DOCSLIB.ORG

Mathematical Monographs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Monographs Translations of MATHEMATICAL ONOGRAPHS M Volume 239 Wavelet Theory I. Ya. Novikov V. Yu. Protasov M. A. Skopina American Mathematical Society 10.1090/mmono/239 Translations of MATHEMATICAL ONOGRAPHS M 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 ΕΙΣΙΤΩ S A O C C I I American Mathematical Society R E E T ΑΓΕΩΜΕ Y 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, Str¨omberg, 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 represen- tation 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 process- ing. 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.
Load more

Recommended publications
  • Refinable Functions with Non-Integer Dilations
    REFINABLE FUNCTIONS WITH NON-INTEGER DILATIONS XIN-RONG DAI, DE-JUN FENG, AND YANG WANG Abstract. Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refinable functions and distributions. We prove that uniform decay can be achieved for any dilation. This leads to the existence of refinable functions that can be made arbitrarily smooth for any given dilation factor. We exploit the connection between algebraic properties of dilation factors and the regularity of refinable functions and distributions. Our work can be viewed as a continuation of the work of Erd¨os[6], Kahane [11] and Solomyak [19] on Bernoulli convolutions. We also construct explicitly a class of refinable functions whose dilation factors are certain algebraic numbers, and whose Fourier transforms have uniform decay. This extends a classical result of Garsia [9]. 1. Introduction In this paper we study the refinement equation m m X X (1.1) f(x) = cjf(λx − dj), cj = |λ| j=0 j=0 where λ ∈ R with |λ| > 1 and all cj, dj are real. It is well known that up to a scalar multiple the above refinement equation has a unique distribution solution f, which is furthermore compactly supported. We shall refer to the distribution solution f(x) of (1.1) with fˆ(0) = 1 the solution to (1.1). For the refinement equation (1.1), the value λ is called the dilation fac- tor of the refinement equation, and {dj} the translation set or simply the translations.
    [Show full text]
  • Spectral Factorization of 2-Block Toeplitz Matrices and Refinement Equations
    Algebra i analiz St. Petersburg Math. J. Tom. 18 (2006), 4 Vol. 18 (2007), No. 4, Pages 607–646 S 1061-0022(07)00963-6 Article electronically published on May 30, 2007 SPECTRAL FACTORIZATION OF 2-BLOCK TOEPLITZ MATRICES AND REFINEMENT EQUATIONS V. YU. PROTASOV Abstract. Pairs of 2-block Toeplitz (N ×N)-matrices (Ts)ij = p2i−j+s−1, s =0, 1, i, j ∈{1,...,N}, are considered for arbitrary sequences of complex coefficients p0,...,pN . A complete spectral resolution of the matrices T0, T1 in the system of their common invariant subspaces is obtained. A criterion of nondegeneracy and of irreducibility of these matrices is derived, and their kernels, root subspaces, and all common invariant subspaces are found explicitly. The results are applied to the study of refinement functional equations and also subdivision and cascade approximation algorithms. In particular, the well-known formula for the exponent of regularity of a refinable function is simplified. A factorization theorem that represents solutions of refinement equations by certain convolutions is obtained, along with a character- ization of the manifold of smooth refinable functions. The problem of continuity of solutions of the refinement equations with respect to their coefficients is solved. A criterion of convergence of the corresponding cascade algorithms is obtained, and the rate of convergence is computed. §1. Introduction The finite-dimensional 2-Toeplitz operators corresponding to a given sequence of com- N plex numbers p0,...,pN are the operators T0,T1 acting in R and determined by the following matrices: (1) (T0)ij = p2i−j−1, (T1)ij = p2i−j,i,j∈{1,...,N} (we set pk =0fork<0andfork>N).
