Generalized Multiresolution Analyses

Lecture Notes in Applied and Numerical Harmonic Analysis

Kathy D. Merrill

Generalized Multiresolution Analyses

Lecture Notes in Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA Editorial Advisory Board Emmanuel Candes Stanford University Stanford, CA, USA Peter Casazza University of Missouri Columbia, MO, USA Gitta Kutyniok Technische Universität Berlin Berlin, Germany Ursula Molter Universidad de Buenos Aires Buenos Aires, Argentina Michael Unser Ecole Polytechnique Federale de Lausanne Lausanne, Switzerland

More information about this series at http://www.springer.com/series/4968

Kathy D. Merrill

Generalized Multiresolution Analyses

Kathy D. Merrill Department of Mathematics The Colorado College Colorado Springs, CO, USA

ISSN 2296-5009 ISSN 2296-5017 (electronic) Applied and Numerical Harmonic Analysis ISSN 2512-6482 ISSN 2512-7209 (electronic) Lecture Notes in Applied and Numerical Harmonic Analysis ISBN 978-3-319-99174-0 ISBN 978-3-319-99175-7 (eBook) https://doi.org/10.1007/978-3-319-99175-7 Library of Congress Control Number: 2018954118 Mathematics Subject Classification: 42-02, 42C40, 42C15, 43A40, 28A80, 20H15, 65T60 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

LN-ANHA Series Preface

The Lecture Notes in Applied and Numerical Harmonic Analysis (LN-ANHA) book series is a subseries of the widely known Applied and Numerical Harmonic Analysis (ANHA) series. The Lecture Notes series publishes paperback volumes, ranging from 80 to 200 pages in harmonic analysis as well as in engineering and scientific subjects having a significant harmonic analysis component. LN-ANHA provides a means of distributing brief-yet-rigorous works on similar subjects as the ANHA series in a timely fashion, reflecting the most current research in this rapidly evolving field. The ANHA book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-theart ANHA series. Our vision of modem harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, bio-medical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet

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theory depends not only on classical Fourier analysis but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in applicable topics such as the following, where harmonic analysis plays a substantial role: Bio-mathematics, bio-engineering, and bio-medical signal processing; Communications and RADAR; Compressive sensing (sampling) and sparse representations; Data science, data mining and dimension reduction; Fast algorithms; Frame theory and noise reduction;

Image processing and super-resolution; Machine learning; Phaseless reconstruction; Quantum informatics; Remote sensing; Sampling theory; Spectral estimation; Time-frequency and time-scale analysis – Gabor theory and wavelet theory.

The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, for example, the concept of “function.” Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in

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Fourier analysis not only characterizes the behavior of the prime numbers but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the adaptive modeling inherent in time-frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’être of the ANHA series! University of Maryland College Park, MD, USA

John J. Benedetto Series Editor

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Wavelets and Multiresolution Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Classes of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 7

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The Invariance of the Core Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Arbitrary Hilbert Spaces and Affine Structures. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Affine Structures with Abelian Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Classical Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 13 18 20

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The Multiplicity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Examples of Multiplicity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Consistency Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Dimension Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Characterizing Multiplicity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 26 29 32 36

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Wavelet Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Wavelet Sets from Multiplicity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Wavelet Sets from the Consistency Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Wavelet Sets from Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 38 40 43 49 53

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Generalized Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Classical Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Extended Uses of Classical Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Generalized Filters in GMRAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Building GMRAs and Parseval Wavelets from Filters in L2 (RN ). . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 58 60 68 73

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Fractal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 GMRAs and Wavelets on Enlarged Fractal Spaces . . . . . . . . . . . . . . . . . . . 6.2 Generalized Filters in Fractal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Fourier Transforms on Enlarged Fractal Spaces . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 81 83

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Composite Dilations and Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . 7.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Haar Type Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Wavelet Set Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Filters and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 93 95 96

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Abstract Constructions of GMRAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.1 Super-Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.2 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.3 A Classifying Space for GMRAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Chapter 1

Introduction

The history of wavelets is a story that demonstrates the power of collaboration between different specialties within mathematics, physics, engineering, and computer science. Many parallel beginnings of wavelets emerged during the twentieth century, as researchers in all of these fields applied and modified Fourier’s 1807 decomposition of functions into canonical pieces. (See [21] or [42] for details.) Applications in signal and image processing drove innovation, while theory from harmonic analysis and theoretical physics broadened and deepened the ideas. One early example of the fruitfulness of this interaction between application and theory coincided with the naming of wavelets in the early 1980s. Geophysical engineer J. Morlet and theoretical physicist A. Grossman collaborated to adapt windowed Fourier transform functions into a tool they called a wavelet [27]. Their paper already incorporated abstract harmonic analysis, a theme of this book, in the form of representations of the ax + b group. A second important example of collaboration between application and theory happened in the mid-1980s. Stéphane Mallat brought his knowledge of pyramidal algorithms in image processing to a collaboration with Yves Meyer, who brought insights from harmonic analysis dating back to the work of Littlewood-Paley and Calderon. The result was a formulation of the concept of multiresolution analyses, the ancestor of this book’s topic, and the basis of a construction method for wavelets [34, 41]. Since these beginnings, tools from classical harmonic analysis involving the standard Fourier transform have been an essential part of the theoretical side of the interactions that built and refined wavelets. Abstract harmonic analysis, while not quite so constant a presence in wavelet theory, has reappeared regularly, after being introduced into the beginnings of the subject by Grossman and Morlet. For example, in 1988 Feichtinger and Gröchenig [24] used group representations to unify Gabor and wavelet transforms into a single theory [28], and later Hartmut Führ extended their work to more general settings [25]. In 1995, Baggett, Carey, Moran, and Ohring [3] introduced a representation theoretic approach to the study of multiresolution structures. This volume will © Springer Nature Switzerland AG 2018 K. D. Merrill, Generalized Multiresolution Analyses, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-319-99175-7_1

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1 Introduction

describe the resulting strand of research, which is centered around the concept of a generalized multiresolution analysis. The use of abstract harmonic analysis to further develop this concept has been carried out by several groups during the late 1990s and into the first two decades of 2000. This work has led to more powerful and more general methods to build new wavelets, and also to an ability to use wavelets in broader contexts.

1.1 Wavelets and Multiresolution Structures Historically, wavelets have used the operations of dilation and translation to decompose functions into canonical pieces, so that each piece has a prescribed location and level of zoom. Ideally, the pieces decay rapidly outside of a narrow focus, and thus overcome the Fourier system’s weakness of being non-localized. Multiresolutions have been used to break the space up according to level of zoom, with a subspace Vj containing elements zoomed in up to level j . The development of multiresolution structures has roots in subband filtering in signal processing and pyramid algorithms in image reconstruction [15], and such structures occur naturally in these and other applied fields. For example, pattern recognition requires the ability to examine an image at different resolutions, since patterns of varying sizes are often being sought [35]. The use of multiresolutions in computer vision applications is encouraged by evidence showing that the human visual system itself uses multiresolution signal processing [1]. Further, the decomposition of a signal into a coarser component together with a detail component at each level enables a recursive method perfectly adapted to computer reconstruction. Conversely, the recursive structure behind dynamical systems such as iterated function systems can be naturally adapted to build multiresolution structures in that context. (See, for example, Chapter 6, and for a different approach, [36].) Wavelets and multiresolution structures were originally developed in L2 (R) with translations by integers and (x) = √ dilation by 2, using the unitary operators2 τn f N ) with to L (R f (x − n) and δf (x) = 2f (2x). Later, they were extended √ translations by the integer lattice and dilation δA f (x) = | det A|f (Ax) by an expansive (all eigenvalues have absolute value greater than 1) integer matrix A. The work of Baggett, Carey, Moran, and Ohring [3] generalized these operations to interrelated operators on an abstract Hilbert space. In particular, they replaced standard translations by a countable, discrete, not necessarily abelian group Γ of unitary operators on a separable Hilbert space H , and replaced dilation by another unitary operator δ on H such that δ −1 Γ δ ⊂ Γ. They called such a collection of operators an Affine Structure. In this context, they defined a wavelet: Definition 1.1 A (orthonormal) wavelet in a Hilbert space H is a finite set {ψ1 , ψ2 , · · · , ψL } ⊂ H such that {δ j (γ (ψi ))} forms an orthonormal basis of H , with −∞ < j < ∞, γ ∈ Γ, and 1 ≤ i ≤ L.

1.1 Wavelets and Multiresolution Structures

3

The authors in [3] then generalized the Mallat/Meyer definition of a multiresolution analysis not only to allow operators on an abstract Hilbert space, but also to replace the requirement of an orthonormal basis of translates in the core subspace by that of the core subspace being invariant under the action of Γ . They called this structure simply a multiresolution; it was later given the name generalized multiresolution analysis in [4], which is the name we will use in this book. We will follow the convention of reserving the name multiresolution analysis for such a structure that also has a scaling vector as defined below. Definition 1.2 A generalized multiresolution analysis (GMRA) of a Hilbert space H , relative to Γ and δ, is a collection {Vj }∞ j =−∞ of closed subspaces of H that satisfy: 1. 2. 3. 4.

Vj ⊆ Vj +1 for all j . δ(Vj ) = Vj +1 for all j . ∪Vj is dense in H and ∩Vj = {0}. V0 (called the core subspace) is invariant under the action of Γ .

A multiresolution analysis (MRA) is a collection {Vj }∞ j =−∞ of closed subspaces of H that satisfy conditions 1–3 above, together with 4 . There exists a scaling vector φ ∈ V0 such that {γ φ : γ ∈ Γ } gives an orthonormal basis for V0 . Condition (4) allows the application of the theory of group representations to prove theorems about the inter-relationship between wavelets and multiresolution structures. In [3], the authors showed that all (orthonormal) wavelets have associated GMRAs. This makes GMRAs a natural tool to pair with wavelets. While they found an example of a GMRA that can have no associated orthonormal wavelet, they also used an analysis of the representation of the group Γ on V0 to show that all abstract MRAs for groups Γ where δ −1 Γ δ is of finite index d in Γ do have associated wavelets, and established that the number L of wavelet functions must equal d − 1. A parallel line of research was begun in 1994 by de Boor, DeVore, and Ron [22], with the study of shift invariant spaces, spaces invariant under integer translations. They exploited the concept of a range function, introduced by Helson [29], to determine conditions under which a finitely generated shift invariant space has a generating set that is orthogonal or stable. Ron and Shen [44] later introduced techniques involving Gramian fibers, functions on RN whose values are nonnegative Hermitian matrices with coefficients that are inner products that sum over translates of L2 functions. They also used what they called quasi-affine systems to overcome the fact that sets of positive dilates of wavelets are not shift invariant [45]. We will point out connections to this parallel approach throughout the book. In the late 1990s and early 2000s, other researchers proposed different generalizations of Mallat and Meyer’s multiresolution analysis that were based on the concept of a frame. Definition 1.3 A frame for a Hilbert space H is a collection of vectors  {v1 , v2 , · · · } for which there exist positive frame bounds A and B with Av2 ≤ i | v, vi |2 ≤ Bv2 for all v ∈ H . A Parseval frame (or normalized tight frame) is a frame with bounds A = B = 1.

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Benedetto and Li [10] defined a frame multiresolution analysis (FMRA) in L2 (R), in which condition (4) of Definition 1.2 is replaced by the requirement that the subspace V0 have a frame consisting of translates of a single function. M. Papadakis [43] later defined a more general concept of a generalized frame multiresolution analysis (GFMRA) for an abelian group of operators acting on an abstract Hilbert space, in which the frame for V0 is allowed to consist of the group orbit of an arbitrary collection of functions in V0 . These authors also considered wavelets that provide frames rather than orthonormal bases for H . Definition 1.4 A Parseval wavelet in a Hilbert space H is a set {ψ1 , ψ2 , · · · } ⊂ H such that {ψi,γ ,j = δ j (γ (ψi ))} forms a Parseval frame for H , with −∞ < j < ∞, γ ∈ Γ, and i ≥ 1. If, in addition, ψi,γ ,j , ψi ,γ ,j = 0 whenever j = j , then {ψ1 , ψ2 , · · · } is called a semiorthogoonal Parseval wavelet.  Parseval wavelets allow reconstruction of elements of H by v = v, ψi,γ ,j ψi,γ ,j , while offering more robustness and mitigation of noise. Baggett, Courter, and Merrill showed in [5] that a quite general class of GMRAs are guaranteed to have Parseval wavelets. Chapter 2 will pursue these ideas by developing the properties of GMRAs that follow from the invariance of V0 under the action of the group. Using these properties, it will describe the relationship between GMRAs and both types of wavelets, as well as how GMRAs relate to the other multiresolution structures. The paper by Baggett, Medina and Merrill [4] that named GMRAs also analyzed more fully the case where the group Γ is abelian. Here they showed the existence of a multiplicity function and measure class that characterize the GMRA. In the particular case of L2 (RN ), the measure class was shown to be Haar. In this case, by comparing the representation of Γ on V0 with its representation on V1 , they determined that the multiplicity function has to obey a consistency equation, a result that is key to many of the applications of GMRA theory to wavelets in L2 (RN ). In the classical setting of dilations and translations in L2 (R), the multiplicity function has been shown to equal the wavelet dimension function [13, 46, 47], thus tying the results obtained by this abstract approach to classical results using the Fourier transform. Work by Bownik, Rzeszotnik, and Speegle [14] on the dimension function, and later by Baggett and Merrill [2] on the multiplicity function, established exactly which functions on the N −dimensional torus can be multiplicity functions for GMRAs in L2 (RN ), thus paving the way for constructions described below that start with a multiplicity function. The theory of the multiplicity function is developed in Chapter 3.

