame, or another type of synthesizing family for which a Banach norm could be encoded with an equivalent coefficient space norm, or a basis. Mathematicians working in this area were continuously being challenged by engineers and scientists in parallel disciplines to develop tools that could lead to practical and useful implementations.

This collection chronicles—through thirty-seven fundamental contributions divided into seven categories—the early development of wavelet theory, primarily in the 1980s and early 1990’s, but dating all the way back to Haar’s 1910 thesis, written under Hilbert’s direction and appearing here in English translation for the first time. Quoting the editors’ preface, “This volume traces the development of modern wavelet theory by collecting into one place many of the fundamental original papers in signal processing, physics, and mathematics that stimulated the rise of wavelet theory, along with many major papers in the early development of the subject.”

The first section, introduced by Jelena Kovačević, documents the development of subband coding by engineers working primarily in speech and image processing, leading to the concept of a quadrature mirror filter. The second and shortest section, introduced by Jean-Pierre Antoine, documents the development of the concept of an affine coherent state. The third paper in this section, Grossmann, Morlet and Paul’s Transforms associated to square integrable group representations I provides a mathematical physicist’s view of the context in which the continuous wavelet transform is a simple but useful example. The third section, introduced by Hans G. Feichtinger, documents through examples of early wavelet bases, including the Haar and Franklin systems, Meyer’s construction of Schwartz class wavelets, and Battle’s block spin construction, various early wavelet constructions. Reading Sects. I and III one sees what the engineers had that the mathematicians did not, and vice-versa. In particular one sees the divergent perspectives that prevented an earlier formulation of the multiresolution analysis structure developed in Sect. V. It might have been tempting for the editors to place the materials of Sects. I, III and V adjacently, but doing so might have given an inaccurate impression of the mathematical context in which discrete wavelets developed. Orthonormal wavelet bases seemed a miracle to mathematicians who were accustomed to thinking in terms of frames and, in particular, of discrete wavelet and Gabor transforms as discretizations of their continuous parameter counterparts. In the case of Weyl-Heisenberg discretizations, bases with good time-frequency localization were known to be impossible. Thus Sect. IV, which develops atom and frame decompositions in the context of harmonic analysis, is placed where it is. This section is introduced by Yves Meyer. Both Sects. V and VI—the latter one addressing wavelet bases for \(L^2(\mathbb{R}^n)\), \(n>1\)—are introduced by Guido Weiss. The final section, introduced by Mladen Victor Wickerhauser, develops some of the early applications of orthonormal wavelets, particularly numerical algorithms, compression and denoising, and image coding. Jaffard’s short paper Hölder exponents at given points and wavelet coefficients is also found in English translation in this section. Besides Haar’s thesis and Jaffard’s note, six additional works appear here in English translation from French: Yves Meyer’s Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs; Lemarié and Meyer’s Ondelettes et bases hilbertiennes; Meyer’s 1987 Zygmund Lectures, titled Wavelets with compact support, which provides an alternative proof of existence of compactly supported regular wavelets—the lectures were given while Daubechies’ work was in press (and were distributed as handwritten notes); A. Cohen’s Ondelettes, analysis multirésolutions et filtres mirroirs en quadrature; Meyer’s Ondelettes, fonctions splines et analyses graduées; and Gröchenig’s Analyse multi-échelle et bases d’ondelettes.

The contributions within each section are ordered so as to give a sense of how the concepts developed over time. Consequently, the ordering is mostly, but not entirely, chronological. Burt and Adelson’s paper The Laplacian pyramid as a compact image code, which appears first in Sect. I, is one exception. As Kovačević explains in her section introduction, the remaining papers in Sect. I did develop a theory conceptually in sequential order, whereas the relevance of Burt and Adelson’s paper to the acoustics and speech signal processing community was not understood until later. Another exception occurs in Sect. V. After Meyer’s remarkable but relatively elementary 1985 construction of an orthonormal basis of wavelets in the Schwartz class (found in Sect. III), the subject developed rapidly. Mallat’s IEEE paper A theory for multiresolution signal decomposition: The wavelet representation—which first quantified the mutiresolution structure of wavelet bases and appears first in Sect. V—was not published until July 1989, two years after it was submitted. By that time it was firmly established that regular orthonormal wavelets can always be associated with a multiresolution analysis, but a lot of fundamental questions remained about the relationship between the two. The papers by Cohen and Lawton in Sect. III went a long way toward clarifying the relationship, though the basic question of when a function \(\psi\) is an orthonormal wavelet for \(L^2(\mathbb{R})\) was not settled until the mid-90’s, as Guido Weiss explains in the introduction to Sect. III. Daubechies’ seminal 1988 Orthonormal bases of compactly supported wavelets, which not only first constructed continuous orthonormal wavelets having compact support, but also quantified how pointwise regularity depends on the subband filters and addressed implementation issues, is placed at the end of this section. Placed here, the broad perspective that Daubechies drew from contributions in earlier sections of this collection becomes evident, and questions about the relationship between MRAs and wavelets are less distracting. The introductions to the sections set the context for the contributions themselves, in most cases providing first hand accounts of the contexts in which those contributions arose, and filling in the picture regarding the roles played by further fundamental contributions that were not included in this volume. Meyer’s introduction to Sect. IV stands out among these. Meyer was front and center of the wavelet revolution in the 1980s and his introduction transcends the section, creating a broad but personal account of the collisions of streams of concepts from geoexploration, group representation theory, and continuous and discrete mathematical representations found in Sects. II–IV, that contributed to his own fundamental discoveries and the fundamental contributions of others, especially those of Mallat and Daubechies in Sect. V. While Meyer’s introduction does not say so much about some of the specific contributions in Sect. IV, Benedetto’s introduction to the full volume does specifically address, or at least touch on, the contributions of Duffin and Schaefer, of Frazier and Jawerth, and of Feichtinger and Gröchenig found in Sect. IV. Both Daubechies’ foreword and Benedetto’s introduction to the full volume also shed a lot of light on the crucial roles played by diverse perspectives in the early development of wavelets.