    [Show full text]
  • Infinite Convolution Products and Refinable Distributions on Lie Groups
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 6, Pages 2913{2936 S 0002-9947(00)02409-0 Article electronically published on March 2, 2000 INFINITE CONVOLUTION PRODUCTS AND REFINABLE DISTRIBUTIONS ON LIE GROUPS WAYNE LAWTON Abstract. Sufficient conditions for the convergence in distribution of an infi- nite convolution product µ1 ∗ µ2 ∗ ::: of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure µ and a dilation automorphism A.The existence of A implies G is nilpotent and simply connected and the exponen- tial map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset K⊂Gsuch that for any open set U containing K; and for any distribution f on G with compact support, there exists an integer n(U;f) such that n ≥ n(U;f) implies supp(T nf) ⊂U: If µ is supported on an A-invariant uniform subgroup Γ; then T is related, by an intertwining opera- tor, to a transition operator W on C(Γ): Necessary and sufficient conditions for T nf to converge to φ 2 L2, and for the Γ-translates of φ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of W to functions supported on Ω := KK−1 \ Γ: 1. Introduction and Statement of Results This paper extends concepts, associated with the theory of refinable functions and wavelets on Rd; to a connected Lie group G: In x2 we consider a sequence of measures µi; whose integrals equal 1, whose total variations jµij are uniformly bounded, and whose supports supp(µi)converge exponentially fast to the identity 1 2G.
    [Show full text]
  • Algebraic Properties of Subdivision Operators with Matrix Mask and Their Applications
    Journal of Approximation Theory 97, 294310 (1999) Article ID jath.1997.3266, available online at http:ÂÂwww.idealibrary.com on Algebraic Properties of Subdivision Operators with Matrix Mask and Their Applications Di-Rong Chen* Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China Communicated by Rong-Qing Jia Received June 19, 1997; accepted in revised form March 17, 1998 Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, espe- cially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case. 1999 Academic Press Key Words: subdivision operator; transition operator; mask; refinable vector; shift-invariant space; approximation order; accuracy; linear independence. 1. INTRODUCTION We denote by l(Z) the linear space of all complex valued sequences and by l0(Z) the linear space of all finitely supported sequences respectively. T For r sequences *k=(*k( j))j # Z # l(Z), 1kr, we identify * := ( *1 , ..., *r) #(l(Z))r with a sequence of components being r_1 vectors by letting T *( j)=(*1( j), ..., *r( j)) , j # Z. Therefore we adopt the notation *=(*( j))j # Z .
    [Show full text]
  • Solutions to Dilation Equations∗
    Solutions to Dilation Equations∗ David Malone <[email protected]> Ph.D., University of Dublin, 2000. A ∗Version: Date : 2000=10=1122 : 38 : 19 Revision : 1:70. LTEXed: September 7, 2001 Declaration There was a maths thesis submitted and for just one degree was admitted. I just won't say don't to the library's wont; and of no other's work it consisted. Or, somewhat less anapæsticlyy: This has not previously been submitted as an exercise for a degree at this or any • other University. This work is my own except where noted in the text. • The library should lend or copy this work upon request. • David Malone (September 7, 2001). ySee http://www.sfu.ca/~finley/discussion.html for more details. ii Summary This thesis aims to explore part of the wonderful world of dilation equations. Dilation equations have a convoluted history, having reared their heads in various mathematical fields. One of the early appearances was in the construction of continuous but nowhere differentiable functions. More recently dilation equations have played a significant role in the study of subdivision schemes and in the construction of wavelets. The intention here is to study dilation equations as entities of interest in their own right, just as the similar subjects of differential and difference equations are often studied. It will often be Lp(R) properties we are interested in and we will often use Fourier Analysis as a tool. This is probably due to the author's original introduction to dilation equations through wavelets. A short introduction to the subject of dilation equations is given in Chapter 1.
    [Show full text]
  • Approximation by Multiple Refinable Functions
    Approximation by Multiple Refinable Functions † Rong-Qing Jia, S. D. Riemenschneider, and Ding-Xuan Zhou Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1 Abstract We consider the shift-invariant space, S(Φ), generated by a set Φ = {φ1, . , φr} of T compactly supported distributions on IRwhen the vector of distributions φ := (φ1, . , φr) satisfies a system of refinement equations expressed in matrix form as X φ = a(α)φ(2 · − α) α∈ZZ where a is a finitely supported sequence of r × r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in S(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lp-approximation order of S(Φ) when the functions in Φ belong to Lp(IR) (see, Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of φ. AMS Subject Classification: Primary 39B12, 41A25, 65F15 Keywords and Phrases: Refinement equations, refinable functions, approximation order, accuracy, shift-invariant spaces, subdivision † Research supported in part by NSERC Canada under Grants # OGP 121336 and A7687 Approximation by Multiple Refinable Functions §1.
    [Show full text]
  • A Systematic Construction of Multiwavelets on the Unit Interval
    A Systematic Construction of Multiwavelets on the Unit Interval by Michelle Michelle A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics Department of Mathematical and Statistical Sciences University of Alberta c Michelle Michelle, 2017 Abstract One main goal of this thesis is to bring forth a systematic and simple con- struction of a multiwavelet basis on a bounded interval. The construction that we present possesses orthogonality in the derivatives of the multiwavelet basis among all scale levels. Since we are mainly interested in Riesz wavelets, we call such wavelets mth derivative–orthogonal Riesz wavelets. Furthermore, we present some necessary and sufficient conditions as to when such a construc- tion can be done. We show that our constructed multiwavelet bases possess many desirable properties such as symmetry, stability, and short support. The second goal of this thesis is to provide some conditions that guarantee a Riesz wavelet in L2(R) can be adapted so that it forms a Riesz wavelet for L2(I), where I is a bounded interval. As the third goal of this thesis, we also evalu- ate the performance of the newly constructed bases in obtaining the numerical solutions to some differential equations to showcase their potential usefulness. More specifically, we show how the resulting coefficient matrices are sparse and have a low condition number. ii Acknowledgements Firstly, I would like to thank my supervisor Dr. Bin Han for his willing- ness to share his knowledge, his guidance, and his unceasing encouragement throughout my entire Master’s program.
    [Show full text]
  • How to Refine Polynomial Functions
    October 1, 2018 21:17 WSPC/WS-IJWMIP paper International Journal of Wavelets, Multiresolution and Information Processing c World Scientific Publishing Company How to refine polynomial functions Henning Thielemann Institut für Informatik Martin-Luther-Universität Halle-Wittenberg 06110 Halle Germany [email protected] Received 28.11.2010 Accepted 26.03.2011 Revised 13.07.2011 Extended 23.11.2011 Research on refinable functions in wavelet theory is mostly focused to localized functions. However it is known, that polynomial functions are refinable, too. In our paper we inves- tigate on conversions between refinement masks and polynomials and their uniqueness. Keywords: Refinable Function; Wavelet Theory; Polynomial. AMS Subject Classification: 42C40, 1. Introduction Refinable functions are functions that are in a sense self-similar: If you add shrunken translates of a refinable function in a weighted way, then you obtain that refinable function again. For instance, see Figure 1 for how a quadratic B-spline can be de- arXiv:1012.2453v3 [math.FA] 29 Jan 2012 composed into four small B-splines and how the so called Daubechies-2 generator function is decomposed into four small variants of itself. All B-splines with successive integral nodes are refinable, but there are many more refinable functions that did not have names before the rise of the theory of refinable functions. In fact we can derive a refinable function from the weights of the linear combination in the refinement under some conditions. Refinable functions were introduced in order to develop a theory of real wavelet functions that complements the discrete sub-band coding theory.6 Following the requirements of wavelet applications, existing literature on wavelets focuses on re- finable functions that are L2-integrable and thus have a well-defined Fourier trans- form, are localized (finite variance) or even better of compact support.
    [Show full text]
  • 7Th International Conference on Computational Harmonic Analysis
    7th International Conference on Computational Harmonic Analysis Vanderbilt University Nashville, TN, USA May 14{18, 2018 Spatio-Temporal Trade-Off for Initial Data Best Approximation Roza Aceska Ball State University [email protected] Coauthors: Alessandro Arsie and Ramesh Karki We present a mathematical framework and efficient computational schemes to solve some classes of PDEs from scarcely sampled initial data. Full knowledge of the initial conditions in an initial value problem (IVP) is essential but often impossible to attain; the way to overcome this impairing is to exploit the evo- lutionary nature of the problem at hand, while working with a reduced number of sensors. Our framework combines spatial samples of various temporal states of the system of interest, thus compensating for the lack of knowledge of the initial conditions. Stable phase retrieval in infinite dimensions Rima Alaifari ETH Zurich, Switzerland [email protected] Coauthors: Ingrid Daubechies (Duke University), Philipp Grohs (University of Vienna), Rujie Yin (Duke University) Phase retrieval is the problem of reconstructing a signal from only the mag- nitudes of a set of complex measurements. The missing information of the phase of the measurements severely obstructs the signal reconstruction. While the problem is stable in the finite dimensional setting, it is very ill- conditioned. This makes it necessary to study the stability and regularization properties in an infinite-dimensional setting. We show that in some sense the ill- conditioning is independent of the redundancy of the measurements. However, the instabilities observed in practice are all of a certain type. Motivated by this observation, we introduce a new paradigm for stable phase retrieval.
    [Show full text]