1.2 Classes of Examples As with most mathematical concepts, much of the value of the GMRA definition lies in interesting and useful examples. All MRAs are also GMRAs, but in the classical setting, the most interesting examples are GMRAs that are not also MRAs. The earliest of these examples were tied to wavelet sets, sets whose characteristic

1.2 Classes of Examples

5

functions are the Fourier transforms of wavelets. The first was discovered by Journé (see [20], p. 136) before a GMRA was defined. He found a wavelet set for dilation by 2 in L2 (R), which he could show has no associated MRA, but which necessarily (by Theorem 2.1) has an associated GMRA. By the L = d − 1 result from [3] mentioned above, single wavelets in L2 (RN ) for dilations of determinant greater than 2 must be non-MRA wavelets, and the earliest examples of these also were wavelet set wavelets. Dai, Larson and Speegle surprised the wavelet community in the late 1990s by showing that single wavelet set wavelets exist for all expansive dilations in all dimensions [16]. In general, wavelet sets for expansive matrix dilations in L2 (RN ) are directly linked to the multiplicity function of their GMRAs, so that examples of wavelet sets and GMRAs in L2 (RN ) are naturally built together. In [4], the multiplicity function was used to develop a construction procedure for all wavelet sets in RN . Later, this procedure led to the discovery of wavelet sets that are finite unions of convex sets for any scalar dilation in L2 (RN ) (see [37, 40]), as well as to a partial determination of which matrix dilations have wavelet sets with this property [39]. Chapter 4 develops the relationship between GMRAs and wavelet sets in L2 (RN ) and shows how it leads to these and other wavelet set results. Although wavelet sets are useful in providing key examples that settle questions in wavelet theory, their associated wavelets are not well-localized, and thus not the most useful for applications. In general, it has been shown that non-MRA (orthonormal) wavelets cannot be both well-localized and smooth. For example, ˆ is continuous and |ψ(ω)| ˆ any orthonormal wavelet ψ ∈ L2 (R) such that |ψ| = − 12 − ) for some positive , must have an associated MRA [30]. Thus, to O(|ω| associate smooth and well-localized wavelets in L2 (RN ) with GMRAs that are not MRAs, we must turn to Parseval wavelets. The technique for building well-localized and smooth Parseval wavelets for GMRAs uses generalized filters. Classical filters, periodic functions which are multiplied by Fourier transforms, were used by Mallat and Meyer [34, 41] to build wavelets from MRAs in L2 (R), and later by Daubechies [19] to build smooth compactly supported wavelets in the same setting. For dilation by 2 in L2 (RN ), A. Ron and Z. Shen [45] used filters with the Fourier transform in what they called the Unitary Extension Principle to build Parseval wavelets from given frames of translates for V0 . Benedetto/Li [10] and Papadakis [43] also used filters coming from given frames in their FMRA and GFMRA settings to build Parseval wavelets. Generalized filters in GMRAs are functions defined on the support of the multiplicity function, and in the case of L2 (RN ), can be viewed in terms of the Fourier transform in a way similar to classical filters. Baggett, Courter, and Merrill [5], and later Baggett, Jorgensen, Merrill, and Packer [6], developed a theory of generalized filters defined in terms of the multiplicity function, and used them to build both the GMRA itself and Parseval wavelets. One result [7] was the construction of a non-MRA Parseval wavelet with the same multiplicity function as the Journé wavelet, yet which has a C ∞ Fourier transform and can be made to be C r for an arbitrary integer r > 0. Smooth well-localized Parseval wavelets were

6

1 Introduction

also built with the same multiplicity functions as well-known wavelet sets in R2 [38]. Chapter 5 describes generalized filters, and shows how they are used to build GMRAs and wavelets. Other noteworthy examples of GMRAs have been MRAs built-in different settings than ordinary translations and dilations in L2 (RN ). In 2006, Dutkay and Jorgensen [23] constructed a fractal Hilbert space from the Hausdorff measure on R associated with the Cantor set, and then built an MRA and (orthonormal) wavelet on this space. Their results were later extended to other fractals, such as the Sierpinski gasket [18], Sierpinski carpet [17], and fractals from non-linear iterated function systems [12]. In [17], Parseval wavelets were also constructed for fractal spaces. All of these constructions use generalized filters in this non-traditional setting. The application of GMRA theory to fractal spaces is developed in Chapter 6. At about the same time, wavelets with composite dilations were introduced by Guo, Labate, Lim, Weiss, and Wilson [26]. These wavelets live in GMRAs on L2 (RN ), with Γ a (not necessarily discrete) subgroup of the affine group of RN and δ a dilation by a compatible expansive matrix. Note that unlike the previous examples, this group Γ is not abelian. Shearlets are examples of this type of construction that are useful in applications. Another important example of the theory of composite dilations is the subclass of the two-dimensional crystallographic groups consisting of those groups that split as semidirect products of translations and finite point groups. Recently, MacArthur and Taylor [33] described how composite dilations fit into GMRA theory, and also worked out how the action of the nonsemidirect product crystallographic groups on RN can be combined with compatible dilations to give GMRAs and build wavelets. A related theory by Larson and Massopust [32] replaced translations entirely by an affine Weyl group of reflections in a family of hyperplanes, and found wavelet sets for that affine structure. Chapter 7 of this volume presents the theory of composite dilations, as well as the actions of crystallographic groups that are not semidirect products, as examples of GMRA theory. The final chapter of this book, Chapter 8, looks at abstract constructions of GMRAs using direct limits and direct sums. Both techniques use multiplicity functions to build filters and then GMRAs, and both give new ways of looking at the examples of GMRAs described above. The direct limit construction was first developed by Larsen and Raeburn [31], and further explored in [8]. These authors showed that there are fewer restrictions on which functions can be multiplicity functions in contexts other than L2 (RN ), and used direct limits to build from the multiplicity function both new examples and familiar ones such as the Journé wavelet and the Cantor fractal space. Bildea, Dutkay, and Picioroaga in [11] used the ideas of abstract GMRAs to develop what they call super-wavelets in a Hilbert space composed of direct sums of L2 (R). Finally, Baggett, Furst, Merrill, and Packer [9] described a direct sum method of building an abstract GMRA from any multiplicity function and filter for which a GMRA exists. They used this construction to form a classification space for GMRAs.

References

7

References 1. Akansu, A., Haddad, R.: Multiresolution Signal Decomposition: Transforms, Subbands and Wavelets. Academic Press, San Diego (2001) 2. Baggett, L., Merrill, K.: Abstract harmonic analysis and wavelets in Rn . Contemp. Math. 247, 17–27 (1999) 3. Baggett, L., Carey, A., Moran, W., Ohring, P.: General existence theorems for orthonormal wavelets, an abstract approach. Publ. Res. Inst. Math. Sci. 31, 95–111 (1995) 4. Baggett, L., Medina, H., Merrill, K.: Generalized multiresolution analyses and a construction procedure for all wavelet sets in Rn . J. Fourier Anal. Appl. 5, 563–573 (1999) 5. Baggett, L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ). Appl. Comput. Harmon. Anal. 13, 201–223 (2002) 6. Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: Construction of Parseval wavelets from redundant filter systems. J. Math. Phys. 46, 1–28 (2005) 7. Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: A non-MRA C r frame wavelet with rapid decay. Acta Appl. Math. 89, 251–270 (2006) 8. Baggett, L., Larsen, N., Merrill, K., Packer, J, Raeburn, I.: Generalized multiresolution analyses with given multiplicity functions. J. Fourier Anal. Appl. 15, 616–633 (2009) 9. Baggett, L., Furst, V., Merrill, K., Packer, J.: Classification of generalized multiresolution analyses. J. Funct. Anal. 258, 4210–4228 (2010) 10. Benedetto, J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389–427 (1998) 11. Bildea, S., Dutkay, D., Picioroaga, G.: MRA super-wavelets. N. Y. J. Math. 11, 1–19 (2005) 12. Bohnstengel, J., Kesseböhmer, M.: Wavelets for iterated function systems. J. Funct. Anal. 259, 583–601 (2010) 13. Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces. Mich. Math. J. 51, 387–414 (2003) 14. Bownik, M., Rzeszotnik, Z., Speegle, D.: A characterization of dimension functions of wavelets. Appl. Comput. Harmon. Anal. 10, 71–92 (2001) 15. Bratteli, O., Jorgensen, P.: Wavelets Through a Looking Glass. Birkhäuser, Boston (2002) 16. Dai, X., Larson, D., Speegle, D.: Wavelet sets in Rn . J. Fourier Anal. Appl. 3, 451–456 (1997) 17. D’Andrea, J.: Constructing fractal wavelet frames. Numer. Funct. Anal. Optim. 33, 906–927 (2012) 18. D’Andrea, J., Merrill, K., Packer, J.: Fractal wavelets of Dutkay-Jorgensen type for the Sierpinski gasket space. Contemp. Math. 451, 69–88 (2008) 19. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988) 20. Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992) 21. Daubechies, I.: Where do wavelets come from?–A personal point of view. Proc. IEEE 84, 510– 513 (1996) 22. de Boor, C., DeVore, R., Ron, A.: The structure of finitely generated shift-invariant subspaces of L2 (Rd ). J. Funct. Anal. 119, 37–78 (1994) 23. Dutkay, D., Jorgensen, P.: Wavelets on fractals. Rev. Math. Iberoamericana 22, 131–180 (2006) 24. Feichtinger, H., Gröchenig, K.: A unified approach to atomic decompositions through integrable group representations. In: Cwikel, M., et al. (eds.) Function Spaces and Applications, pp. 52–73. Springer, Berlin (1988) 25. Führ, Hartmut: Abstract Harmonic Analysis of Continuous Wavelet Transforms. SpringerVerlag, Berlin, Heidelberg (2005) 26. Guo, K., Labate, D., Lim, W., Weiss, G., Wilson, E.: The theory of wavelets with composite dilations. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 231–250. Birkhäuser, Boston (2006)

8

1 Introduction

27. Grossmann, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. 15, 723–736 (1984) 28. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989) 29. Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York (1964) 30. Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996) 31. Larsen, N., Raeburn, I.: From filters to wavelets via direct limits. Contemp. Math. 414, 35–40 (2006) 32. Larson, D., Massopust, P.: Coxeter groups and wavelet sets. Contemp. Math. 451, 187–218 (2008) 33. MacArthur, J., Taylor, K.: Wavelets with crystal symmetry shifts. J. Fourier Anal. Appl. 17, 1109–1118 (2011) 34. Mallat, S.: Multiresolution approximations and wavelet orthonormal bases of L2 (R). Trans. Am. Math. Soc. 315, 69–87 (1989) 35. Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989) 36. Massopust, P.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, Orlando (1995) 37. Merrill, K.: Simple wavelet sets for scalar dilations in L2 (R2 ). In: Jorgensen, P., Merrill, K., Packer, J. (eds.) Wavelets and Frames: A Celebration of the Mathematical Work of Lawrence Baggett, pp. 177–192. Birkhäuser, Boston (2008) 38. Merrill, K.: Smooth well-localized Parseval wavelets based on wavelet sets in R2 . Contemp. Math. 464, 161–175 (2008) 39. Merrill, K.: Simple wavelet sets for matrix dilations in R2 . Numer. Funct. Anal. Optim. 33, 1112–1125 (2012) 40. Merrill, K.: Simple wavelet sets in Rn . J. Geom. Anal. 25, 1295–1305 (2015) 41. Meyer, Y.: Ondelettes, fonctions splines et analyses graduées. Rapport Ceremade 8703 (1987) 42. Meyer, Y.: Wavelets: Algorithms and Applications. Society for Industrial and Applied Mathematics, Philadelphia (1993) 43. Papadakis, M.: Generalized frame multiresolution analysis of abstract Hilbert spaces. In: Benedetto, J., Zayed, A. (eds.) Sampling, Wavelets and Tomography, pp. 179–223. Birkhäuser, Boston (2004) 44. Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of L2 (Rd ). Can. J. Math. 47, 1051–1094 (1995) 45. Ron, A., Shen, Z.: Affine systems in L2 (Rd ): the analysis of the analysis operator. J. Fourier Anal. Appl. 3, 408–447 (1997) 46. Ron, A., Shen, Z.: The wavelet dimension function is the trace function of a shift-invariant system. Proc. Am. Math. Soc. 131, 1385–1398 (2002) 47. Weber, E.: Applications of the wavelet multiplicity function. Contemp. Math. 247, 297–306 (1999)

Chapter 2

The Invariance of the Core Subspace

The invariance of the core subspace V0 of a GMRA under the group Γ of unitary operators implies the existence of a unitary representation of Γ on V0 , thus enabling the use of tools from abstract harmonic analysis. Because of the required condition δ −1 Γ δ ⊂ Γ , the invariance of V0 also gives invariance of all Vj for j ≥ 0, and thus representations of Γ there as well. The invariance of V1 in turn implies the invariance of W0 = V1  V0 , where the representations of Γ are useful in proving the existence of orthonormal or Parseval wavelets. In this chapter, we will explore results that follow from analyzing these representations of Γ . We start with the most general setting of an arbitrary affine structure on a separable abstract Hilbert space, and later consider the case of abelian Γ , and in particular, the classical setting of ordinary translation and expansive matrix dilations in L2 (RN ).

2.1 Arbitrary Hilbert Spaces and Affine Structures Let H be a separable Hilbert space with an affine structure given by a countable, discrete group Γ of unitary operators on H , together with another unitary operator δ, such that δ −1 Γ δ ⊂ Γ . Note that the relationship between Γ and δ implies that to be a part of an affine structure, Γ must have a chain of subgroups {Γi } = {δ −i Γ δ i }, for which the index of Γi in Γi−1 is constant. This gives an implicit restriction on which groups can be part of an affine structure associated with a GMRA. The map α(γ ) = δ −1 γ δ is an isomorphism of Γ onto Γ1 , and it induces a map α ∗ on the dual group Γ, defined by α ∗ (ω) = ω ◦ α. We begin with two theorems from Baggett, Carey, Moran, and Ohring [3] that establish a relationship between wavelets and GMRAs in the general case. Theorem 2.1 Let {ψ1 , · · · ψL } be an orthonormal wavelet in a Hilbert space H relative to the affine structure given by Γ and δ, and define Vj = span{δ k (γ (ψi )) :

© Springer Nature Switzerland AG 2018 K. D. Merrill, Generalized Multiresolution Analyses, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-319-99175-7_2

9

10

2 The Invariance of the Core Subspace

1 ≤ i ≤ L, γ ∈ Γ, −∞ < k < j }. Then {Vj }∞ j =−∞ is a GMRA of H relative to Γ and δ. Proof Properties (1), (2), and the first half of (3) of Definition 1.2 follow immediately from the choice of Vj . The intersection of the Vj is trivial since any element of Vj must be orthogonal to δ k (γ (ψi )) for k ≥ j. To see that the core subspace V0 is invariant under Γ , it will suffice to show that for k ≥ 0, each Wk ≡ Vk+1  Vk is invariant, since V0 is the orthogonal complement of ⊕∞ k=0 Wk . The invariance of Wk follows from γ (δ k (γ (ψi ))) = δ k ((δ −k γ δ k )(γ (ψi ))).

(2.1)  

Remark 2.1 The situation for Parseval wavelets is not quite so straightforward. If H has a semiorthogonal Parseval wavelet, then the same proof shows that H has a GMRA. For other Parseval wavelets, it is not known whether ∩Vj = {0} always holds, where Vj is defined as in Theorem 2.1. The answer is negative for general frame wavelets, i.e.,{ψ1 , · · · , ψL } such that {δ j γ (ψi )} forms a frame for H . In fact, Bownik and Rzeszotnik [8] found an example of a frame wavelet in L2 (R) with frame bounds 1 and 1 + δ that has ∩Vj = L2 (R). A partial converse to Theorem 2.1 follows from a comparison of the representations of Γ on V0 and V1 with the left regular representation Λ of Γ, which acts on ξ ∈ l 2 (Γ ) by Λγ (ξ(γ )) = ξ(γ −1 γ ). Theorem 2.2 Suppose the affine structure on a Hilbert space H is given by a group Γ and operator δ with the property that δ −1 Γ δ is of finite index d in Γ . If there is an MRA of H , then there exists an orthonormal wavelet {ψ1 , · · · , ψd−1 } for H . Proof The existence of an MRA means that the action of the group Γ on V0 is equivalent to the left regular representation Λ of Γ , with the unitary operator U : V0 → l 2 (Γ ) by U (γ (φ)) = 1γ establishing the equivalence. On the other hand, we claim that the action of Γ on V1 is equivalent to the direct sum of d copies of Λ. To see this, first note that {δ(γ (φ)) : γ ∈ Γ } forms an orthonormal basis of V1 . Let Γ = Γ1 ∪ · · · ∪ Γ d be a division of Γ into its d distinct right cosets of δ −1 Γ δ, and define V1,i = { γ ∈Γi cγ δγ (φ)}. Write V1 = d  i=1 V1,i . The action of Γ on each V1,i is equivalent to Λ, since γ ∈Γi cγ δγ (φ) =  −1 −1 −1 γ ∈Γi cγ δγ δ δ(φ), and as γ ranges over a right coset of δ Γ δ in Γ , δγ δ −1 ranges over a right coset of Γ in δΓ δ . As before, we write Wj = Vj +1  Vj . If we let σ be the representation of Γ on W0 , we can also write the representation of Γ on V1 = V0 ⊕ W0 as Λ ⊕ σ , obtaining d  i=1

Λ=Λ⊕σ

(2.2)

2.1 Arbitrary Hilbert Spaces and Affine Structures

11

Since Γ is discrete, the commutant of Λ, and hence also that of the direct sum of d copies of Λ, is a finite von Neumann algebra. Thus, the cancellation property for representations of groups with finite von Neumann algebras applies (see [13] 3.2.3 d−1 Prop. 6), to yield σ = i=1 Λ. Since V0 is a proper subspace of V1 (else H would be trivial), we have W =  0, so that d − 1 > 0. 0 d−1 Write W0 = W , where the action of Γ is equivalent to Λ on each 0,i i=1 W0,i , and denote by Ui : W0,i → l 2 (Γ ) the unitary operator that establishes this equivalence. Let ψi = Ui−1 (1e ), where e is the identity of Γ . Then {γ (ψi ) : γ ∈ Γ, 0 ≤ i ≤ d − 1} forms an orthonormal basis for the subspace W0 , so that {δ j (γ (ψi )) : γ ∈ Γ, 0 ≤ i ≤ d − 1, −∞ < j < ∞} forms an orthonormal basis for Wj = H .   Corollary 2.1 If {ψ1 , . . . ψl } is an orthonormal wavelet associated with an MRA in a Hilbert space H , then l = d − 1, where d is the index of δ −1 Γ δ in Γ . Proof By the cancellation property for representations of groups with finite von Neumann algebras referenced above, d−1  i=1

implies that l = d − 1.

Λ=

l 

Λ

(2.3)

i=1

 

Note that the proofs of both Theorem 2.2 and Corollary 2.1 use the assumption that Γ is a discrete group. Remark 2.2 An example is given in [3] to show that the hypothesis of the existence of a scaling vector cannot be entirely removed. However, MacArthur and Taylor in [22] were able to relax this hypothesis in the following natural way. They define a finite scaling ensemble (FSE) to be {φ1 , · · · , φr } ⊂ V0 such that {γ φi : γ ∈ Γ, 1 ≤ i ≤ r} forms an orthonormal basis of V0 . They show that the same proof as above establishes the existence of a wavelet {ψ1 , · · · ψr(d−1) } when the requirement of an MRA is replaced by that of a GMRA that has an FSE. In addition to Theorem 2.2, the authors in [3] also developed abstract general theorems about building wavelets that belong to desirable dense subspaces of the Hilbert space H , including one that generalizes classical results about the existence of smooth wavelets with compact support. Both the strengths and the limitations of Theorem 2.2 and these variations lie in their abstract generality. While the theorems are not constructive, they can be used to build wavelets in situations where finding an orthonormal basis for W0 from the scaling function(s) is straightforward. This occurs, for example, when the scaling functions are characteristic functions, and is exploited in [16] and [22] to build Haar wavelets for composite dilations and the crystallographic groups. (See Chapter 7.) Here is a simple example of Theorem 2.2 with non-abelian Γ that fits within both the composite and crystallographic classifications. This MRA was mentioned in [7].

12

2 The Invariance of the Core Subspace

Example 2.1 Let H = L2 (R2 ), and take Γ to be the crystallographic group pm, which can be written as the semidirect product of a translation group and a two element point group, Γ = Z2  {I, σy }, where σy is a reflection in the y axis. Γ is a subset of the affine group, and acts on R2 by [n, S] · x = S(x + n), and hence on L2 (R2 ) by 1

[n, S] · f (x) = | det S|− 2 f (S −1 x − n) = f (Sx − n)

(2.4)

for n ∈ Z2 , S ∈ {I, σy }. We take δ to be dilation by 2, δf (x) = 2f (2x),

(2.5)

and observe that δ −1 [n, S]δ = [2n, S], so that δ −1 Γ δ is a subgroup of order 4 in Γ. Let R = [0, 12 ] × [0, 1], which is a fundamental domain for the action of Γ on R2 , √ and define φ = 21R . If we let V0 be the closed linear span of {γ φ : γ ∈ Γ }, and Vj = δ j V0 , we obtain an MRA for L2 (R2 ). Thus Theorem 2.2 applies to guarantee the existence of a 3-wavelet for L2 (R2 ) relative to the action of the group Γ = pm and dilation by 2. For this simple φ, we can find the wavelet explicitly by completing the basis for V0 into a basis for V1 . We note that V1 consists of functions that are constant on rectangles of width 14 and height 12 . Thus it will suffice to include, along with {γ φ}, the Γ orbits of the three functions ψ1 = ψ2 = ψ3 =

√

2(1[0, 1 ]×[0, 1 ] − 1[0, 1 ]×[ 1 ,1] ) 2

√

2

2

2(1[0, 1 ]×[0, 1 ] + 1[ 1 , 1 ]×[ 1 ,1] − 1[ 1 , 1 ]×[0, 1 ] − 1[0, 1 ]×[ 1 ,1] ) 4

√

2

2

4 2

2

4 2

2

4

2

2(1[0, 1 ]×[0,1] − 1[ 1 , 1 ]×[0,1] ). 4

4 2

In general, explicitly finding the wavelet from the scaling function is not so simple. Wavelet sets and filters have often been used to carry out this construction, particularly in the classical setting of ordinary dilations and translations on L2 (RN ). We will discuss the generalizations of these techniques in Chapters 4 and 5. In addition to the use of non-abelian groups in composite and crystallographic examples such as Example 2.1, a currently active strand of research builds multiresolution structures with Γ a non-abelian Lie group. Early work by P. G. Lemarié [21] constructed a variant of a multiresolution analysis and spline wavelets using a stratified nilpotent Lie group. More recently, A. Mayeli built a wavelet frame multiresolution analysis with translation on the Heisenberg group [24]. See also recent papers by B. Currey, A. Mayeli, and V. Oussa, e.g., [11]. Theorem 2.2 and its extension in Remark 2.2 both depend on the existence of scaling functions, as does the traditional use of filters to build wavelets. In order to make GMRAs more useful in building wavelets, the concept of a scaling function

2.2 Affine Structures with Abelian Γ

13

was generalized in [5, 6, 10] and [25] to include scaling functions that provide frames instead of orthonormal bases for the core subspace. Definition 2.1 A set of generalized scaling vectors for a GMRA on a Hilbert space H is a countable (possibly finite) collection {φ1 , φ2 , · · · } ⊂ V0 such that {γ φi : γ ∈ Γ } form a Parseval frame for V0 . We will show below that if we restrict our attention to abelian Γ , we have such scaling vectors in a large class of GMRAs that includes all in the traditional setting of L2 (RN ). By applying a similar idea to the representation of Γ on W0 , we also will prove the existence of semiorthogonal Parseval wavelets for this large class of GMRAs.

2.2 Affine Structures with Abelian Γ We now use decomposition into irreducibles to look more closely at the representations of Γ on V0 and W0 in the case of an abelian Γ . Let π be a unitary representation of Γ on an invariant subspace V ⊂ H . When Γ is abelian, Stone’s theorem generalized to locally compact abelian groups, or equivalently, the spectral theorem for a commuting family of operators (see, e.g., [14] Th. 4.44), shows that there exists a unique projection valued measure (spectral measure) p on the dual group Γ such that for every γ ∈ Γ ,  πγ =

Γ

(2.6)

γ (ω)dp(ω).

To make this characterization of π more useful in our setting, we apply spectral multiplicity theory. The idea is to describe p completely in terms of canonical projection valued measures q, which act in L2 (ν) spaces with finite measures ν on Γ by qE f = 1E f . Note that if K is a cyclic subspace of V for p, with cyclic vector x of norm 1, we can define a measure ν on Γ by ν(E) = pE x, x , and the unitary map U (1E ) = pE (x) extended to L2 (ν) will intertwine the canonical projection valued measure on L2 (ν) with p on K. To use this for our V , first find a maximal subspace K1 ⊂ V that is cyclic for p. To do this, start with a countable dense subset {zi } of vectors in V , and use the Hausdorff Maximality Principle to find a maximal linearly ordered set of cyclic subspaces for p containing z1 . Let K1 be the closure of the union of this set of cyclic subspaces. Then K1 has a cyclic vector y1 and associated measure μ1 defined by μ1 (E) = pE y1 , y1 . By maximality, we have μ1 ≡ p. Let U1 : L2 (μ1 ) → K1 be the intertwining operator. Next use the same process to find a maximal cyclic subspace K2 containing the projection onto K1⊥ of the first vector in {zi } that is not in K1 . The associated measure for K2 will satisfy μ2 ≡ p|K ⊥ , and we will have a map U2 : L2 (μ2 ) → K2 .

1

14

2 The Invariance of the Core Subspace

Repeating this process the separable Hilbert space V into a direct sum of breaks ∞ cyclic subspaces V = K , with associated measures μi and maps Ui , such i i=1 that the measures satisfy μi+1  μi . (The process may terminate in a finite number of steps, the Ki andμi are all trivial after i = n.) The unitary operator  so that −1 2 J = ∞ U maps V to ∞ i=1 i i=1 L (μi ). Since J intertwines the projection valued measure p with the direct sum of canonical projection valued measures on L2 (μi ), it intertwines the direct integral πγ with multiplication by γ (as a function on Γ) in each component. As a result, we have the following, where μ = μ1 and Si is the support of μi : Theorem 2.3 Given a unitary representation π of Γ on V , there exists a unique  Borel measure class [μ] on Γ, Borel subsets S1 2⊇ S2 ⊇ · · · of Γ , which are unique (a.e.μ), and a unitary operator J : V → L (Si ) (not necessarily unique) such that for γ ∈ Γ, f ∈ V , [J (πγ (f ))](ω) = γ (ω)[J (f )](ω).

(2.7)

Proof See [17], [19] (Sec. 1.7), or [23] (Theorem 1.21) for general proofs. See also [9] (Chapter 3) for more details of the specific outline given above.   Remark 2.3 Spectral multiplicity theory is often described as showing that the projection valued measure p is uniquely determined by the measure class [μ] on Γ, and a multiplicity function m : Γ → {0, 1, 2, · · · } ∪ {∞} defined a.e. μ. Chapter 3 will further explore the multiplicity function, which is derived from Theorem 2.3 by m=

∞ 

1 Si ,

(2.8)

i=1

or equivalently, Si = {ω ∈ Γ : m(ω) ≥ i}. We will apply Theorem 2.3 to the invariant subspaces V = V0 and V = W0 . We will follow the convention of using J , μ, Si , and m for the components of Theorem 2.3 in the case of V = V0 , and J,  μ,  Si , and m  for these components in the case of V = W0 . Also, for V = V0 , we will refer to the measure class [μ] of Theorem 2.3 as the core measure class, and the multiplicity function m of the Remark as the core multiplicity function, or just the multiplicity function. We refer to m  as the complementary multiplicity function. In the case where the core measure class [μ] is absolutely continuous with respect to Haar measure, following [2, 5, 10] and [15], we can use the map J to define generalized scaling vectors for the GMRA. Theorem 2.4 Let {Vj } be a GMRA on a Hilbert space H with respect to an affine structure with abelian Γ . Suppose that the core measure class [μ] is absolutely continuous with respect to Haar measure on Γ. Using the map J , and setsSj , from L2 (Si ) Theorem 2.3, define φj = J −1 (1Sj ), where 1Sj denotes the function in

2.2 Affine Structures with Abelian Γ

15

whose j th component is 1Sj and other components are 0. Then {φ1 , φ2 , · · · } are generalized scaling vectors for {Vj }. Proof Write λ for normalized Haar measure on Γ. Since μ  λ, we can take μ = λ|S1 , and in general, μ|Sj = λ|Sj . We have then that {γ 1Sj : γ ∈ Γ } is a Parseval frame for L2 (Sj ) (see, e.g.,[18]), and thus that {γ 1Sj : γ ∈ Γ, j ≥ 1}  2 is a Parseval frame for L (Sj ). The result then follows from the fact that J is a unitary map that intertwines multiplication by γ with the action of πγ .   As mentioned in Chapter 1, M. Papadakis in [25] defined an alternative multiresolution structure called a generalized frame multiresolution analysis (GFMRA). A GFMRA is defined in the same context as GMRAs except that the group Γ is not required to be countable or discrete. A GFMRA replaces condition (4) in the definition of GMRA by the requirement that V0 has a frame consisting of the Γ orbit of a countable set of vectors. Theorem 2.4 shows that in the case of countable discrete Γ , when the core measure μ is absolutely continuous with respect to Haar measure, the definitions of GMRA and GFMRA are equivalent. When the measure μ of Theorem 2.3 is not absolutely continuous with respect to Haar measure, it is possible that there exist GMRAs that are not also GFMRAs. Han and Larson in [18] (Theorem 3.11) and Weber in [27] showed that a countable group Γ acting on a finite set {φ1 , · · · φn } of vectors cannot give a frame for a subspace of a Hilbert space unless the measure μ is absolutely continuous with respect to Haar measure. Thus, a GMRA or GFMRA with core measure not absolutely continuous with respect to Haar measure could only have generalized scaling vectors involving an infinite collection of φi . GMRAs (for countable discrete groups) that do have such a collection are also GFMRAs; GMRAs that do not are exactly those GMRAs that are not also GFMRAs. Examples of multiresolution structures in which the core measure class is not absolutely continuous with respect to Haar are rare in the literature. In particular, we will see in the next section that such examples cannot occur for translations by integers and dilation by expansive matrices in L2 (RN ). Here is a sketch of a 2010 example with singular measure by Baggett [1]: Example 2.2 Let S be the countable set  ∞

1 2n − 1 , S= , 2n 2n

(2.9)

n=1

and define an atomic probability measure ν on S by ν( 12 ) = 12 , and ν( 21n ) = n 1 for n ≥ 2. Set K0 = L2 (ν), and for n ≥ 1, Kn = C2 . Define ν( 2 2−1 n ) = ∞ 2n+1 H = n=0 Kn , so that f ∈ H is of the form f = {f0 , f1 , f2 , · · · },

fn ∈ Kn

(2.10)

16

2 The Invariance of the Core Subspace

Let Γ = {T n : n ∈ Z}, where T is the operator on H given by ⎧ 2π ix f0 (x) ⎨e [T (f )]n (x) = (f1 1 , −f1 2 ) ⎩ fn

n=0 n=1 . n≥2

(2.11)

Define the operator D by [D(f )]n = fn+1 for n ≥ 1,

(2.12)

and ⎧ f1 1 ⎪ ⎪ ⎪ ⎪ ⎨ √2(f (2x) + f ) 0 12 [D(f )]0 (x) = √ ⎪ 2(f0 (2x) − f1 2 ) ⎪ ⎪ ⎪ ⎩√ 2f0 (2x)

x= x= x=

1 2 1 4 3 4

,

(2.13)

else

and take δ = D −1 . Then δ −1 T δ = T 2 , so that Γ and δ form an affine structure for H. We define a GMRA on H relative to Γ and δ as follows: For j ≥ 0, set Vj = j n=0 Kn . For j < 0, we define Vj in terms of a Ruelle operator Sh ,√a class of operators we will discuss more in Chapter 5. Specifically, we let h = 2 1S\{ 1 } , 2

and define the isometry Sh on L2 (ν) by [Sh (f )](x) = h(x)f (2x). Then, for j < 0, −j set Vj = Sh (V0 ). The map J of Theorem 2.3 can be taken to be the identity map on V0 = L2 (ν), so that the core measure class [μ] is that of the given measure ν. See [1] for more details. We now apply the ideas of this section to the representation of Γ on W0 = wavelets for a large class of V1  V0 in order to prove the existence of Parseval GMRAs. Let [ μ] be the measure class and J : W0 → L2 ( Si ) be the map given by Theorem 2.3 for the representation σ of Γ on W0 . The following theorem, a version of which first appeared in [5], is the analog of Theorem 2.4 for this representation. Theorem 2.5 Let {Vj } be a GMRA for the Hilbert space H . Suppose the affine structure on H has an abelian group Γ whose representation σ on W0 has associated measure absolutely continuous with respect to Haar measure on Γ. Then H has a semiorthogonal Parseval wavelet {ψ1 , ψ2 , · · · }, where ψi = J−1 (1 Si ). Proof The same argument as used in the proof of Theorem 2.4 shows that {γ 1 Si :  2 −1   γ ∈ Γ, i ≥ 1} is a Parseval frame for L (Si ). Hence, applying J , we have that {γ ψi : γ ∈ Γ, i ≥ 1} is a Parseval frame for W0 , so that {δ j γ ψi : γ ∈ Γ, i ≥ 1, j ∈ Z} is a Parseval frame for H = ∪Wj . Thus {ψ1 , ψ2 , · · · , ψL } is a Parseval wavelet, which is semiorthogonal because Wj ⊥ Wk .  

2.2 Affine Structures with Abelian Γ

17

Here is an example from [15] of a frequency space GMRA, in which we can explicitly construct both generalized scaling vectors and Parseval wavelets from the maps J and J. Example 2.3 Let E = [− 14 . 12 ) ∪ [ 34 , 1), and define Vj = L2 (2j E). It is easy to check that {Vj } forms a GMRA for L2 (R) with Γ = Z acting by γ · f (ω) = e2π iγ ω f (ω), and δf (ω) = √1 f ( ω2 ). We have Γ = T, which we parametrize as 2 [− 12 , 12 ). Since Γ is already acting on copies of L2 (Γ) in L2 (R) via multiplication by characters, we can take J to be an operator that translates copies of Γ into E in the argument of the function. Since we have two copies of parts of Γ in E, J will map to a direct sum of two L2 (Si ) spaces with Si ⊂ Γ. The set S1 will be the set of points in Γ that have at least one translate in E, and S2 the set of points that have two translates in E. That is, define 1 1 S1 = [− , ) 4 2

1 S2 = [− , 0), 4

and

and let J (f )(ω) = f (ω)1S1 (ω) + f (ω + 1)1S2 (ω). (Recall we write 1Si for the element of L2 (S1 ) ⊕ L2 (S2 ) whose i th component is 1Si and other component is 0.) The core measure class is absolutely continuous with respect to Lebesgue measure restricted to S1 , and {φ1 , φ2 } are generalized scaling vectors, where φ1 = 1[− 1 , 1 ) ,

φ2 = 1[ 3 ,1) .

4 2

4

  We have W0 = L2 (2E \ E) = L2 [− 12 , − 14 ) ∪ [ 12 , 34 ) ∪ [ 32 , 2) . Following the same procedure as with V0 , define 1 1  S2 = [− , − ), 2 4

1  S1 = [− , 0), 2

1 1  S3 = [− , − ), 2 4

and J(f )(ω) = f (ω + 2)1 S1 (ω) + f (ω)1 S2 (ω) + f (ω + 1)1 S3 (ω). The semiorthogonal Parseval wavelet of Theorem 2.5 is thus ψ1 = 1[ 3 ,2) , ψ2 = 1[− 1 ,− 1 ) , ψ3 = 1[ 1 , 3 ) . 2

2

4

2 4

Example 2.3 can be used to illustrate the non-uniqueness of the operator J . We could, for example, define   Jˇ(f )(ω) = f (ω + 1)1[− 1 ,0) (ω) + f (ω)1[0, 1 ) (ω) 1S1 (ω) + f (ω)1S2 (ω), 4

2

and note that Jˇ also satisfies the intertwining requirement of Theorem 2.3 for V = V0 . The scaling vectors φi = Jˇ−1 (1Si ) would become φˇ 1 = 1[0, 1 )∪[ 3 ,1) and φˇ 2 = 2 4 1 1 . Similar variations are possible for J. [− 4 ,0)

18

2 The Invariance of the Core Subspace

We will see in the next section that in the classical case, the measure class of the representation of Zn on W0 is always absolutely continuous with respect to Haar measure, so that semiorthogonal Parseval wavelets exist for every GMRA.

2.3 The Classical Setting Now consider the classical wavelet setting in which H = L2 (RN ), and Γ = ZN acts on H by translation. Here, we have Γ = TN , which we parametrize by ω ∈ [− 12 , 12 )N , with ω(n) = e2π i n,ω for n ∈ ZN . In this case, the operator J of Theorem 2.3 is closely related to the Fourier transform, since both operators intertwine translation with multiplication by exponentials. This leads to the following theorem from [4]: Theorem 2.6 Let {Vj } be a GMRA on H √ = L2 (RN ) relative to Γ = ZN and δA , where γ ·f (x) = f (x −γ ), and δA f (x) = | det A|f (Ax) for an expansive integer matrix A. Then the measure classes of the representations of Γ on V0 and W0 are absolutely continuous with respect to Haar measure. Proof By taking Fourier transforms, the representation of ZN on all of RN is equivalent to multiplication by exponentials ei n,ω . On the other hand, when we parametrize Γ as above, the regular representation Λ of Γ is equivalent to multiplication by exponentials on L2 ([− 12 , 12 )N ). Thus, by writing RN as the disjoint union of translates of [− 12 , 12 )N by elements of Γ , we have that the representation of Γ on all of H is equivalent to an infinite multiple of the regular representation of Γ . Therefore, the projection valued measure associated with the representation of Γ on H is equivalent to Haar measure on Γ, so that the projection valued measure associated with any subrepresentation will be absolutely continuous with respect to Haar measure.   Theorems 2.4, 2.5, and 2.6 together imply that GMRAs in the classical setting always have generalized scaling vectors, and always have semiorthogonal Parseval wavelets. Recall from Chapter 1 that a frame multiresolution analysis (FMRA), defined by Benedetto and Li in [6] for dilation by 2 in L2 (R), adds to the requirements of a GMRA that the core subspace V0 has a frame consisting of the group Γ acting on a single vector φ. We see then that an FMRA is the same as a GMRA in which the operator J of Theorem 2.3 maps V0 to a single L2 space, L2 (μ1 ). Equivalently, we can characterize an FMRA as a GMRA in L2 (R) for which the multiplicity function takes on only the values 0 and 1. Benedetto and Li use filters to build wavelets from FMRAs, a process we will describe for GMRAs in Chapter 5. They show that filters for FMRAs can be narrow band, resulting in better signal reconstruction than with MRAs. The FMRAs of Benedetto and Li, and the GFMRAs of Papadakis, have frames for the core subspace specified in their definition. The multiresolution structure can thus explicitly include particular frames that are useful for the purpose at hand. This

2.3 The Classical Setting

19

also avoids the sometimes difficult task of finding a frame for the core subspace. While Theorem 2.4 gives a way of finding a frame using the operator J , that operator is often elusive. In L2 (RN ) however, the fact that the operator J and the Fourier transform both intertwine translation with multiplication by exponentials, make the frame given by Theorem 2.4 useful even when an explicit formula for J is hard to find. In particular, it was shown in [2] that the Fourier transforms of the generalized scaling vectors defined by Theorem 2.4 satisfy the following orthogonality condition, which gives a concrete connection between these generalized scaling vectors and the multiplicity function. The quantities evaluated in the Lemma are instances of what are called Gramian fibers and bracket products in [12] and [26]. We will make use of this condition to study the multiplicity function in Chapter 3, and also to build filters in Chapter 5. Lemma 2.1 Let {φi } = {J −1 1Si } be the generalized scaling vectors produced by Theorem 2.4 for L2 (RN ). Then  i (ω + γ )φ j (ω + γ ) = δi,j 1Si (ω). φ (2.14) γ ∈ZN

for almost all ω ∈ Γ = [− 12 , 12 )N . Proof Since both sides are in L2 ([− 12 , 12 )N ), it will suffice to show their Fourier coefficients, cγ˜ agree. We have    i (ω+γ )φ j (ω+γ )) = i (ω+γ )φ j (ω + γ )dω φ φ e−2π i γ˜ ,ω cγ˜ ( [− 12 , 12 )N

γ ∈Zn

=

  γ ∈ZN

[− 12 , 12 )N +γ

γ ∈ZN

i (ω)φ j (ω)dω e−2π i γ˜ ,ω φ

j i , e2π i γ˜ ,· φ = φ = φi , πγ˜ φj = 1Si , e2π i γ˜ ,· 1Sj = cγ˜ (δi,j 1Si (ω))   The close relationship between the Fourier transform and the operator J of Theorem 2.3 was also used by V. Furst [15] to generalize the Fourier equations that characterize classical wavelets in L2 (R) to an abstract Hilbert space. The classical result, attributed to Y. Meyer, states that a function ψ ∈ L2 (R), with ψ = 1 is an orthonormal wavelet if and only if the equations

20

2 The Invariance of the Core Subspace



(2j ω)|2 = 1 |ψ

(2.15)

j ∈Z

and ∞ 

(2j ω)ψ (2j (ω + n)) = 0 for every n ∈ 2Z + 1 ψ

(2.16)

j =0

hold for a.e. ω ∈ R. (See [20].) This result also characterizes Parseval wavelets if we omit the requirement that ψ = 1. Furst’s generalization uses the following equation, in which the Fourier transform is replaced by the operator J : ∞    1  −j  J δ ψl (ω) J δ −j ψl i (ωξ ) = δi,i δξ,e 1Si (ω) i dj l∈L j =1

for every ξ ∈ ker(α ∗ ), (2.17) where d is the index of δ −1 Γ δ in Γ , and e is the identity in Γ. Furst’s result is that for GMRAs with core measure absolutely continuous with respect to Haar measure, and whose multiplicity function is integrable, any two of the three statements 1. The set {ψ1 , ψ2 , · · · } is a Parseval wavelet. 2. The set {ψ1 , ψ2 , · · · } is semiorthogonal. 3. The set {ψ1 , ψ2 , · · · } satisfies Equation (2.17) for a.e. ω ∈ Γ about a countable collection of vectors in H imply the third [15]. This result points out both the possibilities and the limitations involved in generalizing wavelet results from L2 (R) to an abstract Hilbert space. Note, for example, that since J is only defined on V0 , only negative powers of δ can appear in Equation (2.17). The result replaces the two classical characteristic equations with one, and brings semiorthogonality into the mix.

References 1. Baggett, L.: Non-Lebesgue multiresolution analyses. Colloq. Math. 118, 133–145 (2010) 2. Baggett, L., Merrill, K.: Abstract harmonic analysis and wavelets in Rn . Contemp. Math. 247, 17–27 (1999) 3. Baggett, L., Carey, A., Moran, W., Ohring, P.: General existence theorems for orthonormal wavelets, an abstract approach. Publ. Res. Inst. Math. Sci. 31, 95–111 (1995) 4. Baggett, L., Medina, H., Merrill, K.: Generalized multiresolution analyses and a construction procedure for all wavelet sets in Rn . J. Fourier Anal. Appl. 5, 563–573 (1999) 5. Baggett, L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ). Appl. Comput. Harmon. Anal. 13, 201–223 (2002) 6. Benedetto, J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389–427 (1998)

References

21

7. Blanchard, J., Steffen, K.: Crystallographic Haar-type composite dilation wavelets. In: Cohen, J., Zyed, A.I. (eds.) Wavelets and Multiscale Analysis: Theory and Applications, pp 83–108. Birkhäuser, Boston (2011) 8. Bownik, M., Rzeszotnik, Z.: On the existence of multiresolution analyses for framelets. Math. Ann. 332, 705–720 (2005) 9. Courter, J.: Construction of dilation d orthonormal wavelets. Ph.D. thesis, University of Colorado at Boulder (1999) 10. Courter, J.: Construction of dilation d wavelets. Contemp. Math. 247, 183–206 (1999) 11. Currey, B., Mayeli, A., Oussa, V.: Characterization of shift invariant spaces on a class of nilpotent Lie groups with applications. J. Fourier Anal. Appl. 20, 384–400 (2014) 12. de Boor, C., DeVore, R., Ron, A.: The structure of finitely generated shift-invariant subspaces of L2 (Rd ). J. Funct. Anal. 119, 37–78 (1994) 13. Dixmier, J.: Von Neumann Algebras. North-Holland, Amsterdam (1981) 14. Folland, G.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995) 15. Furst, V.: A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces. J. Geom. Anal. 17, 569–591 (2007) 16. Guo, K., Labate, D., Lim, W., Weiss, G., Wilson, E.: The theory of wavelets with composite dilations. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 231–250. Birkhäuser, Boston (2006) 17. Halmos, P.: Introduction to Hilbert Spaces and the Theory of Spectral Multiplicity. Chelsea, New York (1957) 18. Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147 (2000) 19. Helson, H.: The Spectral Theorem. Springer, Berlin (1986) 20. Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996) 21. Lemarié, P.G.: Base d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. France 117, 211–232 (1989) 22. MacArthur, J., Taylor, K.: Wavelets with crystal symmetry shifts. J. Fourier Anal. Appl. 17, 1109–1118 (2011) 23. Mackey, G.: The Theory of Unitary Group Representations. University of Chicago Press, Chicago (1976) 24. Mayeli, A.: Shannon multiresolution analysis on the Heisenberg group. J. Math. Anal. Appl. 348 671–684 (2008) 25. Papadakis, M.: Generalized frame multiresolution analysis of abstract Hilbert spaces. In: Benedetto, J., Zayed, A. (eds.) Sampling, Wavelets and Tomography, pp. 179–223. Birkhäuser, Boston (2004) 26. Ron, A., Shen, Z.: The wavelet dimension function is the trace function of a shift-invariant system. Proc. Am. Math. Soc. 131, 1385–1398 (2002) 27. Weber, E.: Frames and single wavelets for unitary groups. Can. J. Math. 54, 634–647 (2002)

Chapter 3

The Multiplicity Function

In the last chapter, we made use of the fact that a GMRA whose group Γ of unitary operators is abelian has a  unique associated measure class [μ] on Γ, and an associated map J : V0 → L2 (Si ), where {Si } is a nested sequence of  subspaces S1 ⊇ S2 ⊇ · · · of Γ . We saw that the operator J is often hard to make explicit and is not itself uniquely determined by the representation. In this chapter, following [4], we focus on an alternative way of describing the result of Theorem 2.3 that is sometimes easier to access. Remark 2.3 points out that the representation π of Γ is completely determined by the measure class [μ] and a unique multiplicity function m : Γ → {0, 1, 2, · · · } ∪ {∞}. Recall that the function ∞ m is related to the sets Si in Equation (2.8) by m = i=1 1Si , or equivalently, Si = {ω ∈ Γ : m(ω) ≥ i}. Roughly speaking, the multiplicity function counts the number of times each irreducible in Γ is included in the representation π of V0 . Since every orthonormal wavelet, and every semiorthogonal Parseval wavelet, has an associated GMRA, each also has an associated multiplicity function. We will now explore the use of the multiplicity function as a tool to analyze and build wavelets, and see how it is related to classical tools such as the dimension function. Throughout this chapter we will consider only GMRAs with affine structures that have abelian groups whose representations on both V0 and W0 have associated measure class absolutely continuous with respect to Haar measure.

3.1 Examples of Multiplicity Functions A GMRA is an MRA if and only if its core measure class [μ] is that of Haar measure and its core multiplicity function satisfies m(ω) = 1 for a.e. ω ∈ Γ. This follows since each of the two statements holds if and only if the representation π is equivalent to the regular representation of Γ . Thus, by Theorem 2.6, a GMRA in L2 (RN ) is an MRA if and only if its multiplicity function is identically 1. In © Springer Nature Switzerland AG 2018 K. D. Merrill, Generalized Multiresolution Analyses, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-319-99175-7_3

23

24

3 The Multiplicity Function

particular, all of the classical MRA wavelets for dilation by 2 in L2 (R), such as the Haar wavelet, the Shannon wavelet, and the Daubechies wavelets, have multiplicity function m ≡ 1. By Theorem 2.2, a wavelet in L2 (RN ) for dilation by A can have multiplicity function m ≡ 1 only if | det A| = 2. Baggett, Medina and Merrill developed the theory of multiplicity functions for GMRAs in [4], and showed how it is related to wavelet set wavelets in L2 (RN ), i = 1Fi , Fi ⊂ RN . Since that is, wavelets {ψ1 , ψ2 , · · · , ψL } that are of the form ψ the Fourier transform turns translation by the integer lattice into multiplication by  = L2 (∪L Fi ). Since dilation on the exponentials, a wavelet set wavelet has W i=1  0 2 0 =  Fourier side is inverse dilation, V j 1. Sj,i is formed by taking those elements of j −1

the previously unused part of Sj that have at least j translates in A∗i (∪k=1 Ek ) ∩ Δ. This ensures we can choose the pieces Ej,i to be piecewise translates of Sj,i that are j −1 disjoint from ∪k=1 Ek , and with Ej,i ⊂ A∗ Ej,i−1 .

3.4 Characterizing Multiplicity Functions

35

This construction is quite complicated and difficult to carry out in general. However, when Δ = RN , it can often be accomplished by starting with the set F ⊂ A∗ F , and building Ej ≡ Sj (mod 1) by just paying attention to the   required disjointness and containments Ej ∩Ej = ∅ and ∪kj =1 Ej ⊂ A∗ ∪kj =1 Ej . Versions of the following examples first appeared in [3]. We will explore other examples related to wavelet sets in Chapter 4.

Example 3.3 (1) For the trivial case of m ≡ 1, we have Δ = RN , and conditions (i), (ii), and Equation (3.30) are satisfied for any dilation A. If [− 12 , 12 )N ⊂ A∗ ([− 12 , 12 )N ), we can take F = E1 = [− 12 , 12 )N , and the construction ends with n = L2 (A∗n [− 1 , 1 )N ). E = E1 . In this case, the GMRA is V 2 2 1 1 N If, on the other hand, [− 2 , 2 )  A∗ ([− 12 , 12 )N ), the construction must take E1,0 = F to be a proper subset of [− 12 , 12 )N , and there will be other nonempty   2 3 E1,i ’s. For example, if A = , then F can be taken to be the parallelogram −2 −2 with vertices at ±( 13 , 16 ), ±( 13 , 12 ). Since Δ = RN , at each stage of the construction, j −1 E1,j is just the part of A∗ E1,j −1 that is not congruent mod 1 to points in ∪i=0 E1,i . Thus, E1,1 = A∗ (E1,0 ) \ E1,0 , E1,2 = A∗ (E1,1 ) \ (E1,1 ∪ E1,0 ) ∩ {(x, y) : |x| ≤ 12 }, and the process terminates with E1,3 to yield E1 =the parallelogram with vertices n = L2 (A∗n E1 ), with at ±( 12 , 1), ±( 12 , 0). The GMRA (FMRA) is given by V  generalized scaling function φ = 1E1 . (2) For another simple example that again works for any dilation, take F ⊆ [− 12 , 12 )N to be a neighborhood of the origin such that F ⊂ A∗ F . Define m = 1F . This satisfies (i) and (ii) since A∗−1 F ⊂ F , and so is a multiplicity function. Since n } = F ⊂ Δ, and ∪j ∈Z A∗j F = RN , m satisfies Equation (3.30) as well. Thus {V {L2 (A∗n F )} gives a GMRA for all of L2 (RN ). (3) To get GMRAs with higher multiplicities, we can, for example, take

m(ω) =

2 if x ∈ B , 1 if x ∈ /B

where B is any subset of [− 12 , 12 )N . Condition (i) is satisfied since there are always at least two terms on the right-hand side, each of which is at least 1. Condition (ii) and Equation (3.30) are immediate because Δ = RN . In particular, if we again let   2 3 A= , and let B be the parallelogram with vertices at ±( 13 , 16 ), ±( 13 , 12 ), −2 −2 then the construction as before gives E1 =the parallelogram with vertices at ±( 12 , 1), ±( 12 , 0). The next and final level of construction, E2 , consists of two parallelograms R and −R, where R has vertices (1, 76 ), (1, 56 ), ( 23 , 12 ), and ( 23 , 56 ). n = L2 (A∗n (E1 ∪ E2 )), with generalized scaling vectors The GMRA is given by V 2 = 1E2 . 1 = 1E1 and φ φ

36

3 The Multiplicity Function

References 1. Auscher, P.: Solution of two problems on wavelets. J. Geom. Anal. 5 181–236 (1995) 2. Baggett, L.: An abstract interpretation of the wavelet dimension function using group representations. J. Funct. Anal. 173, 1–20 (2000) 3. Baggett, L., Merrill, K.: Abstract harmonic analysis and wavelets in Rn . Contemp. Math. 247, 17–27 (1999) 4. Baggett, L., Medina, H., Merrill, K.: Generalized multiresolution analyses and a construction procedure for all wavelet sets in Rn . J. Fourier Anal. Appl. 5, 563–573 (1999) 5. Bownik, M.: On characterizations of multiwavelets in L2 (Rn ). Proc. Am. Math. Soc. 129, 3265–3274 (2001) 6. Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces. Mich. Math. J. 51, 387–414 (2003) 7. Bownik, M., Rzeszotnik, Z., Speegle, D.: A characterization of dimension functions of wavelets. Appl. Comput. Harmon. Anal. 10, 71–92 (2001) 8. Calogero, A.: A characterization of wavelets on general lattices. J. Geom. Anal. 10, 597–622 (2000) 9. Dai, X., Larson, D.: Wandering vectors for unitary systems and orthogonal wavelets. Memoirs Amer. Math. Soc. 134 (1998) 10. Dai, X., Larson, D., Speegle, D.: Wavelet sets in Rn . J. Fourier Anal. Appl. 3, 451–456 (1997) 11. Dai, X., Larson, D., Speegle, D.: Wavelet sets in Rn II. Contemp. Math. 216, 15–40 (1998) 12. Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992) 13. Dixmier, J.: Von Neumann Algebras. North-Holland, Amsterdam (1981) 14. Gripenberg, G.: A necessary and sufficient condition for the existence of a father wavelet. Stud. Math. 114, 207–226 (1995) 15. Gu, Q., Han, D.: On Multiresolution Analysis Wavelets in Rn . J. Fourier Anal. Appl. 6, 437– 447 (2000) 16. Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996) 17. Lemarié-Rieusset, P.G.: Sur l’existence des analyses multi-résolutions en théorie des ondelettes. Rev. Mat. Iberoamericana 8, 457–474 (1992) 18. Paluszy´nski, M., Šiki´c, H., Weiss, G., Xiao, S.: Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties. Adv. Comput. Math. 18, 297–327 (2003) 19. Ron, A., Shen, Z.: The wavelet dimension function is the trace function of a shift-invariant system. Proc. Am. Math. Soc. 131, 1385–1398 (2002) 20. Rzeszotnik, Z.: Characterization theorems in the theory of wavelets. Ph.D. thesis, Washington University, St. Louis (2000) 21. Weber, E.: Applications of the wavelet multiplicity function. Contemp. Math. 247, 297–306 (1999)

Chapter 4

Wavelet Sets

The term wavelet set was first used by Dai and Larson [10] in the late 1990s to describe a set W ⊂ RN such that 1W is the Fourier transform of a single (orthonormal) wavelet ψ for dilation by a scalar in L2 (RN ). At about the same time, Fang and Wang [14] coined the term minimally supported frequency (MSF) wavelet to describe dyadic wavelets in L2 (R) whose Fourier transforms are supported on sets of the smallest possible measure. They noted that such a wavelet would necessarily, because of Equation (2.15) (see [18] Corollary 2.4), have the absolute value of its Fourier transform equal to a characteristic function. Since any multiple of an MSF wavelet by a function of absolute value 1 is still an MSF wavelet, the study of wavelet sets and of MSF wavelets coincide. As we saw in the last chapter, wavelet sets are intimately related to the multiplicity function of a GMRA. In particular, for every wavelet, there is an MSF wavelet with the same multiplicity function, and the multiplicity function can be easily determined from a wavelet set. In this chapter, we will explore how to build wavelet sets from multiplicity functions, as well as directly from the consistency equation and from their geometric properties. We will include wavelet sets for all expansive integer matrix dilations in L2 (RN ), and also consider multiwavelet sets and Parseval wavelet sets. Definition 4.1 A set W is an L−wavelet set if W = ∪L l=1 Wl for a disjoint collection l = 1Wl , the collection {ψ1 , · · · ψL } is an of sets {W1 , · · · WL } such that when ψ orthonormal wavelet for L2 (RN ). The set W is a Parseval L−wavelet set if the collection {ψ1 , · · · ψL } is a Parseval wavelet for L2 (RN ). We start with a brief discussion of the essential properties of wavelet sets, their history, and their importance to wavelet theory.

© Springer Nature Switzerland AG 2018 K. D. Merrill, Generalized Multiresolution Analyses, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-319-99175-7_4

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4 Wavelet Sets

4.1 Preliminaries Even before wavelet sets were named, they were an important source of examples. One of the earliest and simplest is the Shannon or Littlewood-Paley wavelet for dilation by 2 in L2 (R), with wavelet set W = [−1, − 12 ) ∪ [ 12 , 1). This wavelet derives its names from its important connections to the Shannon Sampling Theorem [30] and to Littlewood-Paley theory [25]. Later, wavelet sets became essential test cases for wavelet theory. As mentioned in Section 3.1, a wavelet set due to Journé provided the first known example of a non-MRA wavelet. In the late 1990s, Dai, Larson, and Speegle [11] used wavelet sets to show that single wavelets exist for an arbitrary expansive matrix in any dimension. This result was somewhat surprising since the first dyadic wavelet examples known in dimension N > 1 were given by tensor products, and required L = 2N − 1 wavelets. Single wavelets for matrices whose determinant is not ±2 are necessarily non-MRA wavelets by Theorem 2.2. Although useful in developing the theory of wavelets, MSF wavelets are not directly useful for applications. This is because the lack of smoothness of their Fourier transforms causes these wavelets to be not well-localized. However, wavelet sets have been used as building blocks for more well-localized wavelets, beginning with the smoothing of the Shannon wavelet set by Lemarié and Meyer [23] in the mid-1980s. Since then, many authors (e.g., [3, 7, 9, 19, 20, 27]) have smoothed the filters associated with MSF wavelets to make the Fourier transform of the wavelet arbitrarily smooth. In the non-MRA case, this has necessarily resulted in Parseval wavelets. We will discuss the use of filters in Chapter 5. Other techniques not involving the smoothing of filters have also been used to connect MSF wavelets to more well-behaved wavelets. Dai and Larson [10] developed a procedure they call interpolation to build wavelets that lie between two MSF wavelets in an operator theoretic sense. Lim, Packer, and Taylor ([24]) used a wavelet set as a space over which to decompose wavelet representations as direct integrals of irreducibles. In addition to their wavelet uses, wavelet sets also have an important geometric property. The single wavelet set version of the following theorem first appeared in [15] and [11], and the multiwavelet set version in [4] and [8]. This theorem is a consequence of the fact that the dilation and translation operators interact with the Fourier transform F(f )(ω) = R2 f (x)e−2π i x,ω dx via the formulas: −1 f (ω) FδA F −1 f (ω) = δA

and

Fτk F −1 f (ω) = e−2π i k,ω f (ω).

Theorem 4.1 A set W is an L−wavelet set for dilation by an expansive integer matrix A if and only if  k∈ZN

1W (ω + k) = L

a.e. ω ∈ RN ,

(4.1)

4.1 Preliminaries

39

and 

1W (A∗j ω) = 1

a.e. ω ∈ RN .

(4.2)

j ∈Z

Proof First we claim that Equation 4.1 holds if and only if W can be written as a disjoint union W = L l=1 Wl , where 

1Wl (ω + k) = 1

a.e. ω ∈ RN

(4.3)

k∈ZN

for each 1 ≤ l ≤ L. The sufficiency of this condition is immediate. To see the opposite implication, order ZN in a norm non-decreasing manner {0 = k0 , k1 , · · · }, and build W1 = ∪∞ W as follows: Let W1,0 = W ∩ [− 12 , 12 )N . For j ≥ 1, let   j =0 1,j  j −1  W1,j = W ∩ ([− 12 , 12 )N \ ∪i=0 W1,i − ki + kj . Use the same procedure to l−1 build Wl for l > 1, except with W replaced by W \ ∪i=1 Wi . j

We must now show {ψj,k,l ≡ | det A| 2 ψl (Aj · −k) : −∞ < j < ∞, k ∈ N l = 1Wl , forms an orthonormal basis of L2 (RN ) if and Z , 1 ≤ l ≤ L}, with ψ only if (4.3) and (4.2) hold. For a fixed l, the set {ψ0,k,l : k ∈ ZN } is an orthonormal family if and only if k∈ZN 1Wl (ω + k) = 1 for a.e. ω. Since the Wl are pairwise disjoint, the set {ψ0,k,l : k ∈ ZN , 1 ≤ l ≤ L} is an orthonormal family under exactly the same conditions, with the closed span of their Fourier transforms given N by L2 (∪L l=1 Wl ). Thus, the set {ψj,k,l : k ∈ Z , 1 ≤ l ≤ L, −∞ < j < ∞} is an orthonormal family if and only if (4.1) holds and the sets A∗j (∪L l=1 Wl ) are pairwise L ∗j disjoint. The closed span of its Fourier transforms is L2 (∪∞ j =−∞ ∪l=1 A (Wl )) = L2 (RN ).   This theorem means that a set is an L-wavelet set for dilation by A exactly when it tiles RN under dilation by A∗ and gives an L-fold tiling under translation by the integer lattice. Another way of saying this is that W must be dilation congruent to A∗ [− 12 , 12 )N \ [− 12 , 12 )N and each of the Wl ’s in the disjoint union W = L l=1 Wl must be translation congruent mod 1 to [− 12 , 12 )N . For an example of the theorem for L = 1, picture a tiling of R2 under both translation and dilation by 2 by the left hand graph in Example 3.1. A related result in [13] (see [17] for the single wavelet set version) shows that a set W is a Parseval L−wavelet set if and only if W is dilation congruent to A∗ [− 12 , 12 )N \ [− 12 , 12 )N and each Wl in the disjoint union is translation congruent mod 1 to a subset of [− 12 , 12 )N . The inverse Fourier transform of Example 2.3 gives a Parseval 3-wavelet for which this result is easily seen. This geometric characterization of Parseval L-wavelet sets highlights a weakness of the definition.



That is, it is possible to write W as a different disjoint union W = L l=1 Wl with





 = 1Wl , the collection {ψ , . . . ψ } also forms a Parseval L = L such that taking ψ l 1 L wavelet. Thus a Parseval L− wavelet set can also be a Parseval L − wavelet set. A

40

4 Wavelet Sets

trivial example of this is to take W to be a wavelet set for an orthonormal single wavelet, and then build a Parseval 2-wavelet set by writing W as the disjoint union of W1 and W2 . For a refinement of the definition that avoids ambiguity, see [12]. The equivalence given by Theorem 4.1 makes the fact that single wavelet sets exist for any dilation in any dimension geometrically remarkable, since these are sets that tile under both translation and dilation. The first examples for dilation by 2 in dimension N > 1, such as the wedding cake set in Figure 3.1, were built by an iterative process that repeatedly corrects for the one of these two types of tilings not yet achieved, while preserving the other. All of them have a geometric structure that shows the fingerprints of this process, in that they consist of the union of an infinite number of scaled pieces that do not fit smoothly together. Early researchers (see, e.g., [6, 31]) conjectured that this complicated structure was inevitable, at least for dilation by 2 in dimension greater than one. That is, they believed that no simple single wavelet sets exist in that context, where a simple wavelet set or L-wavelet set is one that consists of a finite union of convex sets. Benedetto and Sumetkijakan proved a result in this direction by showing that a wavelet set for dilation by 2 in RN cannot be the union of N or fewer convex sets [6]. However, Gabardo and Yu found a simple single wavelet set for dilation by 2 in R2 in 2004 [16], and the current author [26, 28, 29] later independently found simple wavelet sets in all dimensions for all dilations that have a power equal to a scalar multiple of the identity. The simple wavelet sets for dilation by 2 included in these papers consist of 2N + 1 convex sets, compared to the lower bound of N + 1 given by the Benedetto/Sumetkijakan theorem. We will see more about these examples and the processes used to build them in the remainder of this chapter. Simple wavelet sets are attractive from a geometric point of view, and also easier to smooth with filters, as we will see in Chapter 5. To partially answer the question of which dilations have simple wavelet sets, the idea of Z-similarity classes is useful. Two matrices A and B are called Z-similar if there exists an integer matrix C with det C = ±1 such that A = CBC −1 . Using Theorem 4.1, it is easy to see that if W is an L-wavelet set for B, then CW is an L-wavelet set for A. Thus the property of having a simple (L-)wavelet set is shared by all members of a Zsimilarity class. These similarity classes are completely characterized for matrices with small determinant in [21] and [22]. When A is a scalar matrix, the Z-similarity class has only one element. Thus if W is a simple wavelet set for the scalar matrix A, then CW will also be for any integer matrix C with det C = ±1.

4.2 Wavelet Sets from Multiplicity Functions We saw in Chapter 3 that given a multiplicity function m that satisfies the conditions n = L2 (A∗n E), where of Theorem 3.3, there always exists a GMRA of the form V  N E ⊂ R is called a generalized scaling set and satisfies m(ω) = 1E (ω + k). If m also satisfies the consistency equation (3.14), then A∗ E \E is an L−wavelet set. The algorithm described after Theorem 3.3 gives an explicit technique for constructing the generalized scaling set, and thus the wavelet set.

4.2 Wavelet Sets from Multiplicity Functions

41

1

1

0.5 -1

-0.5

0.5

1

-1

1

-1

1 -0.5

-1

Fig. 4.1 Wavelet sets for

-1

 0 ±2   1 −1   1 −1  1 0 , ± 1 1 and ± 2 0

Building a wavelet set in RN by this technique requires first knowing a multiplicity function. The simplest possible m ≡ 1 is a multiplicity function for a single wavelet when the dilation has determinant of absolute value 2, or for an Lwavelet when the determinant has absolute value L + 1. Thus, using the algorithm described after Theorem 3.3, we can explicitly construct L-wavelet sets from the function m ≡ 1 for any dilation whose determinant has absolute value L + 1. For A of determinant ±2, if [− 12 , 12 )N ⊂ A∗ [− 12 , 12 )N , we can use scaling set   E = [− 12 , 12 )N , with wavelet set W = A∗ [− 12 , 12 )N \[ 12 , 12 )N . This is the case for   1 −1   1 −1   the six 2-dimensional matrices 10 ±2 0 , ± 1 1 , and ± 2 0 , whose resulting wavelet sets are shown in Figure 4.1. Since Lagarias and Wang showed in [22] that all 2 × 2 matrices of determinant 2 are Z-similar to one of these, we know that all expansive integer dilations of determinant ±2 in dimension 2 have simple wavelet sets. In R3 , there are fourteen Z-similarity classes of expansive integer matrices with determinant ±2 [22]. For ten of these classes, it is easy to find a representative A with [− 12 , 12 )3 ⊂ A∗ [− 12 , 12 )3 . The wavelet sets for these matrices turn out to be cylinder sets of the 2-dimensional examples in Figure 4.1. It is unknown whether such representatives exist for the other four classes. To show that there are simple wavelet sets for all matrices of determinant ±2, we use the algorithm developed in Theorem 3.3 to find a scaling set, working to ensure that it is the finite union of convex sets at each stage. We have the following generalized form of this result: Theorem 4.2 Let A be an N × N expansive integer matrix of determinant d. Then A has a (d − 1)-wavelet set that is a finite union of convex polytopes. Proof We use the method of the proof of Theorem 3.3 to build a generalized scaling set E = ∪Jj=0 Ej that is a finite union of convex polytopes, and which satisfies  E ⊂ A∗ E and m(ω) = k∈ZN 1E (ω + k) = 1. To build E0 , we employ the following result from matrix theory, which has been used extensively in the wavelet literature. By looking first at individual Jordan blocks, it is possible to find nonsingular complex matrices D and S such that

42

4 Wavelet Sets

√ D −1  < 1 and A∗ = S −1 DS. Then, taking T = !(S ∗ S), and C = T A∗ T −1 , we have real matrices T and C such that A∗ = T −1 CT and C −1  < 1. (For a detailed proof, see [33]). Now choose r > 0 such that Br , the neighborhood of the origin in RN of radius r satisfies T (Br ) ⊂ [− 12 , 12 )N . Since C −1 is norm decreasing, there is a δ > 0 such that C −1 (Br ) ⊂ Br−δ . Build a convex polytope P such that Br−δ ⊂ P ⊂ Br . Then if x ∈ P , C −1 x ∈ Br−δ ⊂ P . Thus C −1 P ⊂ P , so that T −1 (P ) ⊂ T −1 (CP ) = A∗ (T −1 P ).

(4.4)

We take E0 = T −1 P , and note that it is a convex polytope that is mapped into ∗j itself under A∗ . Since C −n  → 0 as n → ∞, we have ∪∞ j =0 A (E0 ) =     j −1 ∪∞ C j B N T −1 ∪∞ r =R . j =0 C P ⊃ T j =0 Next, we recursively build the sets Ej , j > 0 to be finite unions of convex polytopes, and so that Ej includes exactly one point of A∗ Ej −1 in each j −1 congruence class mod 1 represented in A∗ Ej −1 , but not represented in ∪i=0 Ei . N Assume E0 , E1 , · · · Ej −1 have already been constructed. Order Z in a norm nondecreasing manner {0 = k0 , k1 , · · · }, and let Qp = [− 12 , 12 )N + kp . Define      j −1 Ej,0 = A∗ Ej −1 \ ∪k∈ZN ∪i=0 Ei − k ∩ Q0 .

(4.5)

Then, for each p > 0, let        p−1 j −1 Ej,p = A∗ Ej −1 \ ∪i=0 Ej,i \ ∪k∈ZN ∪i=0 Ei − k ∩ Qp .

(4.6)

Note that only a finite number of Ej,p can be nonempty since A∗ Ej −1 is a bounded set. Thus Ej is a finite union of convex polytopes. As constructed, Ej includes exactly one point of A∗ Ej −1 in each congruence class mod 1 not represented in j −1 ∪i=0 Ei . ∗j N Since ∪∞ j =0 A E0 = R , it follows from a compactness argument that there exists an integer J > 0 such that ∪Jj=0 A∗j E0 ⊃ [− 12 , 12 )N . Thus, since ∪Jj=0 Ej contains exactly one point congruent mod 1 to each point in ∪Jj=0 A∗j E0 , the set  E = ∪Jj=0 Ej will satisfy k∈ZN 1E (ω + k) = 1. By construction E is a finite union of convex polytopes, and since Ej ⊂ A∗ Ej −1 , E also satisfies E ⊂ A∗ E. Define W = A∗ E \ E. Then W is also a finite union of convex polytopes, and is a (d − 1)-wavelet set by the consistency equation 3.14.   The wavelet sets we have found so far in this section have been built from the multiplicity function that is identically 1. To use other multiplicity functions, we first must find multiplicity functions that satisfy the conditions of Theorem 3.3 as well as the consistency equation 3.14. Some, such as the following example, have been found from partial information using educated guesses.

4.3 Wavelet Sets from the Consistency Equation

m=2

-

1 2

m=2

-1 2

1 2

m=2 m=1

1 1 2 3

1 3

1 2

m=2

43

2 1 1 - - 3 2 3

1

1 1 2 3 2 3

-1

-1 2

^ Ψ2 ^ Ψ1 1

-1

Fig. 4.2 Multiplicity function, scaling set, and corresponding simple 2-wavelet set

Example 4.1 The 2-wavelet set in Figure 4.2, which first appeared in [2], was the first known simple L−wavelet set for dilation by 2 in R2 with L < | det A| − 1 = 3. It was found by experimenting with multiplicity function discontinuities at x = ± 13 , y = ± 13 in R2 , motivated in part by the fact that Equation 3.26 shows that the integral of a multiplicity function of an L-wavelet for dilation by 2 in R2 must be L3 . The multiplicity function shown on the left in Figure 4.2 satisfies all the conditions of Theorem 3.3, as well as the consistency equation (3.14). A scaling set E for this multiplicity function, shown in the center of Figure 4.2 can be found using the method described after that theorem. The resulting 2-wavelet set, formed by W = 2E \ E, is shown on the right, with the support of the Fourier transforms of the two wavelets shown in different colors. In general, finding simple multiplicity functions directly is quite difficult, particularly in higher dimensions. We will see in the next section that it is easier to build a generalized scaling set directly from the consistency equation rather than building the multiplicity function first.

4.3 Wavelet Sets from the Consistency Equation Building a generalized scaling set directly from the consistency equation has the advantage of avoiding the need to check the Δ-condition that the multiplicity function is required to satisfy (condition (ii) of Theorem 3.3). We have the following theorem from [1]: Theorem 4.3 Let A be an expansive N ×N integer matrix. Suppose the measurable set E ⊂ RN is invariant under A∗−1 , contains a neighborhood of the origin, and satisfies the consistency equation 1+

 k∈ZN

1E (ω + k) =



1E (A∗−1 (ω + k)) a.e..

k∈ZN

Then W = A∗ E \ E is a wavelet set for dilation by A in L2 (RN ).