While each section is laid out so as to expose how key concepts evolved, the ordering of sections within the volume serves to build a picture of what Daubechies called the “cross-fertilization between the signal analysis applications and the purely mathematical aspects …” (see p. 448). Presenting the buildup to quadrature mirror filters in Sect. I shows that the signals community basically had it all except for some obscure but crucial details about orthogonality and regularity of iterated function systems. Setting contributions on affine coherent states in Sect. II shows that (continuous) wavelets are but one example of reproducing systems associated with square-integrable group representations. Recalling early mathematical constructions (Sect. III) indicates that mathematicians had all the tools but lacked intuition as to how they might be used. The atoms and frame decompositions of Sect. IV remind us that, over time, mathematicians developed uses for the tools they forgot they already had, and so on.

While Mallat’s A theory for multiresolution signal decomposition… and Daubechies’ Orthonormal bases of compactly supported wavelets and several other papers included here are cornerstones of the subject, other watershed moments in the development of wavelets are less obvious. Quoting from the editors’ preface: “We tried to choose papers that represented a significant leap forward or breakthrough that led to a significant development in the theory, and did not necessarily choose papers in which a concept first appeared.” An example of this is Coifman and Weiss’s 1977 work, Extensions of Hardy spaces and their use in analysis in Sect. IV on atom and frame decompositions. The notion of a Hardy space atom appeared earlier in separate of works of R.R. Coifman [Stud. Math. 51, 269–274 (1974; Zbl 0289.46037)] and R.H. Latter [Stud. Math. 62, 93–101 (1978; Zbl 0398.42017)] and of C. Herz in the martingale case [Trans. Am. Math. Soc. 193, 199–215 (1974; Zbl 0321.60041)]. But Coifman and Weiss’s 1977 work illustrated the role that the atomic characterization could play in extending Hardy space theory to broader settings and later would play in the operator theory of those spaces. A lot of other fundamental subtopics had to be omitted entirely due to lack of space. A partial list can be found in Daubechies’ foreword, which suggests a “missing chapter” on key concepts relating subdivision and refinement to computer graphics.

While wavelets are not the subject of intense focus they once were, it is generally agreed that wavelets were at the center of an early renaissance in applied and computational harmonic analysis that brought about sustained interaction between pure mathematics and concrete applications. Niels Abel famously said, “It appears to me that if one wishes to make progress in mathematics, one should study the masters …” Wavelet theory has an extraordinary number of monographical accounts both by undisputed masters including [I. Daubechies, Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics (1992; Zbl 0776.42018)]; [S. Mallat, A wavelet tour of signal processing. The sparse way. 3rd ed. Amsterdam: Elsevier/Academic Press (2009; Zbl 1170.94003)]; [Y. Meyer, Ondelettes et opérateurs I: Ondelettes. Actualités Mathématiques. Paris: Hermann, Éditeurs des Sciences et des Arts (1990; Zbl 0694.41037)] and [E. Hernández and G. Weiss, A first course on wavelets. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press (1996; Zbl 0885.42018)] among others, and by pupils too many to mention. But the collection under review contains the original works, unfiltered except for translations, guided by first hand accounts from several of the masters themselves. Even if one is not interested in proving new results about wavelets, this volume is valuable in a sense that amounts to an extrapolation of Abel’s maxim to the context of interactions between mathematics and applications: if one wants to learn how to do applied harmonic analysis well, one should study this volume.

##### MSC:

00B60 | Collections of reprinted articles |

42-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to harmonic analysis on Euclidean spaces |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

65T60 | Numerical methods for wavelets |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |