Mathematical Monographs

Translations of MATHEMATICAL MONOGRAPHS

Volume 239

Wavelet Theory

I. Ya. Novikov V. Yu. Protasov M. A. Skopina

American Mathematical Society 10.1090/mmono/239

Translations of MATHEMATICAL MONOGRAPHS

Volume 239

Wavelet Theory I. Ya. Novikov V. Yu. Protasov M. A. Skopina

Translated by Evgenia Sorokina

M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N Ω Τ Ι Σ Ι Ε S A O

C C I

I American Mathematical Society

R E

E T

ΑΓΕΩΜΕ

M A Providence, Rhode Island

F O 8 U 88 NDED 1 EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheﬀ (Chair) ASL Subcommittee Steﬀen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) I. . Novikov, V. . Protasov, M. A. Skopina TEORI VSPLESKOV M.: Fizmatlit, 2005 This work was originally published in Russian by Fizmatlit under the title “Teori vspleskov” c 2005. The present translation was created under license for the American Mathematical Society and is published by permission. Translated by Evgenia Sorokina

2010 Mathematics Subject Classiﬁcation. Primary 42C40.

For additional information and updates on this book, visit www.ams.org/bookpages/mmono-239

Library of Congress Cataloging-in-Publication Data Novikov, I. IA. (Igor IAkovlevich), 1958– [Teoriia vspleskov. English] Wavelet theory / I. Ya. Novikov, V. Yu. Protasov, M.A. Skopina ; translated by Evgenia Sorokina. p. cm. — (Translations of mathematical monographs ; v. 239) Includes bibliographical references and index. ISBN 978-0-8218-4984-2 (alk. paper) 1. Wavelets (Mathematics) 2. Harmonic analysis. I. Protasov, V. IU. (Vladimir IUrevich), 1970– II. Skopina, M. A. (Mariia Aleksandrovna), 1958– III. Title. QA403.3.N6813 2010 515.2433—dc22 2010035110

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 To my beloved wife and children. I. Ya. Novikov

To my father Yurii Ivanovich Protasov who taught me to love hard sciences. V. Yu. Protasov

My mathematical dynasty was founded by my grandfather I. A. Skopin. It was further continued by my father A. I. Skopin. I dedicate this book to them. M. A. Skopina i Contents

Preface ix Chapter 1. Wavelets on the Line 1 1.1. Riesz bases 1 1.2. MRA and scaling functions 7 1.3. Wavelet spaces 15 1.4. Haar, Battle-Lemarie, Strömberg, and Meyer systems 23 1.5. Uncertainty constants 30 1.6. Computational algorithms 40 1.7. Convergence of wavelet expansions 42 1.8. Wavelet frames 52 Chapter 2. Multivariate Wavelets 63 2.1. Separable MRA 63 2.2. Matrix dilation 66 2.3. Nonseparable MRAs 69 2.4. Construction of refinable functions 74 2.5. Conditions of biorthogonality 79 2.6. Construction of wavelet functions 90 2.7. Wavelet bases 98 2.8. Haar MRAs 105 Chapter 3. Compactly Supported Refinable Functions 115 3.1. Existence, uniqueness, and weak convergence 115 3.2. Strang-Fix conditions 117 3.3. Approximation by shifts of refinable functions 125 3.4. Linear independence, stability, and orthogonality of integer shifts 130 3.5. Examples and applications 140 Chapter 4. Wavelets with Compact Support 143 4.1. Construction of orthogonal wavelets 143 4.2. Wavelets generated by a compactly supported scaling function 152 4.3. Time-frequency localization 156 4.4. Asymptotics of zeros of Bernstein’s polynomials 175 Chapter 5. Fractal Properties of Wavelets 191 5.1. Refinable functions and fractal curves 191 5.2. Fractal curves in the space Lp 198 r r 5.3. Smoothness of fractal curves in the spaces Wp and C 200 5.4. Local smoothness of fractal curves 204 5.5. Examples 216

v vi CONTENTS

Chapter 6. Factorization of Reﬁnement Equations 219 6.1. Operators corresponding to the clean mask 219 6.2. Mask cleaning procedure 222 6.3. Space A˜n and the general form of the operators T0,T1 on it 226 6.4. Factorization theorems 231

Chapter 7. Smoothness of Compactly Supported Wavelets 233 7.1. Matrix method 233 7.2. Local smoothness of wavelets 236 7.3. Special cases and examples 241 7.4. Method of pointwise estimation of the Fourier transform 256 7.5. Estimation by invariant cycles 262

Chapter 8. Nonstationary Wavelets 271 8.1. General theory of nonstationary wavelets 271 8.2. Nonstationary infinitely differentiable orthonormal wavelets with compact support 280 8.3. Decay rate of the Fourier transforms of elements of a nonstationary scaling sequence 288 8.4. Uncertainty constants for Ψ 294 8.5. Nonstationary wavelets with modified Daubechies masks 299 8.6. Uncertainty constants for Ψa 302 8.7. Nonstationary wavelets bases in Sobolev spaces 305

Chapter 9. Periodic Wavelets 315 9.1. PMRA and scaling sequence 315 9.2. Construction of wavelet functions 325 9.3. Wavelet packets 330 9.4. Generating function 332 9.5. Kotelnikov-Shannon system 338

Chapter 10. Approximation by Periodic Wavelets 345 10.1. Convergence of wavelet expansions in norm 345 10.2. Convergence of wavelet expansions almost everywhere 349 10.3. Direct and inverse theorems 358 10.4. Convergence of wavelet expansions at a point 370

Chapter 11. Remarkable Properties of Wavelet Bases 381 11.1. Unconditional wavelet bases 381 11.2. Optimal polynomial bases in the space C(T) 390 11.3. Optimal polynomial bases in the space C[−1, 1] 393 11.4. Wavelet bases in Besov and Lizorkin-Triebel spaces 407 11.5. Linear operators in the Lizorkin-Triebel spaces 431

Appendix A. Auxiliary Facts of the Theory of Functions and Functional Analysis 453 A.1. Bases 453 A.2. Linear functionals in normed spaces 454 A.3. Distributions 455 A.4. Marcinkiewicz interpolation theorem 458 CONTENTS vii

A.5. Spectral radius 458 A.6. Joint spectral radius and the Lyapunov exponent 459 A.7. Smoothness and the decay rate of the Fourier transform 462 A.8. Wiener theorem for L2 464 A.9. Lebesgue sets 465 A.10. Absolutely continuous functions 469 A.11. Nonnegative trigonometric polynomials, Riesz’s lemma 469 A.12. The Enestrem-Kakey theorem about zeros of polynomials 470 A.13. Sobolev spaces 471 A.14. Moduli of continuity 471 A.15. Approximation by trigonometric polynomials 472 A.16. Multidimensional Fejer means 473 A.17. Self-similar sets 474 A.18. Diﬀerence equations 475 A.19. Landau-Kolmogorov inequality 478 A.20. Legendre polynomials 478 Appendix B. Historical Comments 481 Bibliography 493 Index 505

Preface

Wavelet theory lies at the intersection of pure mathematics and computational mathematics, as well as audio and graphic signal processing and compression and transmission of information. The English word wavelet is a translation of the French “ondelette” originally introduced by A. Grossman and J. Morlet. Under the wavelet system is usually understood dilations and shifts of a single function that form a system of representation in some sense (for example, orthogonal basis in L2(R)). In some situations the wavelet systems consist of shifts and dilations of several functions or an entire sequence. In our monograph the notion of “wavelet” as such is not introduced; a specific meaning is given to such phrases as “wavelet function”, “wavelet space”, “wavelet expansion”, etc. Interest in the study of wavelet systems emerged long before the appearance of the terminology and the laying of the foundation of the theory and was primarily attributable to the need of using them for signal processing. In connection with these tasks wavelet analysis was formed (in some sense as an alternative to classical Fourier analysis) in the late 1980s–early 1990s in the works of S. Mallat, Y. Meyer, P. J. Lemarie, I. Daubechies, A. Cohen, R. De- Vore, W. Lawton, C. K. Chui, and others. Wavelet bases have several advantages compared to other bases which are used as tools of approximation. They have the so-called time-frequency localization; i.e., these basis functions as well as their Fourier transforms rapidly decrease at infinity. Due to this property, when we ex- pand in such bases signals whose frequency characteristics vary over time and space (such as speech, music, and seismic signals as well as images) many coefficients of harmonics that are not essential for a space or time region turn out to be small and can be neglected, which leads to a data compression. Permissibility of this deletion is explained by another important property: wavelet expansions are uncondition- ally convergent series. In addition, there are efficient algorithms that allow the fast calculation of coefficients of wavelet expansions. All this attracts numerous special- ists in various fields of applied and engineering mathematics to the use of wavelets. On the other hand, wavelet systems have proved useful for solving some problems of approximation theory and functional analysis. So, the wavelets provide a rare example of where the theory and its practical implementation develop in parallel. The impact on the development of mathematical wavelet theory was made by basic works of Y. Meyer and S. Mallat that introduced the notion of multiresolution analysis, described a method of its construction based on a given (suitable) function, and found explicit formulas for finding an appropriate wavelet function whose shifts and dilations form an orthonormal basis. Due to this theory many examples of wavelet systems have been found whose basic functions are smooth and

ix xPREFACE have good time-frequency localization, in particular, smooth wavelets with compact support. Just such examples were required for applications. In the fifteen-year pe- riod thousands of papers have been written on the development of wavelet theory. Major monographs include books by Y. Meyer [5], I. Daubechies [2], C. K. Chui [6], P. Wojtaszczyk [8], and E. Hernandez and G. A. Weiss [9]. In the Russian literature there are translations of the monographs [2]and[6], a textbook by A. P. Petukhov [10], chapters in monographs by B. S. Kashin and A. A. Sahakian [1], V. I. Berdy- shevandL.V.Petrak[11], as well as surveys by I. Ya. Novikov and S. B. Stechkin [43], [44] and N. M. Astafeva [45]. The above list is not exhaustive; in particular, we are not citing books dedicated to narrow special issues and books in the engineering field. The book which we offer to the reader is the first monograph in the Russian literature which is devoted entirely to a systematic exposition of modern wavelet theory. It details the construction of orthogonal and biorthogonal systems of wavelets and it studies their structural and approximation properties, begin- ning with basic theory and ending with special issues and problems. We present some largely theoretical applications of wavelets. In the first two chapters there are the basic facts of the theory for the one-dimensional and multidimensional cases, respectively. Their presentation, however, does not duplicate any of the above- mentioned books. Many of the known fundamental assertions are represented in the most general form or have new proofs. The majority of the material presented in subsequent chapters was not given in earlier monographs. In particular, for the first time we provide a general theory of periodic multiresolution analysis for the multidimensional case with matrix dilation, the theory of nonstationary wavelets, and offer the results of the local smoothness of wavelet functions. The theory of compactly supported wavelets is offered in detail; the relevant refinement equations are investigated; the fractal properties of the wavelets and their relationship to the classical fractal curves are examined. We discuss different ways of measuring the smoothness of wavelets with compact support. In particular, in an explicit way the moduli of continuity of such wavelets are found, as well as orders of their approximation. A separate chapter is devoted to the time-frequency localization of wavelets. A new construction of modified Daubechies wavelets is given, which pre- serves the localization with a growth of their smoothness. Much attention is paid to convergence of wavelet expansions in various senses, as well as to the evaluation of the order of approximation by wavelets in various functional spaces. Note, however, that this monograph cannot claim to be a complete presentation of all aspects of modern wavelet theory. For example, we did not touch upon the subject of continuous wavelet transform, applications of wavelets to differential equations, as well as many practical applications. All the facts presented in the book are provided with complete proofs except for a small number of inserts, printed in small type. Especially detailed are the proofs given in the basic chapters. Auxiliary facts are contained in the appendix also with proofs (with the exception of only those facts that can be found in widely available monographs). Histori- cal commentary is provided for all chapters, and most chapters have exercises for the reader. This book is intended for readers who are familiar with the basics of classical and functional analysis on the level of a standard university course. Much of the material is available to engineers and, perhaps, could attract their interest. In conclusion, we would like to note that the authors are very grateful to PREFACE xi

S. B. Stechkin—one of the ﬁrst in Russia who has shown interest in wavelet theory and has in various ways contributed to the involvement of all three authors in this subject matter. The authors also express their gratitude to A. P. Petukhov, who read the manuscript fragments and made some helpful observations.

Basic notation N is the set of positive integers. Rd is the d-dimensional Euclidean space, x =(x1,...,xd), y =(y1,...,yd) are its d elements (vectors), 0 =(0,...,0) ∈ R ,(x, y)=x1y1 + ···+ xdyd, |x| = (x, x). R = R1. Zd is the integer lattice in Rd. Z = Z1. Zd { ∈ Zd ≥ } + = x : xk 0,k=1,...,d . Z Z1 + = +. Td =[0, 1)d is the d-dimensional torus. μ is the Lebesgue measure in Rd. d χe is the characteristic function of a set e ⊂ R ; it takes the value 1 at the points t ∈ e and 0 at all other points.

δlk is the Kronecker delta, which is equal to 1 for l = k;otherwise,itisequalto0. If a ∈ R,then[a]:=max{n ∈ Z : n ≤ a}. span M is the linear span of a set M, i.e., the set of all ﬁnite linear combinations of elements from M. supp f is the support of a function f, i.e., the minimal (with respect to inclusion) closed set such that f is equal to zero almost everywhere on the complement of this set. f(k)= f(t)e−2πi(k,t) dt is the k-th Fourier coeﬃcient, k ∈ Zd, of a function Td f ∈ L(Td) with respect to the trigonometric system. f(x)= f(t)e−2πi(x,t) dt is the Fourier transform of a function f from L(Rd); f Rd d denotes also the Fourier transform of a function f from L2(R )orfromthespace of temperate distributions. F and F −1 are the operators taking a function to its direct and inverse Fourier transforms, respectively. L log L(Td)istheclassoffunctionsf ∈ L(Td) such that

|f| max{0, log |f|} < ∞.

Td xii PREFACE

≤ ≤∞ { }∞ p,1 p , is the space of sequences of complex numbers c = cn n=1 with ∞ 1/p p norm cp = |cn| . n=1 f,g is the scalar product of elements of the Hilbert space. { ∈ N} { }∞ span fn,n is the set of ﬁnite linear combinations of a system fn n=1 with complex coeﬃcients. T ∗ is the operator conjugate to an operator T ; T −1 is its inverse operator. If A is a d × d matrix, then A is its Euclidean operator norm from Rd to Rd, AT is its transpose, A∗ is its Hermitian conjugate matrix, det A is the determinant of A.

Id is the unity d × d matrix. For any trigonometric polynomial m having no positive powers (i.e., m(ξ)= N N −2πikξ k cke ), m(z)= ckz is the corresponding algebraic polynomial. Thus k=0 k=0 m(ξ)=m(e−2πiξ).

A sequence ...x−k,...,x−1,x0,x1,...,xk,... is denoted by {xk}k∈Z (sometimes we omit k ∈ Z)or(x). S is the space of infinitely differentiable and rapidly decreasing functions on R: ∞ (m) k S = {f ∈ C (R) |∀m, k ≥ 0 f (x)(1 + |x|) ∞ < ∞}. The topology of the space S is defined as follows: → ⇔∀ ≥ (m) | | k → fj 0 m, k 0 fj (x)(1 + x ) ∞ 0.

S is the space of linear continuous functionals on the space S (the space of tempered distributions). D is the space of infinitely differentiable, compactly supported functions on R with the topology defined as follows: → ⇔∀ ≥ (m) → ∃ ∀ ⊂ − fj 0 m 0 fj (x) ∞ 0, M>0: j supp fj [ M,M]. D is the space of linear continuous functionals on this space. 1 is the function taking the value 1 everywhere on R.

αf is the Hölder coefficient of a function f on a given segment [a, b]: (k) (k) α αf = k +sup α |f (x1) − f (x2)|≤Cα|x1 − x2| ,x1,x2 ∈ [a, b] , where k is the maximal integer such that f ∈ Ck[−1, 1]. The Hölder coefficient in the space Lp is defined analogously: (k) (k) α αf,p = k +sup α f (· + h) − f (·)≤Cαh ,h>0 .

Local smoothness (local Hölder coefficient) of a function f at a point x is defined as (k) (k) α αf (x)=sup α |f (x + h) − f (x)|≤Cαh ,h>0 . PREFACE xiii

s Wp is the Sobolev space. (s) Bpq is the Besov space. (s) Fpq is the Lizorkin-Triebel space. p pα sp(f)=sup α |f(ξ)| (1 + |ξ| ) dξ < ∞ is the Sobolev coeﬃcient of smooth- R ness.

Bibliography

BOOKS

[1] Kashin, B. S. and Sahakian, A. A., Ortogonalnye ryady (in Russian) [Orthogonal series]. 2nd ed. Izdatelstvo Nauchno-Issledovatelskogo Aktuarno-Finansovogo Tsentra (AFTs), Moscow, 1999; English transl., Translations of Mathematical Monographs, Vol. 75, Amer. Math. Soc., Providence, RI, 1989. [2] Daubechies, I., Ten lectures on wavelets, CBMS-NSR Series in Appl. Math., SIAM, 1992. [3] Daubechies, I., Ten lectures on wavelets, Izhevsk: NIZ “Regulyarnaya and khaotich- eskaya dinamika”, 2001 (Russian translation of [2]). [4] Meyer, Y., Ondelettes. Herman, Paris, 1990. [5] Meyer, Y., Wavelets and operators, Cambridge University Press, Cambridge, 1992 (English translation of [4]). [6] Chui, C.K., An introduction to wavelets, Academic Press, New York, 1992. [7] Chui, C., An introduction to wavelets, M.: Mir, 2001 (Russian translation of the book [6]). [8] Wojtaszczyk, P., A mathematical introduction to wavelets, London Math. Soc. Stu- dent Texts 37, 1997. [9] Hernandes, E. and Weis, G. A., A first course of wavelets, CRC Press, Boca Raton, FL, 1996. [10] Petukhov, A. P., Vvedenie v teoriyu bazisov vspleskov (in Russian). [Introduction to the theory of wavelet bases], St. Petersburg Univ., St. Petersburg, 1999. [11] Berdyshev, V. I. and Petrak, L.V., Approksimatsiya funktsi˘ı, szhatie chislennoi in- formatsii, prilozheniya (in Russian). [Approximation of functions, compression of numerical information, applications], Inst. Mat. Mekh., Ekaterinburg, 1999. [12] Sadovnichii, V. A., Teoriya operatorov (in Russian). [Theory of operators], M.: Vysshaya Shkola, 1999. [13] Rudin, W., Real and complex analysis, McGraw-Hill, New York, 1974. [14] Zygmund, A., Trigonometric series. 2nd ed., Vols. I and II, Cambridge University Press, New York, 1959. [15] Stein, E. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces. Vol. 32 of Princeton Mathematical Series, Volume 1 of Monographs in Harmonic Analysis, Princeton University Press, 1971. [16] Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton, 1970. [17] Singer, I., Bases in Banach spaces. I, Springer-Verlag, Berlin, 1970. [18] De Guzman, M., Differentiation of integrals in Rn, with appendices by Antonio Cor- doba and Robert Fefferman and two by Roberto Moriyon. Lecture Notes in Mathe- matics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. [19] Timan, A. F., Teoriya priblizheni˘ı funktsi˘ıde˘ıstvitelnogo peremennogo (in Russian). [Theory of approximation of functions of a real variable], M.: Fizmatgiz, 1960; Eng- lish transl., Macmillan, New York, 1963.

493 494 BIBLIOGRAPHY

[20] Natanson, I. P., Theory of functions of a real variable, New York, 1964. [21] Hardy, H., Littlewood, J. E., and Polya, G., Inequalities, 2nd ed, Cambridge Univ. Press, Cambridge, 1952. [22] Suetin, P. K., Klassicheskie ortogonalnye mnogochleny (in Russian). [Classical orthogonal polynomials], Nauka, Moscow, 1976. [23] Szeg˝o, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. XXIII, Providence, R.I., 1975. [24] Pólya, G. and Szeg˝o, G., Problems and theorems in analysis, Vol. I: Series, integral calculus, theory of functions, Springer-Verlag, Berlin, 1998. [25] Pólya, G. and Szeg˝o, G., Problems and theorems in analysis. Vol. I: Theory of functions, zeros, polynomials, determinants, number theory, geometry, Springer-Verlag, Berlin, 1998. [26] Schaefer, H. H., Topological vector spaces, Macmillan, New York; Collier-Macmillan, London, 1966. [27] Mirolyubov, A. A. and Soldatov, M. A., Line˘ınye neodnorodnye raznostnye uravneniya (in Russian). [Linear inhomogeneous difference equations], Nauka, Moscow, 1986. [28] Godunov, S. K. and Ryabenki˘ı, V. S., Raznostnye skhemy. Vvedenie v teoriyu (in Russian). [Difference schemes. An introduction to the underlying theory], Nauka, Moscow, 1977; English transl., Studies in Mathematics and its Applications, Vol. 19, North-Holland, Amsterdam, 1987. [29] Kirillov, A. A. and Gvishiani, A. D., Teoremy i zadachi funktsionalnogo analiza (in Russian). [Theorems and problems of functional analysis], 2nd ed. Nauka, Moscow, 1988; French transl., Mir, Moscow, 1982 [30] Barnsley, M., Fractals everywhere. Academic Press, Boston, 1988. [31] Mandelbrot, B. B., Fractals and multifractals. Selecta, Vol. 1, Springer, New York, 1991. [32] Gelfand, I., Raikov, D., and Shilov, G., Kommutativnye normirovannye koltsa (in Russian). [Commutative normed rings], Fizmatgiz, Moscow, 1960; English transl., Chelsea, New York, 1964. [33] Strang, G. and Fix, G. J., An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, XIV, Prentice-Hall, Englewood Cliffs, New Jersey, 1973. [34] Cordes, H. O., Elliptic pseudo-differential operators: An abstract theory, Springer- Verlag, Berlin, 1979. [35] Taylor, M., Pseudodifferential operators, Princeto University Press, Princeton, 1981. [36] Lions, J.-L. and Magenes, E., Nonhomogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 181, Springer-Verlag, New York-Heidelberg, 1972. [37] Ladyzhenskaya, O. A., Solonnikov, V. A., and Uraltseva, N. N., Line˘ınye i kvazi- line˘ınye uravneniya parabolicheskogo tipa (in Russian). [Linear and quasilinear equations of parabolic type], Nauka, Moscow, 1967; English transl., Translations of Math- ematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. [38] Nikolskii, S. M., Priblizhenie funktsi˘ımnogikh peremennykh i teoremy vlozheniya (in Russian). [Approximation of functions of several variables and imbedding theorems], Nauka, Moscow, 1977; English transl., Die Grundlehren der Mathematischen Wissenschaften, Vol. 205, Springer-Verlag, New York-Heidelberg. 1975. [39] Besov, O. V., Ilin, V. P., and Nikolskii, S. M., Integralnye predstavleniya funktsi˘ıi teoremy vlozheniya (in Russian). [Integral representations of functions, and embedding theorems], 2nd ed., Nauka, Moscow, 1975; English transl. of Chapters IV–VI, Scripta Series in Mathematics. Edited by Mitchell H. Taibleson, V. H. Winston and BIBLIOGRAPHY 495

Sons, Washington, D.C.; Halsted Press [John Wiley and Sons], New York-Toronto- London, 1979. [40] Tribel, Kh., Theory of function spaces. II, Monographs in Mathematics, Vol. 84, Birkha”user Verlag, Basel, 1992. [41] Gelfand, I. M. and Shilov, G. E., Generalized functions. Vol. 2. Spaces of fundamental and generalized functions. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. [42] Black, H. C., Modulation Theory. New York, 1953.

SURVEYS [43] Novikov, I. Ya. and Stechkin, S. B., Basic constructions of wavelets (in Russian). Fundam. Prikl. Mat. 3 (1997), no. 4, 999–1028. [44] Novikov, I. Ya. and Stechkin, S. B., Fundamentals of wavelet theory (Russian). Uspekhi Mat. Nauk 53 (1998), no. 6(324), 53–128; translation in Russian Math. Surveys 53 (1998), no. 6, 1159–1231. [45] Astafeva, N. M., Wavelet analysis, foundations of the theory, and application examples. Usp. Fiz. Nauk., Vol. 166, No. 11, 1998. [46] Cavaretta, D., Dahmen, W., and Micchelli, C., Stationary subdivision. Mem. Amer. Math. Soc., Vol. 93, 1991, pp. 1–186.

ARTICLES [47] Bari, N. K., Biorthogonal systems and bases in Hilbert space (in Russian). Moskov. Gos. Univ. Ucenye Zapiski Matematika 148(4), (1951), 69–107. [48] Schweinler, H. C. and Wigner, E. P., Orthogonalization methods. J. Math. Phys. 1970, Vol. 11., pp. 1693–1694. [49] Meyer, Y., Ondelettes and fonctions splines. Seminaire EDP. Paris, December 1986. [50] Mallat, S., Multiresolution representation and wavelets. Ph. D. Thesis, University of Pennsylvania, Philadelphia, PA, 1988. [51] Mallat, S., An efficient image representation for multiscale analysis. In Proc. of Machine Vision Conference, Lake Taho, 1987. [52] Mallat, S., A theory of multiresolution signal decomposition: the wavelets representation. IEEE Trans. Pattern Anal. Machine Intell. 1989. Vol. 11, pp. 674–693. [53] Mallat, S., Multiresolution approximation and wavelets. Trans. Amer. Math. Soc. 1989, Vol. 315, pp. 69–88. [54] Lemarié, P. G. and Meyer, Y., Ondelettes et bases Hilbertiennes. Rev. Math. Iber. 1987, Vol. 2, N. 1/2, pp. 1–18. [55] De Boor, C., DeVore, R., and Ron, A., On construction of multivariate (pre) wavelets. Constr. Approx. 1993, Vol. 9, pp. 123–166. [56] Lorents, R. A., Madych, W. R., and Sahakian, A. A., Translation and dilation invariant subspaces of L2(R) and multiresolution analysis, Appl. Comput. Harmon. Anal. 1998, Vol. 5, No. 4, pp. 75–388. [57] Ouscher, P., Ondelettes fractales et applications. Ph. D. Thesis, UniversitéParis Dauphine, Paris, France, 1989. [58] Chui, C. K. and Wang, J. Z., A cardinal spline approach to wavelets. Proc. Amer. Math. Soc. 1991. [59] Cohen, A., Daubechies, I., and Feauveau, J. C., Biorthogonal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics. 1992, Vol. XLV, pp. 2485–560. [60] Gripenberg, G., A necessary and sufficient condition for the existence of father wavelet. Studia Mathematica, 1995, 114, 3, pp. 207–226. [61] Haar, A., Sur Theorie de orthogonalen Funktionensysteme. Math. Ann. 1910, Vol. 69, pp. 331–371. 496 BIBLIOGRAPHY

[62] Strömberg, J.-O, A modified Franklin system and higher-order spline on Rn as unconditional basis for Hardy spaces. In “Conference in Harmonic Analysis in Honor of A. Zygmund”, Chicago, 1981, Vol. 2, 1983, pp. 475–494. [63] Battle, G., A block spin construction of ondelettes. Part 1: Lemarié functions. Comm. Math Phys. 1987, Vol. 110, pp. 601–615. [64] Lemarié, P.-G., Ondelettesa ´ localization exponentialle. J. Math, Pures Appl. (9) 1988. Vol. 67, No. 3, pp. 227–236. [65] Shannon, C. E., Communication in the presence of noise. Proc. of the IRE. 1949, Vol. 37, pp. 10–21. [66] Kotelnikov, V. A., On the transmission capacity of the ‘ether’ and of cables in electrical communications (in Russian). Proceedings of the first All-Union Conference, Upravlenie svyazi RKK, 1933. [67] Meyer, Y., Principle d’incertitude, bases hilbertiennes et algebres d’operateurs. Sem- inaire Bourbaki. 1985–1986, Vol. 38, No. 662. [68] Walter, G., Approximation of the delta-function by wavelets. Local convergence for wavelet expansions. Preprint, 1992. [69] Walter, G., Pointwise convergence of wavelet expansions. Local convergence for wavelet expansions. Preprint, 1992. [70] Kelly, S. E., Kon, M. A., and Raphael, L. A., Pointwise convergence of wavelet expansions. Bull. Amer. Math. Soc. 30, 1994, pp. 87–94. [71] Battle, G., Phase space localization theorem for ondelettes. J. Math Phys. 1989, Vol. 30, pp. 2195–2196. [72] Battle, G., Heisenberg inequalities for wavelet states. ACHA. 1997, Vol. 4, pp. 119– 146. [73] Duffin, R. J. and Schaeffer, A. S., A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 1952, Vol. 72. pp. 341–366. [74] Kashin, B. S. and Kulikova, T. Y., Springer Mathematical Notes, vol. 72, no. 2, 2002, pp. 281–284. [75] Chui, C. K. and He, W., Compactly supported tight frames associated with refinable functions. Appl. and Comp. Harm. Anal. 2000, Vol. 8, pp. 293–319. [76] Petukhov, A., Explicit construction of framelets. Appl. and Comp. Harm. Anal. 2001, Vol. 11, pp. 313–327. [77] Lawton, W., Tight frames of compactly supported affine wavelets. J. Math. Phys. 1990, Vol. 31, pp. 1898–1901. [78] Bownik, M., Tight frames of multidimensional wavelets. J. Fourier Anal. Appl. 1997, Vol. 3, pp. 525–542. [79] Gröchenig, K and Madych, W. R., Maltiresolution analysis, Haar bases and self- similar tillings of Rn. IEEE Trans. Inform. Theory. 1992, Vol. 38, No. 2, pp, 556–568. [80] Madych, W. R., Some elementary properties of multiresolution analysis – a tutiroial in theory and applications. C. K. Chui, ed. Academic Press. 1992, pp. 259–294. [81] Cohen, A. and Daubechies, I., Nonseparable bi-dimensional wavelet bases. Revista Mat. Iberoamericana (I). 1993, pp. 51–137. [82] Cohen, A., Ondelettes, analyses multirésolutions et filtres miroir en quadrature. Ann Inst.H.Poincaré, Anal. Non linéaire. 1990, Vol. 7, pp. 439–459. [83] Maksimenko, I. E., Biorthogonality of multivariable refinable functions (in Russsian). “Voprosy sovremennoi teorii approksimazii”, Izd. SPbGU. 2004, pp. 132–145. [84] Lawton, W., Necessary and sufficient conditions for constructing orthonormal wavelet bases. J. Math. Phys. 1991, Vol. 32, pp. 57–81. [85] Strang, G., Eigenvalues of ( 2)H and convergence of the cascade algorithm. IEEE. Trans. SP 1996, Vol. 44. [86] Lawton, W., Lee, S. N., and Shen, Z., Stability and orthonormality of multivariate refinable functions. SIAM J. Math. Anal. 1997, Vol. 28, No. 4, pp. 999–1114. BIBLIOGRAPHY 497

[87] Lawton, W., Lee, S. N., and Shen, Z., Convergence of multidimensional cascade algorithm. Numerische Mathematik, 1998, Vol. 78, No. 3. pp. 427–438. [88] Gröchenig, K, Analyse multi-échelles et bases d’ondelettes. C. R. Acad. Sci. Paris, Sér. I Math. 1987, Vol. 305, No. 1, pp. 13–15; 1992, Vol. 38, No. 2, pp. 556–568. [89] Jia, R. Q and Micchelli, C. A., Using the refinement equations for the construction of pre-wavelets II: Powers of two. Curves and Surfaces (P. J. Laurent, A. Le Méhauté, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 209–246. [90] Jia, R. Q and Micchelli, C. A., Using the refinement equations for the construction of pre-wavelets V: extensibility of trigonometric polynomials. Computing. 1992, Vol. 48, pp. 61–72. [91] Jia, R. Q. and Shen, Z., Multiresolution and wavelets, Proceedings of the Edinburgh Mathematical Society. 1994, Vol. 37, pp. 271–300. [92] James, I. M., The topology of Stiefel manifolds. LMS Lecture Note Series 24. Cam- bridge University Press, Cambridge, 1976. [93] Lawton, W., Lee, S. L., and Shen, Z., An algorithm for matrix extension and wavelet construction. Math. Comp. 1996, Vol. 37, pp. 271–300. [94] Shen, Z., Extension of matrices with Laurent polynomial entries. Proceedings of the 15th IMACS World Congress on Scientific Computation Modeling and Applied Mathematics, Ashim Syclow, eds. 1997, pp. 57–61. [95] Riemenschneider, S. D. and Shen, Z. W., Construction of compactly supported s biorthogonal wavelets in L2(R ). I. Physics and modern topics in mechanical and electrical engineering. Scientific and Engineering Society Press. 1999, pp. 201–206. [96] Ji, H., Riemenschneider, S. D., and Shen, Z., Multivariate compactly supported fundamental refinable functions, dual and biorthogonal wavelets. Studies of Applied Mathematics. 1999, Vol. 102, pp. 173–204. [97] Riemenschneider, S. D. and Shen, Z. W., Construction of compactly supported s biorthogonal wavelets in L2(R ). II. Wavelet applications signal and image processing VII. Proceedings of SPIE. 1999, Vol. 3813, pp. 264–272. [98] Lam, T. Y., Serre’s conjecture. Lecture Notes in Mathematics, 635, Springer-Verlag, New York, 1978. [99] Suslin, A. A., The structure of the special linear group over rings of polynomials (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477. [100] Park, H. and Woodburn, C., An algorithmic proof of Suslin’s stability theorem for polynomial rings. Journal of Algebra. 1995, Vol. 178, pp. 277–298. [101] Packer, J. A. and Rieffel, M. A., Wavelet filter functions, matrix completion problem, and projective modules over C(Tn). J. Fourier. Anal. Appl. 2003, Vol. 9, No. 3, pp. 101–116. [102] Ron, A. and Shen, Z., Gramian analysis of affine bases and affine frames. In Approx- imation Theory VIII, Vol. 2: Wavelets (C. K. Chui and L. Schumaker, eds). World Scientific Publishing Co. Inc, Singapore, 1995, pp. 375–382. d [103] Ron, A. and Shen, Z., Frame and stable bases for shift-invariant subspaces of L2(R ). Canad. J. Math. 1995, Vol. 47, No. 5, pp. 1051–1094. d [104] Ron, A. and Shen, Z., Affine systems in L2(R ): the analysis of the analysis operator. J. Func. Anal. 1997, Vol. 148, pp. 408–447. d [105] Ron, A. and Shen, Z., Affine systems in L2(R ): dual systems. J. Fourier. Anal. Appl. 1997, Vol. 3, pp. 617–637. [106] Han, B., On dual wavelet tight frames. ACHA. 1997, Vol. 4, pp. 380–413. [107] Han, B., Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. J. Comput. Appl. Math. 2003, Vol. 155, pp. 43–67. [108] Cohen, A. and Daubechies, I., A stability criterion for biorthogonal wavelet bases and their related subband coding schemes. Duke Math. J. 1992, Vol. 68, pp. 313–335. 498 BIBLIOGRAPHY

[109] Jia, R. Q., Approximation properties of multivariate wavelets. Math. Comp. 1998, Vol. 67, pp. 647–655. [110] Lawton, W., Lee, S. L., and Shen, Z., Characterization of compactly supported refinable splines. Adv. Comput. Math. 1995, Vol. 3, No. 1-2, pp. 137–145. [111] Chaikin, G. M., An algorithm for high speed curve generation. Computer Graphics and Image Processing, 1974, Vol. 3, pp. 346–349. [112] Dyn, N., Gregory, J. A., and Levin, D., Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 1991, Vol. 7, pp. 127–147. [113] Dubuc, S., Interpolation through an iterative scheme. J. Math. Anal. Appl. 1986, Vol. 114, pp. 185–204. [114] Deslauriers, G. and Dubuc, S., Symmetric iterative interpolation processes. Constr. Approx. 1989, Vol. 5, pp. 49–68. [115] Protasov, V. Yu., A generalized joint spectral radius. A geometric approach (in Russian). Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 5, 99–136; translation in Izv. Math. 61 (1997), no. 5, 995–1030. [116] Zhow, D. X., The p-norm joint spectral radius and its applications in wavelet analysis. International conference in wavelet analysis and its applications, AMS/IP Studies in Advanced Mathematics, 2002, Vol. 25, pp. 305–326. [117] Zhow, D. X., The p-norm joint spectral radius for even integers. Methods Appl. Anal. 1998, Vol. 5, pp. 39–54. [118] Blondel, V. D. and Nesterov, Yu. P., Computationally efficient approximations of the joint spectral radius. Preprint, 2004. [119] Protasov, V. Yu., The joint spectral radius and invariant sets of linear operators (in Russian). Fundam. Prikl. Mat. 2 (1996), no. 1, 205–231. [120] Protasov, V. Yu., On the smoothness of de Rham curves (in Russian). Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 3, 139–180; translation in Izv. Math. 68 (2004), no. 3, 567–606. [121] Oseledets, V. I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow. Math. Soc. 1968, Vol. 19, pp. 197-231. [122] Ravishankar, K., Power law scaling of the top Lyapunov exponent of a product of random matrices. J. Stat. Phys. 1989, Vol. 54, No.1/2, pp. 531–537. [123] Mandelbrot, B. B., Fisher, A. J., and Calvet, L. E., A Multifractal Model of Asset Returns. Cowles Foundation Discussion Paper, 1997, No. 1164, http:// ssrn.com/abstract=78588. [124] Nikitin, P., The Hausdorff dimension of the harmonic measure on a de Rham curve (in Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. [125] Protasov, V., Refinement equations and corresponding linear operators. Interna- tional Journal of Wavelets, Multiresolution and Information Processing 4. 2006, No. 3, pp. 461-474. [126] Protasov, V., Fractal curves and wavelets, Izv. Math. 70 (2006), no. 5, 975-1013. [127] Protasov, V., The stability of subdivision operator at its fixed point. SIAM J. Math. Analysis, 2001, Vol. 33, No. 2, pp. 448–460. [128] Protasov, V. Yu., Piecewise-smooth refinable functions (in Russian). Algebra i Analiz 16 (2004), no. 5, 101–123; translation in St. Petersburg Math. J. 16 (2005), no. 5, 821–835. [129] Derfel, G. A., Dyn, N., and Levin, D., Generalized refinement equations and subdivision processes. Journal of Approx. Theory, 1995, Vol. 80, pp. 272–297. [130] Protasov, V., Refinement equations with nonnegative coefficients. J. Fourier Anal. Appl., 2000, Vol.6 , No. 6, pp. 55–77. [131] Micchelli, C. A. and Prautzsch, H., Uniform refinement of curves. Linear Alg. Appl. 1989, Vol. 114/115, pp. 841–870. [132] Collela, D. and Heil, C., Characterization of scaling functions. I. Continuous solutions. SIAM J. Matrix Anal. Appl. 1994, Vol. 15, pp. 496–518. BIBLIOGRAPHY 499

[133] Collela, D. and Heil, C., Dilation equations and the smoothness of compactly supported wavelets. In Wavelets: Mathematics and applications, J. Benedetto and M. Frazier, eds. CRC Press, Boca Raton, FL, 1993, pp. 161–200. [134] Lagarias, J. C. and Wang, Y., The ﬁniteness conjecture for the generalized spectral radius. Linear Alg. Appl. 1995, Vol. 214, pp. 17–42. [135] Blondel, V. D., Theys, J., and Vladimirov, A. A., An elementary counterexample to the ﬁniteness conjecture. SIAM J. Matrix Anal. 2003, Vol 24, No. 4, pp. 963–970. [136] Blondel, V. D. and Tsitsiklis, J. N., Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard. IEEE Trans. Autom. Control. 2000, Vol. 45, No. 9, pp. 1762–1765. [137] Blondel, V. D. and Tsitsiklis, J. N., The boundedness of all products of a pair of matrices is undecidable. Syst. Control Lett. 2000, Vol. 41, No. 2, pp. 135–140.

[138] Daubechies, I. and Lagarias, J. Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM. J. Math. Anal. 1992, Vol. 23, pp. 1031–1079. [139] Gripenberg, G., Computing the joint spectral radius. Lin. Alg. Appl. 1996, Vol. 234, pp. 43–60. [140] Maesumi, M., An efficient lower bound for the generalized spectral radius. Linear Alg. Appl. 1996, Vol. 240, pp. 1-7. [141] Hutchinson, J. E., Fractals and self-similarity. Indiana Univ. Math. J. 1981, Vol. 30, No. 5, pp. 713–747. [142] Berger, M. A. and Wang Y., Bounded semi-groups of matrices. Linear Alg. Appl. 1992, Vol. 166, pp. 21–27. [143] Daubechies, I. and Lagarias, J., Corrigendum/addendum to: Sets of matrices all infinite products of which converge. Linear Alg. Appl. 2001, Vol. 327, pp. 69–83. [144] Strang, G., The joint spectral radius. Commentary by Gilbert Strang on paper number 5 in “Collected works of Gian-Carlo Rota”, 2001. [145] Ando, T. and Shih, M., Simultaneous contractibility. SIAM J. Math. Anal. 1998, Vol. 19, No. 2, pp. 487–498. [146] Barabanov, N. E., On the Lyapunov exponent of discrete inclusions. III (in Russian). Avtomat. i Telemekh. 1988, no. 5, 17–24; translation in Automat. Remote Control 49 (1988), no. 5, part 1, 558–565. [147] Gurvits, L., Stability of discrete linear inclusions. Linear Alg. Appl. 1995, Vol. 231, pp. 47–85. [148] Vladimirov, A. A., Elsner, L., and Beyn, W.-J., Stability and paracontractivity of discrete linear inclusions. Linear Alg. Appl. 2000, Vol. 312, pp. 125–134. [149] Wirth, F., The generalized spectral radius and extremal norms. Linear Alg. Appl. 2002, Vol. 342, pp. 17–40. [150] Kozyakin, V. S., Algebraic unsolvability of a problem on the absolute stability of desynchronized systems (in Russian). Avtomat. i Telemekh. 1990, no. 6, 41–47; translation in Automat. Remote Control 51 (1990), no. 6, part 1, 754–759. [151] Tsitsiklis, J. N., The stability of the products of finite set of matrices. Open problems in communication and computation, T. M. Cover and B. Copinath, eds., 1987, Springer-Verlag, New York, pp. 161–163. [152] Guglielmi, N. and Zennaro, M., On the zero-stability of variable stepsize multistep methods: the spectral radius approach. Numer. Math. 2001, Vol. 88, pp. 445–458. [153] Moision, B. E., Orlitsky, A., and Siegel, P. N., On codes that avoid specified differ- ences. IEEE Trans. Inform. Theory, 2001, Vol. 47, No. 1, pp. 433–442. [154] Dyn, N., Gregory, J. A., and Levin, D., Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 1991, Vol. 7, pp. 127–147. [155] Yu, T., Han, B., and Overton, M., Design of Hermite subdivision schemes aided by spectral radius optimization. Preprint, 2004. 500 BIBLIOGRAPHY

[156] Protasov, V. Yu., Asymptotics of the partition function (in Russian). Mat. Sb. 191 (2000), no. 3, 65–98; translation in Sb. Math. 191 (2000), no. 3-4, 381–414. [157] Protasov, V. Yu., On the problem of the asymptotics of the partition function (in Russian). Mat. Zametki 76 (2004), no. 1, 151–156; translation in Math. Notes 76 (2004), no. 1-2, 144–149. [158] de Rham, G., Une peu de mathematique a propos d’une courbe plane (in French). Revue de Mathematiques Elementaires, 1947, II, Nos. 4, 5, pp. 678–689. [159] de Rham, G., Sur une courbe plane (in French). J. Math. Pur. Appl. 1956, 35, pp. 25–42. [160] de Rham, G., Sur les courbes limit de polygones obtenus par trisection (in French). Enseign. Math. 1959, II, 5, pp. 29–43. [161] Erdös, P., On a family of symmetric Bernuolli convolutions. Amer. J. Math. 1939, Vol. 61 , pp. 974–975. [162] Erdös, P., On the smoothness properties of Bernuolli convolutions. Amer. J. Math. 1940, Vol. 62, pp. 180–186. [163] Rvachev, V. L. and Rvachev, V. A., On a function with compact support (in Rus- sian). Dokl. Akad. Nauk. USSR, 1971. [164] Derfel, G. A., A probabilistic method for studying a class of functional-differential equations (in Russian). Ukrain. Mat. Zh. 41 (1989), no. 10, 1322–1327, 1436; translation in Ukrainian Math. J. 41 (1989), no. 10, 1137–1141 (1990). [165] Dyn, N. and Levin, D., Interpolatory subdivision schemes for the generation of curves and surfaces. Multivariate approximation and interpolation, 1990 (Duisburg 1989), pp. 91–106, Birkhäuser, Basel. [166] Daubechies, I. and Lagarias, J., Two-scale difference equations. I. Global regularity of solutions. SIAM. J. Math. Anal. 1991, Vol. 22, pp. 1388–1410. [167] Fix, G. and Strang, G., Fourier analysis of the finite element method in Ritz-Galerkin theory. Stud. Appl. Math. 1969, Vol. 48, pp. 265–273. [168] Protasov, V., A complete solution characterizing smooth refinable functions. SIAM Journal of Math. Analysis, 2000, Vol. 31, No. 6, pp. 1332–1350. [169] Jia, R.-Q., Refinable shift-invariant spaces: From splines to wavelets. C. K. Chui et al., eds., Approximation theory VIII. Vol. 2. Wavelets and multilevel approximation. Papers from the 8th Texas international conference, College Station, TX, USA, Jan- uary 8–12, 1995. Singapore: World Scientific. Ser. Approx. Decompos. 1995, Vol. 6, pp. 179–208. [170] De Boor, C., DeVore, R., and Ron, A., The structure of finitely generated shift- d invariant spaces in L2(R ). J. Funct. Anal. 1994, Vol. 119, No. 1, pp. 37–78. [171] De Boor, C., DeVore, R., and Ron, A., Approximation from shift-invariant subspaces d of L2(R ). Trans. Amer. Math. Soc. 1994, Vol. 341, No. 2, pp. 787–806. [172] De Boor, C., DeVore, R., and Ron, A., Approximation orders of FSI spaces in d L2(R ). Constructive Approximation, 1998, Vol. 14, No. 3, pp. 411–427. [173] Villemoes, L., Wavelet analysis of refinement equations. SIAM J. Math. Anal. 1992, Vol. 25, No. 5, pp. 1433–1460. [174] Jia, R. Q. and Wang, J., Stability and linear independence associated with wavelet decomposition. Proc. Amer. Math. Soc. 1993, Vol. 117, pp. 1115–1124. [175] Lawton, W. M., Necessary and sufficient conditions for constructing orthonormal wavelets. J. Math. Phys. 1991, Vol. 32, pp. 57–61. [176] Cohen, A., Ondelettes, analyses multirésolutions et filtres miroir en quadrature. Ann. Inst. H. Poincaré, Anal. linéaire, 1990, Vol. 7, pp. 439–459. [177] Cohen, A., Ondelettes, analyses multirésolutions et traitement numérique du signal. Ph.D. Thesis, 1990, Université Paris, Dauphine. [178] Zhou, D.-X., Stability of refinable functions, multiresolution analysis, and Haar bases. SIAM J. Math. Anal. 1996, Vol. 27, No. 3, pp. 891–904. BIBLIOGRAPHY 501

[179] Mallat, S., Multiresolution approximation and wavelets. Trans. Amer. Math. Soc. 1989, Vol. 315, pp. 69–88. [180] Protasov, V., Fractal curves and their applications to wavelets. Proceeding of the International Workshop on Self-similar Systems, July 30–August 7, 1998, Dubna, Russia, 1999, pp. 120–125. [181] Berg, L. and Plonka, G., Spectral properties of two-slanted matrices. Results Math. 1999, Vol. 35, No. 3-4, pp. 201–215. [182] Berg, L. and Plonka, G., Some notes on two-scale difference equations. Functional equations and inequalities, Math. Appl. 2000, No. 518, pp. 7–29. [183] Protasov, V., The correlation between the convergence of subdivision processes and solvability of refinement equations. “Algorithms for Approximation IV, Proceedings of the 2001 International Symposium”, Huddersfield, England, July 15–20, 2001, pp. 394–401, J. Levesley, I. J. Anderson, and J. C. Mason, eds., Huddersfield University, ISBN No. 186-218-04-07. [184] Rioul, O., Simple regularity criteria for subdivision schemes. SIAM J. Math. Anal. 1992, Vol. 23, pp. 1544–1576. [185] Rota, G. C. and Strang, G., A note on the joint spectral radius. Kon. Nederl. Acad. Wet. Proc. 1960, Vol. 63, pp. 379–381. [186] Cohen, A. and Conze, J. P., Régularité des bases d’ondelettes et mesures ergodiques. Rew. Math. Iberoamer. 1992, Vol. 8, pp. 351–366. [187] Lau, K. S., Ma, M. F., and Wang, J., Of some sharp regularity estimations of L2 scaling functions. SIAM J. Math. Anal. 1996, Vol. 27, pp. 835–864. [188] Lau, K. S. and Wang, J., Characterization of Lp solutions for two-scale dilation equations. SIAM J. Math. Anal. 1995, Vol. 26, pp. 1018–1046. [189] Battle, G., Heisenberg Inequalities for Wavelets States. Appl. Comp. Harm. Analysis 1997, 4, pp. 119–146. [190] Novikov, I. Ya, Wavelets (brief survey of the foundations of the theory) (in Russian). Materialy 12 Sibirskoi Shkoly, Novosibirsk, 18–23 July, 1998. Novosibirsk: Izd-vo instituta matematiki im. S. L. Soboleva, 1999. pp. 92–111. [191] Lemarie-Rieusset, P. G., Existence de “fonction-pere” pour le ondelettes a support compact. C. R. Acad. Sci. Paris I. 1992, Vol. 314, pp. 17–19. [192] Deslauriers, G. and Dubuc, S., Interpolation dyadique. Fractals, dimensions non entiéres et applications. G. Cherbit, ed., Masson, Paris, 1987, pp. 44–55. [193] Daubechies, I., Orthonormal basis of compactly supported wavelets. Comm. Pure Appl. Math. 1988, 46, pp. 909–996. [194] Taswell, C., The Systematized Collection of Wavelet Filters Computable by Spectral Factorization of the Daubechies Polynomial. Technical Report CT-1998-08. [195] Lemarie-Rieusset, P. G. and Zahrouni, E., More regular wavelets. Applied and Com- putational Harmonic Analysis. 1998, Vol. 5, pp. 92–105. [196] Villemoes, L., Energy moments in time and frequency for two-scale difference equation solutions and wavelets. SIAM J. Math. Anal. 1992. Vol. 23, 6, pp. 1519–1543. [197] Volkmer, H., Asymptotic regularity of compactly supported wavelets. SIAM J. Math. Anal. 1995, Vol. 26, 4, pp. 1075–1087. [198] Volkmer, H., On the regularity of wavelets. IEEE Trans. Inf. Theory. 1992, Vol. 38, 2, pp. 872–876. [199] Ojanen, H., Orthonormal compactly supported wavelets with optimal Sobolev regularity. Technical Report math.CA/9807089, 1998. [200] Chui, C. K. and Wang, J., High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization. CAT Report #322. 1994, pp. 1–24. [201] Novikov, I. Ya., Modified Daubechies wavelets preserving localization with growth of smoothness. East J. Approximation. 1995, Vol. 1, No. 3, pp. 341–348. 502 BIBLIOGRAPHY

[202] Novikov, I. Ya., Uncertainty constants for modified Daubechies wavelets (in Rus- sian). Proceedings of the International Conference “Approximation Theory and Har- monic Analysis” (in Russian) (Tula, 1998). Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 4 (1998), no. 1, 107–111, 165. [203] Shen, J. and Strang, G., The zeros of the Daubechies polynomials. Proc. Amer. Math. Soc. 1995, Vol. 124, No. 12, pp. 3819–3833. [204] Temme, N. M., Asymptotics and numerics of zeros of polynomials that are related to Daubechies wavelets. Appl. Comp. Harmonic Anal. 1997, Vol. 4, No. 4, pp. 414–428. [205] Temme, N. M., Asymptotic invertion of the incomplete beta function. J. Comp. Appl. Math. 1992, Vol. 41, pp. 145–157. [206] Kantorovich, L. V., On the convergence of a sequence of Bernstein polynomials outside the main interval (in Russian). Izv. Akad. Nauk SSSR, 1931. [207] Berkolaiko, M. Z. and Novikov, I. Ya., Infinitely smooth almost-wavelets with compact support (in Russian). Dokl. Akad. Nauk 326 (1992), no. 6, 935–938; translation in Russian Acad. Sci. Dokl. Math. 46 (1993), no. 2, 378–382. [208] Berkolaiko, M. Z. and Novikov, I. Ya., Infinitely smooth compactly supported near- wavelets (in Russian). Mat. Zametki 56 (1994), no. 3, 3–12, 157; translation in Math. Notes 56 (1994), no. 3-4, 877–883 (1995). [209] Novikov, I. Ya., On the construction of nonstationary orthonormal infinitely differentiable compactly supported wavelets. Functional Differential Equations. 1994, Vol. 2, pp. 145–156. [210] Cohen, A. and Dyn, N., Nonstationary subdivision schemes and multiresolution analysis. SIAM J. Math. Anal. 1996, Vol. 27, pp. 1745–1769. [211] Novikov, I. Ya., Nonstationary orthonormal infinitely differentiable compactly supported wavelets with uniformly bounded uncertainty constants. Proceedings of the International Workshop (July 30–August 7, 1998, Dubna, Russia). (JINR, E5-99-38, Dubna, 1999), pp. 110–116. [212] Rvachev, V. L. and Rvachev, V. A., Nonclassical methods of approximation theory in boundary value problems. Naukova Dumka, Kiev, 1979. [213] Rvachev, V. A., Compactly-supported solutions of functional-differential equations and their applications (in Russian). Uspekhi Mat. Nauk 45 (1990), no. 1(271), 77– 103, 222; translation in Russian Math. Surveys 45 (1990), no. 1, 87–120. [214] Dyn, N. and Ron, A., Multiresolution analysis by infinitely differentiable compactly supported functions. Applied and Computational Harmonic Analysis. 1995, 2, pp. 15–20. [215] Chui, C. K. and Shi, X., Continuous two-scale equations and dyadic wavelets. Ad- vances in Comp. Math. 1994, 2, pp. 185–213. [216] Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 1946, 4, pp. 45–99, 112–141. [217] Soman, A. K. and Vaidyanathan, P. P., Orthonormal wavelets and paraunitary filter banks. IEEE Trans. on Sign. Proc. 1993, Vol. 41. No. 3. pp. 1170–1183. [218] Chui, C. K. and Mhaskar, H. N., On trigonometric wavelets. Constr. Approx. 1993, Vol. 9, pp. 167–190. [219] Chui, C. K. and Wang, J., A general framework of compact supported splines and wavelets. J. Approx. Th. 1992, Vol. 71, pp. 263–304. [220] Zheludev, V. A., Periodic splines and wavelets. Proc. of the Conference “Math. Analysis and Signal Processing”, Cairo, January 1994, pp. 2–9. [221] Petukhov, A. P., Periodic wavelets (in Russian). Mat. Sb. 188 (1997), no. 10, 69–94; translation in Sb. Math. 188 (1997), no. 10, 1481–1506. [222] Skopina, M., Multiresolution analysis of periodic functions. East J. Approximation. 1997, Vol. 3, No. 2, pp. 203–224. BIBLIOGRAPHY 503

[223] Maksimenko, I. E. and Skopina, M. A., Multidimensional periodic wavelets (in Rus- sian). Algebra i Analiz 15 (2003), no. 2, 1–39; translation in St. Petersburg Math. J. 15 (2004), no. 2, 165–190. [224] Petukhov, A., Trigonometric rational wavelet bases. Proceedings of the International Workshop (July 30–August 7, 1998, Dubna, Russia) (JINR, E5-99-38, Dubna, 1999), pp. 116–119. [225] Coifman, R. R., Meyer, Y., and Wickerhauser, V. M., Size properties of wavelet packets. In “Wavelets and their applications”, Beylkin et al., eds., Jones and Bartlett, 1992, pp. 453–470. [226] Skopina, M., Local convergence of Fourier series with respect to periodized wavelets. J. Approx. Theory. 1998, Vol. 94, pp. 191–202. [227] Novikov, I. Ya., Wavelets of Y. Meyer—an optimal basis in C(0, 1) (in Russian). Mat. Zametki 52 (1992), no. 5, 88–92, 143; translation in Math. Notes 52 (1992), no. 5-6, 1137–1140 (1993). [228] Skopina, M., Wavelet approximation of periodic functions. J. Approx. Theory, 2000, Vol. 104, pp. 302–329. [229] Belinskii, E. S., Summability of multiple Fourier series at Lebesgue points (in Rus- sian). Teor. Funkcii Funkcional. Anal. i Prilozen. No. 23 (1975), 3–12, 169. [230] Skopina, M. A., The generalized Lebesgue sets of functions of two variables. Pro- ceedings. Colloquia Math. Societ. Janos Bolyai. 58, Conference on Approximation Theory. Hungary. Kecskemet. August 6–11, 1990, pp. 615–625. [231] Saks, S., On the strong derivatives of functions of intervals. Fund. Mat. 1935, Vol. 25, pp. 245–252. [232] Jia, R. Q., A Bernshtein-type inequality associated with wavelet decomposition. Constr. Approx. 1993, Vol. 9, pp. 299–318. [233] Jia, R. Q. and Lei, J., On approximation by multi-integer translates of functions having global support. J. Approx. Th. 1993, Vol. 72, pp. 2–23. [234] Lei, J., Jia, R. Q., and Cheny, E. W., Approximation from shift invariant spaces by integral operators. SIAM Journal of Approx. Th. 1997, Vol. 28, pp. 481–498. [235] Calderón, A. P. and Zygmund, A., Local properties of solutions of elliptic partial differential equation. Studia Math. 1961, Vol. 20, pp. 171–227. [236] Skopina, M., Localization principle for wavelet expansions. Proceedings of the In- ternational Workshop (July 30–August 7, 1998, Dubna, Russia) (JINR, E5-99-38, Dubna, 1999), pp. 125–130. [237] Wojtaszczyk, P., Wavelets as unconditional bases in Lp(R). J. Fourier Anal. Appl. 1999, Vol. 5, No. 1, pp. 73–85. [238] Wojtaszczyk, P. and Figiel, T., Special bases in function spaces. Chapter 14 in Handbook of the Geometry of Banach Spaces, Vol. I, Elsevier Science, 2001. [239] Ulyanov, P. L., On some results and problems from the theory of bases (in Russian). Zapiski nauchnyh seminarov LOMI. Vol. 170, 1989. [240] Faber, G., Uber¨ die interpolatorische Darstelung stetiger Functionen. Jahresber Deutsch. Math.-Verein, 1914, Vol. 23, pp. 192–210. [241] Privalov, Al. A., The growth of the powers of polynomials, and the approximation of trigonometric projectors (in Russian). Mat. Zametki 42 (1987), no. 2, 207–214, 343. (Reviewer: L. Gogoladze) [242] Privalov, Al., A. Growth of powers of polynomial bases (in Russian). Mat. Zametki 48 (1990), no. 4, 69–78, 159; translation in Math. Notes 48 (1990), no. 3-4, 1017–1024 (1991). [243] Offin, D. and Oskolkov, K., A note on orthonormal polynomial bases and wavelets. Constr. Appr. 1993, Vol. 9, No. 1, pp. 319–325. [244] Lorentz, R. A. and Sahakian, A. A., Orthogonal trigonometric Shauder bases of optimal degree for C(0, 2π). J. Fourier Anal. Appl. 1994, Vol. 1, No. 1, pp. 103–112. 504 BIBLIOGRAPHY

[245] Skopina, M. A., On polynomial bases in the space C[−1, 1] (in Russian). Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), Issled. po Linein. Oper. i Teor. Funkts. 27, 223–226, 236; translation in J. Math. Sci. (New York) 110 (2002), no. 5, 3027–3028. [246] Skopina, M. A., Orthogonal polynomial Schauder bases in C[−1, 1] with optimal growth of degrees (in Russian). Mat. Sb. 192 (2001), no. 3, 115–136; translation in Sb. Math. 192 (2001), no. 3-4, 433–454. [247] Wo´zniakowski, K., On an orthonormal polynomial bases C[−1, 1]. Studia Math. 2001, Vol. 144, No. 2, pp. 181–196. [248] Chanillo, S. and Muckenhoupt, B., Weak type estimates for Cesaro sums of Jacobi polynomial series. Mem. Amer. Math. Soc. 1993, Vol. 102, No. 487, pp. 1–90. [249] Lemarie, P. G. and Meyer, Y., Ondelettes et bases hilbertiennes. Revista Matematica Iberoamericana, 1986, 2, 1-2, pp. 1–18. r [250] Lizorkin, P. I., Bases and multipliers in the spaces Bp,θ(Π) (in Russian). Analytic number theory, mathematical analysis and their applications (dedicated to I. M. Vinogradov on his 85th birthday). Trudy Mat. Inst. Steklov. 143 (1977), 88–104, 209. r [251] Orlovskii, D. G., On multipliers in the spaces Bp,θ (Russian summary). Anal. Math. 5 (1979), no. 3, 207–218. [252] Bochkarev, S. V., Bases in function spaces and the Franklin system (Russian). Trans- lated in Proc. Steklov Inst. Math. 1992, no. 1, 19–37. Theory of functions (in Russian) (Amberd, 1987). Trudy Mat. Inst. Steklov. 190 (1989), 22–39. [253] Meyer, Y., Principe d’incertitude, bases hilbertiennes et algebres d’operateurs. As- terisque, 1987, 145–146, pp. 206–223. [254] Friezier, M. and Jawerth, B., A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 1990, 93, No. 1, pp. 34–170. [255] Friezier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS-AMS Regional Conf. Ser., No. 79, 1991. [256] Torres, R. H., Boundedness results for operators with singular kernels on distribution spaces. Mem. Amer. Math. Soc. 1991, 90, 442, pp. 1–172. [257] Kateb, D., Lemarie-Rieusset P. G. Asymptotic behavior of the Daubechies filters. Appl. Comp. Harmonic Anal. 1995, Vol. 2, No. 4, pp. 398–399. [258] Berkolaiko, M. Z., Traces of functions in generalized Sobolev spaces with a mixed norm on an arbitrary coordinate subspace. II (in Russian). Trudy Inst. Mat. (Novosi- birsk) 9 (1987), Issled. Geom. “v tselom” i Mat. Anal., 34–41, 206. [259] Berkolaiko, M. Z., Convexity and concavity of Banach ideal spaces, and embedding theorems (in Russian). Sibirsk. Mat. Zh. 31 (1990), no. 3, 11–18, 216; translation in Siberian Math. J. 31 (1990), no. 3, 373–379 (1991). 0 [260] Lizorkin, P. I., Fourier transformation in Besov spaces. The zero scale of Bp,θ (in Russian). Dokl. Akad. Nauk SSSR 1965, 163, pp. 1318–1321. [261] Strömberg, J.-O., A modified Franklin system and higher order spline systems on Rn as unconditional basis of Hardy spaces. Wadsworth Math. Ser. 1982, 2, pp. 475–493. [262] Seeger, A., A note on Triebel-Lizorkin spaces. Approx. and Func. Spaces. Banach Center Publ., 1989, 22, pp. 391–400. [263] Riviere, N. M., Singular integrals and multiplier operators. Ark. Mat. 1971, 9, pp. 243–278. [264] Beylkin, G., Coifman, R., and Rokhlin, V., Fast wavelet transforms and numerical algorithms. I. Res. Rep. JALEU/DCS/RR-696, 1989, pp. 1–46. [265] Meyer, J., Wavelets and operators. Lect. Notes Math. 1989, Vol. 137. [266] Marshall, J., Weighted parabolic Triebel spaces of product type. Forum Math. 1991, 3, pp. 479–511. r [267] Lizorkin, P. I., Properties of functions in the spaces Λp, θ. (in Russian). Trudy Mat. Inst. Steklov. 131 (1974), 158–181. Index

0-1 set, 209 Fejer means, 324, 472, 473 2-radius, 461 filter, 9 fractal, 474 a-cube, 417 frame, 52 affine fractal, 475 almost diagonal operator, 431 generalized Lebesgue set, 353, 466 anisotropic distance, 410 generating (a PMRA) function, 332 anisotropic order of multi-index, 410 generating scaling function, 8 anisotropy vector, 410 autocorrelation function, 157 Haar MRA, 24, 108 Hahn-Banach theorem, 454 B-spline, 24, 140 Hausdorff distance, 474 basis in a Banach space, 453 Battle-Lemarie MRA, 25 interpolation MRA, 23 Besov space, 369, 407 invariant cycle, 262 Bessel system, 2 irreducible collection of operators, 192 best approximation, 359, 472 Jackson’s theorem, 472 binary tree, 123 joint spectral radius, 191, 459 biorthonormal systems, 6 joint spectral radius along a sequence, 204 block, 413 blocking set, 124 Koch curve, 217 bounded functional, 454 box spline, 74, 96 p-stability, 134 Lawton’s criterion, 87 cardinal B-spline, 140 layer, 413 center of function, 30 linear continuous functional, 454 Chaikin’s algorithm, 140 Lipschitz spaces, 368 clean mask, 219 Lizorkin-Triebel spaces, 407 Cohen’s compact set, 82 lower spectral radius, 191, 459 Cohen’s criterion, 82 Lyapunov exponent, 209, 461 complete system, 453 contraction mapping principle, 474 Marcinkiewicz interpolation theorem, 458 curve of binomial distribution, 216 mask, 70, 94 cycle of polynomial, mask, 133 mask cleaning, 222 cyclic set, 133 mean smoothness, 209, 410 Meyer MRA, 28 Daubechies masks, 151 Meyer-David wavelets, 407 Daubechies wavelets, 151 minimal system, 453 de Rham’s curves, 141, 217, 255 modulus of continuity, 359, 471 difference equation, 475 MRA, 7, 69 Dirac delta function, 456 multiplicity of blocking set, 124 distributions, 455 multiresolution analysis, 7, 69 dual space, 454 dual system, 453 NMRA, 276 Dubuc’s interpolation scheme, 254 nonseparable MRA, 70

505 506 INDEX nonstationary multiresolution analysis, 276 uncertainty constant, 31 nonstationary orthonormal wavelet basis, unconditional basis, 453 271 unimodular matrix, 95 nonstationary wavelet basis, 271 unimodular row, 95 nonstationary wavelet systems, 271 normal number, 204 Vallee Poussin means, 362, 472 Vandermonde determinant, 476 operator of strong type, 458 vectors congruent modulo A,66 operator of the class APDO, 431 wavelet function, 20, 327, 329 operator of weak type, 458 wavelet packet, 332 order of approximation, 126 wavelet polynomial of best approximation, orthogonal scaling function, 143 359 orthogonal wavelet function, 143 wavelet space, 15, 327, 329 orthogonalized scaling sequence, 326 wavelet system, 20 p-radius, 198, 461 pair of symmetric roots, 133 Paley-Wiener theorem, 464 periodic multiresolution analysis, 315 periodic zero, 130 PMRA, 315 polyphase matrices, 92 radius of function, 30 refinable function, 74 refinement equation, 9, 70 regular functions, 456 Riesz basis, 1, 134 Riesz system, 1 Riesz’s lemma, 469 Riesz’s theorem about continuous functionals in C[0, 1] , 454 scaling function, 8, 70 scaling sequence, 276, 316 Schwartz space, 457 self-similar set, 474 separable MRA, 65 set of digits of matrix, 66 set of wavelet functions, 90 sets congruent modulo A,67 sets congruent modulo Zd,67 Shannon-Kotelnikov MRA, 29 singular functions, 456 Sobolev spaces, 471 spline wavelets, 27 stable function, 134 Strömberg wavelets, 27 Strang-Fix condition, 117 strong Lebesgue points, 354, 466 sum rule, 138 symmetric roots, 133 tempered distributions, 457 test functions, 455 tight frame, 52 transition operator, 75 trivial cycle, 133 trivial cyclic set, 133 Wavelet theory lies on the crossroad of pure and computational mathematics, with connections to audio and video signal processing, data compression, and information transmission. The present book is devoted to a systematic exposition of modern wavelet theory. It details the construction of orthogonal and biorthogonal systems of wavelets and studies their structural and approximation properties, starting with basic theory and ending with special topics and problems. The book also presents some applications of wavelets. Historical commentary is supplied for each chapter in the book, and most chapters contain exercises. The book is intended for professional mathematicians and graduate students working in functional analysis and approximation theory. It is also useful for engineers applying wavelet theory in their work. Prerequisites for reading the book consist of graduate courses in real and functional analysis.

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