(4.7)

44

4 Wavelet Sets

Proof Equation 4.7 can be rewritten 1=

 k∈ZN

=





1A∗ E (ω + k) −

1E (ω + k)

k∈ZN

1A∗ E\E (ω + k)

k∈ZN

=



1W (ω + k) a.e..

k∈ZN

This shows that W tiles RN under translation.   Suppose j < k and consider A∗j W ∩ A∗k W = A∗(j +1) E \ A∗j E ∩   ∗(k+1) E \ A∗k E . Since E is invariant under A∗−1 and A∗ is one-to-one, we have A A∗(j +1) E ⊆ A∗k E, so that this intersection is empty. Thus the dilates of W under A∗ are disjoint. The dilates of W will also cover RN a.e. because E contains a neighborhood of the origin. Thus, W tiles under dilation, and the result follows from Theorem 4.1.   Note that Equation (4.7) is equivalent to the consistency equation (3.14) for the multiplicity function for a single wavelet. In [1], Theorem 4.3 is proved using GMRA theory rather than the tiling condition. The theorem is then used to develop an explicit technique for constructing all wavelet sets in RN . Like the method for building multiplicity functions, this technique gives an iterative procedure for building a scaling set E = ∪∞ i=1 Ei . Here the iteration is used to repeatedly correct for how each new piece affects the consistency equation (4.7). Specifically, E1 is taken to be any set that satisfies 1=



1E1 (A∗−1 (ω + k)) a.e..

(4.8)

k∈ZN

This can be accomplished by choosing any measurable one-to-one map T that takes each point ω ∈ [− 12 , 12 )N to a translate of A∗−1 (ω) by the lattice A∗−1 ZN ,   and then letting E1 = T [− 12 , 12 )N . Comparing Equation (4.8) with the required consistency equation (4.7), we see that E1 will fail to satisfy the consistency equation because of the missing second term on the left-hand side of Equation (4.7). The set E2 is chosen to correct for this; that is, E2 is chosen to be disjoint from E1 and satisfy  k∈ZN

1E1 (ω + k) =



1E2 (A∗−1 (ω + k)) a.e..

(4.9)

k∈ZN

In order to also meet the condition E ⊂ A∗ E, we also require A∗−1 E1 ⊂ E1 ∪ E2 . This step can be described in terms of choosing a second map T , which has range

4.3 Wavelet Sets from the Consistency Equation

45

disjoint from E1 , which takes each point ω ∈ E1 to a translate of A∗−1 (ω) by the lattice A∗−1 ZN , and which satisfies T (ω) = A∗−1 (ω) whenever A∗−1 (ω) ∈ / E1 . We then define E2 = T (E1 ). The construction is completed by extending the map T recursively to Ei , i > 2,

requiring again that the range of T is disjoint from ∪i−1 j =1 Ej , that T takes each point ω ∈ Ei to a translate of A∗−1 (ω) by the lattice A∗−1 ZN , and that T (ω) = A∗−1 (ω)

/ ∪i−1 whenever A∗−1 (ω) ∈ j =1 Ej . The set Ei+1 = T (Ei ) will then satisfy  k∈ZN

1Ei (ω + k) =



1Ei+1 (A∗−1 (ω + k)) a.e.,

(4.10)

k∈ZN

and A∗−1 Ei−1 ⊂ ∪ij =1 Ej .

(4.11)

The resulting disjoint union E = ∪∞ i=1 Ei will satisfy all of the hypotheses of Theorem 4.3 except possibly the requirement that E contain a neighborhood of the identity. To ensure that E satisfies this condition as well, it suffices, for example, to choose E1 to contain a neighborhood of the identity. In this case, the set W = A∗ E \ E will be a wavelet set for L2 (RN ). In [1] it is shown that even without including the requirement that E contain a neighborhood of the identity, this procedure will produce a wavelet set for a subspace (possibly all) of L2 (RN ). See [1] for more details and a proof that all (subspace) wavelet sets can be found in this way. Subspace wavelet sets can be combined to form multiwavelet sets for all of RN . The construction just described is a method for building all wavelet sets. Thus, we can retroactively find (non-unique) maps T and T for any existing wavelet set. For example, the scaling set for the wedding cake of Dai, Larson, and Speegle in the center of Figure 3.1 can be built using T (ω) = ω2 and alternating between T (ω) = ω2 + ( 12 , 0) and T (ω) = ω2 in the first and fourth quadrants, and between T (ω) = ω2 − ( 12 , 0) and T (ω) = ω2 in the second and third. If we wish to use this technique to build simple wavelet sets, we must choose T and T so that the pieces Ej of the iterative process match up smoothly. In particular, in one dimension, we need endpoints of the intervals of the Ej to coincide. One family of such examples appears in [10], and a method to create them, which involves solving simultaneous systems of equations involving the endpoints of the intervals in E1 and the translation amounts used in T , is described in [26]. To build simple wavelet sets in higher dimensions involves choosing the shapes carefully as well, so that the pieces join without corners. This idea is the basis of the construction of simple (single) wavelet sets for all scalar dilations and many matrix dilations in R2 that appeared in [26] and [28]. This construction starts with E1 a truncated diamond centered on the diagonal through the first and third quadrants. Such an E1 can be built from [− 12 , 12 )2 by recentering if necessary, and then using the translation part of T to move each of four triangles in the lower right and upper left corners of the dilated square to its

46

4 Wavelet Sets

E2

E3 E1

E1

E3

(4/3,4/3) 1 (1.2,.8) (.6,.4) -1

1 (.4,-.4)

(-2/3,-2/3) -1 Fig. 4.3 Building a simple scaling set, and the resulting wavelet set for dilation by 2

opposite edges as shown in the upper left graph of Figure 4.3. The map T builds E2 by translating the dilate of the truncated diamond out by an element of A∗−1 ZN , and then builds E3 by translating the two trapezoids that make up the dilate of E2 back to join up with the ends of the original truncated diamond. (See the upper right graph of Figure 4.3.) The match-ups are guaranteed by solving simultaneous equations involving the dilation, the center, the side lengths, and the translation amounts. We define T on all the rest of the E2i+1 to be the same as it was on E1 , and on all the E2i to be the same as on E2 . This guarantees that the match-up will continue to happen for the whole iterative process. The graph in the bottom left of Figure 4.3 shows the finished scaling set for dilation by 2, and the bottom right shows the resulting wavelet set. (For details, and more examples, see [26] and [28].) The same construction works for any real scalar dilation dI for d > 1 in R2 . Non-integer dilations produce a wavelet set that is not related to a GMRA. For d = 2, the scaling set is exactly as shown in Figure 4.3; for d > 2 the two diamonds are separated rather than adjacent along the diagonal. The wavelet set in all cases is

4.3 Wavelet Sets from the Consistency Equation

1

(63/80,87/80)

47

(9/8,9/8)

(16/7,16/7) 2

(3/4,3/4) 1 (29/80,21/80) -1

-2

1

-1 (-1/2,-1/2)

(4/9,5/9) 1

2

(4/9,-4/9) -1

(-3/4,-3/4)

(21/80,-51/80)

(-2,-2)

-1

-2

1

-1

1

-1

Fig. 4.4 A wavelet set for dilation by 3, and Journé-like and curved variations for dilation by 2

formed by W = dE \ E, and all have the basic shape of the d = 3 example shown on the left in Figure 4.4, with d = 2 having the larger diamond centered at the origin as in Figure 4.3. Many variations on this construction are also explored in [26]. For example, the wavelet set can be made to be symmetric about the origin, and also to be Journé-like in having higher multiplicities, as shown in the middle graph in Figure 4.4. A final curved variation, shown on the right in Figure 4.4, is built in the same way, but is not a finite union of convex sets, and so not simple. In [28], it is shown that this construction can also produce simple wavelet sets for some non-scalar dilations in R2 . In particular, if A is an expansive integer matrix such that Aq = dI for positive integers √ q and d (A is scalarpotent), and if all the singular values of A are greater than 2, then the process described above produces a scaling set for A that consists of 2q diamonds. The resulting simple wavelet set W = dE \ E will look much the same as in the scalar case, with an annular notched

48

4 Wavelet Sets

0.6 1.0 0.4 0.5

0.2

-0.2 -0.1

0.1

0.2

-0.5

0.3

-0.2

Fig. 4.5 Non-simple scaling set and wavelet set for

0.5

1.0

-0.5 2 0 03

diamond and one satellite diamond. The condition on the singular values, which measure growth of the image size in a way similar to eigenvalues, is needed to prevent overlap of the pieces in the construction process. For matrices that are not scalarpotent, the construction described above does not produce simple wavelet sets. The reason for this is that neither the matrix A itself, nor any of its powers, will preserve slopes of lines. Thus, while the edges and lengths of the truncated diamonds can be made to match up, the slopes of the sides will change from piece to piece. Figure 4.5 shows the curling sides of the nonsimple scaling and sets that are produced by the diamond construction for  wavelet  the matrix A = 20 03 .   It is not known whether the matrix 20 03 has a simple wavelet set that can be made using a different procedure. However, the fact that slopes are not preserved by this dilation makes the existence of a simple wavelet set unlikely. It is shown in [26] that a dilation in R2 has a simple wavelet set if and only if it has a wavelet set that is a finite union of convex polygons. This is used in [28] to prove that an expansive integer matrix that has the property that none of its eigenvalues have a power in R cannot have a simple wavelet set. Thus, for example, the matrix 34 −3 4 cannot have a simple wavelet set. The proof proceeds by first showing that if the wavelet set is a finite union of convex polygons, then the regions of constant value of the multiplicity function must be as well. Because of the importance of preserving slope in both the positive and negative results, it is conjectured in [28] that the class of integer matrices with determinant greater than 2 having simple wavelet sets is exactly those with a nonzero power equal to a multiple of the identity. The method of building diamonds using the consistency equation described in this section can be extended to R3 and R4 to find simple wavelet sets there as well. However, to find a general form for simple wavelet sets in all dimensions, a geometric technique is needed, which we will explore in the next section.

4.4 Wavelet Sets from Geometry

49

4.4 Wavelet Sets from Geometry The first known wavelet sets in R2 were built using the geometric characterization contained in Theorem 4.1. For example, the wedding cake set in Figure 3.1 was constructed by starting with a set that tiles under translation, and then translating pieces so that the resulting set tiles by dilations as well. (For other early examples, see [31, 34].) The geometric characterization of wavelet sets was also the basis for the Dai/Larson/Speegle proof [11] that single wavelet sets exist for every expansive dilation in every dimension. Benedetto, Leon, and Sumetkijakan [4–6] used a similar idea to develop a general wavelet set construction technique they called a neighborhood-mapping construction, in which intermediate steps produce Parseval wavelet sets. Geometric ideas and the characterization of Theorem 4.1 were also used by Gabardo and Yu [16] to build the first simple wavelet set for dilation by 2 in R2 . They used self-affine tiles, which are compact sets K with |K| = 1, such that A∗ K = ∪d∈D K + d, where D is a set of coset representatives for Z2 /A∗ Z2 . Self-affine tiles most naturally produce wavelet sets for determinant 2 matrices. If det A = ±2, |K| = 1, K ∩ Z2 = 0, and D = {0, d1 }, then K + d1 is a wavelet set for dilation by A. However, a simple K that leads to a simple set is not  wavelet  always easy to find. For example, for the quincunx matrix, ± 11 −1 , whose simple 1 wavelet set is shown in Figure 4.1, the self-affine tile with digit set {(0, 0), (1, 0)} is the twin dragon fractal. To build the first simple wavelet set for dilation by 2 in R2 , Gabardo and Yu modified the determinant 2 construction by introducing a multiplier matrix and using a specially constructed digit set for it. The resulting wavelet set is shown on the left in Figure 4.6. Note that though they were developed independently and by different techniques, this wavelet set and the one described in the previous section have very similar shapes. In fact, the wavelet set W in Figure 4.3 the Gabardo/Yu  1and 0 W , an example wavelet set W on the left in Figure 4.6 are related by W = −1 1 1.0 0.5

0.5

-1.0

-0.5

0.5 -0.5

-1.0

1.0

1.5

-0.5

0.5 -0.5

-1.0

Fig. 4.6 Simple wavelet sets built from self-affine tiles for dilation by 2 and by −2

1.0

1.5

50

4 Wavelet Sets

of the phenomenon described at the end of Section 4.1. The graph on the right of Figure 4.6 is a wavelet set for dilation by −2 that was built by the same self-affine tile technique. For more details, see [16]. More recently, the geometric characterization of wavelet sets in terms of tilings was used by the current author to generalize the simple wavelet sets presented in the previous section to an arbitrary dimension [29]. Just as in the early constructions in this section, this was done by starting with a set that tiles under translation, and then translating pieces of this set to achieve tiling under dilation. By using the following lemma by S. Stein to build the initial set, only one simple piece must be translated. Lemma 4.1 Given a real number 0 < α < 1, let L be the lattice spanned by cyclic permutations of the vector (1, −α, 0, . . . , 0). Then the translates by vectors in L of the notched unit cube K = {(x1 , x2 , . . . , xn ) : 0 ≤ xi ≤ 1} \ {(x1 , x2 , . . . , xn ) : 1 − α ≤ xi ≤ 1} tile RN . Proof See [32]. Lemma 4.1 implies that A(K) tiles under translation by ZN , where A is the inverse of the matrix whose columns generate L. We write {a1 , a2 , . . . aN } for the columns of A, which will be the cyclic permutations of the vector 1 (1, α, α 2 , · · · , α N −1 ). Then, if we let Nα be the notched parallelotope spanned 1−α N by these vectors, {x1 a1 +x2 a2 +. . . xn an : 0 ≤ xi ≤ 1} \ {x1 a1 +x2 a2 +. . . xn an : 1−α ≤ xi ≤ 1}, we have that Nα will tile RN under translation by ZN for any 0 < α < 1. To achieve tiling by dilation as well, we must translate (by an integer vector) a dilated central parallelotope out from Nα so that its dilate will fill the notch. We must coordinate the translation amount, the size α of the notch, and the position of Nα . This can be done by solving simultaneous linear equations. One straightforward solution is presented in the following theorem. (For details and other possibilities, see [29].) We write Pα = {x1 a1 +x2 a2 +. . . xn an : 0 ≤ xi ≤ 1} for the un-notched version of Nα , and τt for translation by (t, t, . . . , t). Theorem 4.4 The set       1 1 τ −d N 1 \ τ −d P 1 τ −d P 1 τ1 d+1 d d+1 d 2 d d+1 d 2 d2

(4.12)

formed from the N -dimensional notched parallelotope is a wavelet set in RN for any real dilation d > 1 and any positive integer N . Proof See [29]

 

4.4 Wavelet Sets from Geometry

y -0.5

0.0

0.5

51

1.0

y -0.5

1.0

0.5

1.0

0.0

0.5 z

0.0

0.5

z -0.5

0.0

-1.0

-0.5 -0.5

0.0 x

-0.5 0.5

1.0

0.0 x

Fig. 4.7 Simple 3-dimensional wavelet sets for dilation by 3 and by

0.5

3 0

1 03 0 0 0 −3

1.0



The left-hand graph in Figure 4.7 shows the wavelet set produced by Theorem 4.4 for dilation by 3 in R3 . The theorem works for any expansive scalar dilation, although non-integer dilations are not associated with a GMRA. The theorem can also be adjusted to produce wavelet sets for any dilation that has a power equal to a scalar times the identity, as long as the dilation has all singular values greater √ than  N . The  right-hand graph in Figure 4.7 shows a wavelet set for dilation by 30 1

A = 0 3 0 , which satisfies A2 = 9I , and has singular values 3.54, 3, and 2.54. 0 0 −3 (See [29] for more on such examples.) As discussed at the end of Section 4.1, any of these wavelet sets can be multiplied by an integer matrix S with det S = ±1 to give other wavelet sets. For example, taking S to be the matrix with 1’s on the diagonal, −1’s on the subdiagonal and 0’s elsewhere, produces an N -dimensional wavelet set centered on the x1 −axis, as is the Gabardo/Yu 2-dimensional set in the left-hand graph of Figure 4.6. As the second Gabardo/Yu graph in Figure 4.6 suggests, notched parallelotopes can themselves be wavelet sets for negative scalar dilations. We have the following general theorem from [29]: Theorem 4.5 The N -dimensional notched parallelotope τ −d 2 N 1 is a wavelet set in

RN

d 2 −1

d

for dilation by −d, for any real d > 1 and any positive integer N .

Proof Tiling by translation follows from Lemma 4.1, with α = d1 . To establish tiling by dilation, write W = τ −d 2 N 1 , and note that W ∪ −1 d W is the un-notched d  d 2 −1      parallelotope shell τ −d 2 P 1 \ d12 τ −d 2 P 1 . For more detail, see [29]. d 2 −1

d

d 2 −1

d

The left-hand graph in Figure 4.8 shows a 3-dimensional notched parallelotope wavelet set for dilation by −2.

52

4 Wavelet Sets

0.5

-1

-0.5

x

0

0.5

0 0.5

z

0.0 z -1

-1

0.5 0 y

-0.5

-2

-1 x

0.5 0.0 y -0.5

0 1 2

Fig. 4.8 A single wavelet set for dilation by −2, and a 2-wavelet set for dilation by 2 in R3

As a final application of Lemma 4.1, we build a two-wavelet set with the shape of a parallelotope shell for any scalar dilation in any dimension. The idea is to first stretch the notched parallelotope in the direction of the notch, so that the notch can then be moved to the center of the parallelotope via an integer translation. Theorem 4.6 For any positive integer N and any real d > 1, the N-dimensional parallelotope shell     1 −d −d MP 1 + ( , 0, 0, . . . , 0) \ , 0, 0, . . . , 0) MP 1 + ( d d d −1 d d −1 is a 2-wavelet set for dilation by d in RN , where M is the matrix that results from replacing the first column of the identity by (2, −1, −1, . . . . − 1). Proof The set clearly tiles by dilation since it is obtained by removing a central region scaled by d1 from an original set that contains this central region. To establish 2-fold tiling under translation, we start by noting again that Lemma 4.1 shows that the N −dimensional notched parallelotope N 1 tiles under d   translation by ZN . Thus M N 1 tiles under translation by the lattice spanned by d the columns of M. Since this is the same as the lattice spanned by {2e  1 , e2 , . . . , eN }, N where {e1 , e2 , . . . , eN } is the usual basis for R , we have that M N 1 gives a 2d   −d N fold tiling under translation by Z . Hence, M N 1 + ( d−1 , 0, 0, . . . , 0) also gives d a two-fold tiling under integer translations. It remains to show that this set can be transformed into the proposed 2-wavelet set via an integer translation of a subset. The original notched parallelotope, N 1 , is centered on the line x1 = x2 = · · · = d

d d d xN , with the parallelotope P 1 extending from (0, 0, . . . , 0) to ( d−1 , d−1 , . . . , d−1 ), d

d d d and the notch extending from (1, 1, . . . , 1) to ( d−1 , d−1 , . . . , d−1 ). Thus,

References

53

Table 4.1 Known existence of simple wavelet sets Dimension N N =1 Arbitrary

N =2

Dilation A A∈Z | det A| = d, d ∈ Z+ A = dI , d ∈ R, |d| > 1 A = dI , d ∈ R, d > 1 Ap = dI , |d| > 1, p ∈ Z+ , √ Singular values(A) > n λp ∈ / R for all eigenvalues λ and all p ∈ Z+

Simple L-wavelet set L=1 L=d −1 L=1 L=2 L=1

References [10, 26] Theorem 4.2 Theorems 4.4, 4.5 Theorem 4.6 [29]

Does not exist

[28]

  −d M N 1 + ( d−1 , 0, 0, . . . , 0) is centered on the x1 −axis, with the parallelotope d

−d d extending from ( d−1 , 0, 0, . . . , 0) to ( d−1 , 0, 0, . . . , 0), and the notch extending 1 d from (1− d−1 , 0, 0 . . . , 0) to ( d−1 , 0, 0 . . . , 0). Translating the central parallelotope     1 −d d M P d1 + ( d−1 , 0, 0, . . . , 0) by (1, 0, 0, . . . , 0) will therefore just fit it into   −d , 0, 0, . . . , 0) into the proposed 2-wavelet the notch, and transform M N 1 + ( d−1 d set.  

The right-hand graph in Figure 4.8 shows a dilation by 2 example of this 2wavelet set in R3 . A summary of what is known about the existence of simple wavelet sets is shown in Table 4.1. Filling in the many gaps for 1-wavelet sets in this table is an area of active research. For example, the condition on singular values is an artifact of the construction, and would be satisfying to remove. Also, as mentioned earlier, the existence  of simple 1-wavelet sets is unknown even for diagonal non-scalar matrices such as 20 03 . A more ambitious aim is to settle the conjecture that a matrix A with | det A| = 2 has a simple 1-wavelet set if and only if Ap = dI for p ∈ Z.

References 1. Baggett, L., Medina, H., Merrill, K.: Generalized multiresolution analyses and a construction procedure for all wavelet sets in Rn . J. Fourier Anal. Appl. 5, 563–573 (1999) 2. Baggett, L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ). Appl. Comput. Harmon. Anal. 13, 201–223 (2002) 3. Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: A non-MRA C r frame wavelet with rapid decay. Acta Appl. Math. 89, 251–270 (2006) 4. Benedetto, J.J., Leon, M.: The construction of multiple dyadic minimally supported frequency wavelets on Rd . Contemp. Math. 247, 43–74 (1999) 5. Benedetto, J.J., Leon, M.: The construction of single wavelets in d-dimensions. J. Geom. Anal. 11, 1–15 (2001) 6. Benedetto, J.J., Sumetkijakan, S.: Tight frames and geometric properties of wavelet sets. Adv. Comput. Math. 24, 35–56 (2006)

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7. Bownik, M., Speegle, D.: Meyer type wavelet bases in R2 . J. Approx. Theory 116, 49–75 (2002) 8. Bownik, M., Rzeszotnik, Z., Speegle, D.: A characterization of dimension functions of wavelets. Appl. Comput. Harmon. Anal. 10, 71–92 (2001) 9. Calogero, A.: Wavelets on general lattices, associated with general expanding maps of Rn . Electron. Res. Announc. Am. Math. Soc. 5, 1–10 (1999) 10. Dai, X., Larson, D.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Am. Math. Soc. 134 (1998) 11. Dai, X., Larson, D., Speegle, D.: Wavelet sets in Rn . J. Fourier Anal. Appl. 3, 451–456 (1997) 12. Dai, X., Diao, Y., Li, Z.: Intrinsic s-elementary Parseval frame multiwavelets in L2 (Rd ). J. Math. Anal. Appl. 367, 677–684 (2010) 13. Dutkay, D.: Some equations relating multiwavelets and multiscaling functions. J. Funct. Anal. 226, 1–20 (2005) 14. Fang, X., Wang, X.: Construction of minimally supported frequency wavelets. J. Fourier Anal. Appl. 2, 315–327 (1996) 15. Frazier, M.W., Garrigós, Wang, K., Weiss, G.: A characterization of functions that generate wavelet and related expansions. J. Fourier Anal. Appl. 3, 883–906 (1997) 16. Gabardo, J-P, Yu, X.: Construction of wavelet sets with certain self-similarity properties. J. Geom. Anal. 14, 629–651 (2004) 17. Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147 (2000) 18. Hernandez, E., Weiss, G.: A First Course on Wavelets. CRC Press, Boca Raton (1996) 19. Hernández, E., Wang, X., Weiss, G.: Smoothing minimally supported frequency wavelets. J. Fourier Anal. Appl. 2, 329–340 (1996) 20. Hernández, E., Wang, X., Weiss, G.: Smoothing minimally supported frequency wavelets II. J. Fourier Anal. Appl. 3, 23–41 (1997) 21. Kirat, I., Lau, K.S.: Classification of integral expanding matrices and self-affine tiles. Discrete Comput. Geom. 28, 49–73 (2002) 22. Lagarias, J, Wang, Y.: Haar-type orthonormal wavelet bases in R2 . J. Fourier Anal. Appl. 2, 1–14 (1995) 23. Lemarié, P.G., Meyer, Y.: Ondelettes et bases Hilbertiennes. Rev. Mat. Iberoamericana 2, 1–18 (1986) 24. Lim, L.-H., Packer, J., Taylor, K.: Direct integral decomposition of the wavelet representation. Proc. Am. Math. Soc. 129, 3057–3067 (2001) 25. Littlewood, J.E., Paley, R.E.A.C.: Theorems on Fourier series and power series. J. Lond. Math. Soc. 6, 230–233 (1931) 26. Merrill, K.: Simple wavelet sets for scalar dilations in L2 (R2 ). In: Jorgensen, P., Merrill, K., Packer, J. (eds.) Wavelets and Frames: A Celebration of the Mathematical Work of Lawrence Baggett, pp. 177–192. Birkhäuser, Boston (2008) 27. Merrill, K.: Smooth well-localized Parseval wavelets based on wavelet sets in R2 . Contemp. Math. 464, 161–175 (2008) 28. Merrill, K.: Simple wavelet sets for matrix dilations in R2 . Numer. Funct. Anal. Optim. 33, 1112–1125 (2012) 29. Merrill, K.: Simple wavelet sets in Rn . J. Geom. Anal. 25, 1295–1305 (2015) 30. Shannon, C.E.: Communications in the presence of noise. Proc. Inst. Radio Eng. 37, 10–21 (1949) 31. Soardi, P.M., Weiland, D.: Single wavelets in n-dimensions. J. Fourier Anal. Appl. 4, 299–315 (1998) 32. Stein, S.K.: The notched cube tiles Rn . Discrete Math. 80, 335–337 (1990) 33. Yu, X.: Cube-approximating bounded wavelet sets in Rn . Proc. Am. Math. Soc. 134, 491–499 (2005) 34. Zakharov, V.: Nonseparable multidimensional Littlewood-Paley like wavelet bases. Centre de Physique Théorique, CNRS Luminy 9 (1996)

Chapter 5

Generalized Filters

Filters in a GMRA encode the containments V−1 ⊂ V0 and W−1 = δ −1 W0 ⊂ V0 as relationships between scaling functions, wavelet functions, and their dilates. Classical filters were defined in L2 (R) in terms of Fourier transforms of these functions, and were used to build MRAs and orthogonal wavelets with desirable properties. Later, these Fourier filters were used in non-MRAs and to build Parseval wavelets as well. Generalized filters go beyond this extension of the use of classical filters, taking better advantage of the GMRA structure by using the unitary operator J given by spectral multiplicity in Theorem 2.3. We begin with a brief look at the history and accomplishments of classical filters.

5.1 Classical Filters Suppose {Vj } is an MRA for dilation by 2 in L2 (R) with scaling function φ. Then  the requirement that V−1 ⊂ V0 implies that δ −1 (φ)(x) = √1 φ( x2 ) = k∈Z ak φ(x− 2 k), or equivalently that φ(x) =

√  2 ak φ(2x − k)

(5.1)

k∈Z

for some l 2 sequence {ak }. By taking Fourier transforms, we obtain (ω), or equivalently, h(ω)φ ω ω 1 ( ), (ω) = √ h( )φ φ 2 2 2 where h(ω) =



k∈Z ak e

2π ikω

√ (2ω) = 2φ

(5.2)

is a function with period 1.

© Springer Nature Switzerland AG 2018 K. D. Merrill, Generalized Multiresolution Analyses, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-319-99175-7_5

55

56

5 Generalized Filters

If this MRA has an orthonormal wavelet, then W−1 ⊂ V0 similarly implies the existence of a second periodic function g such that ω ω 1 ( ). (ω) = √ g( )φ ψ 2 2 2

(5.3)

For example, for the Haar wavelet, which has φ = 1[0,1) and ψ = 1[0, 1 ) − 1[ 1 ,1) , 2 2 using Equation (5.1) and its counterpart for ψ shows that these functions are h(ω) = √1 (e−2π iω + 1) and g(ω) = √1 (e−2π iω − 1). For the Shannon wavelet, which has 2 2  = 1 1 1 and ψ =1 φ 1 [− 2 , 2 ) [−1,− 2 )∪[ 12 ,1) , using Equations (5.2) and (5.3) shows they √ √ are h(ω) = 21[− 1 , 1 ) and g(ω) = 21[− 1 ,− 1 )∪[ 1 , 1 ) . Note that, while we describe 4 4

2

4

4 2

the filters in terms of functions on [− 12 , 12 ), they are really periodic functions on R. The functions h and g are called low- and high-pass filters because they typically act by filtering out all but low (for h) or high (for g) frequencies. The use of such filters has historical roots in the quadrature mirror filters invented by Galand and Esteban in signal processing ([18, 26]), and is a key example of the fertile collaboration between engineering and mathematics that fueled wavelet development. Because of the orthonormality conditions satisfied by translates of φ and ψ, these filters are forced to satisfy their own orthonormality-like conditions, which we call the filter equations: 1 |h(ω)|2 + |h(ω + )|2 = 2 2 1 |g(ω)|2 + |g(ω + )|2 = 2 2

(5.4)

1 1 h(ω)g(ω) + h(ω + )g(ω + ) = 0. 2 2 S. Mallat and Y. Meyer discovered in the late 1980s that beginning with functions that satisfy these orthonormality conditions, it is sometimes possible to turn this process around, and create MRAs and wavelets from filters. This process finds scaling functions by iterating Equations (5.1) or (5.2), and then uses Equation (5.3) to find the wavelet. In particular, the cascade algorithm starts with an initial seed such as φ = 1[0,1] , and iterates Equation (5.1) to get successive approximations to φ. Under appropriate conditions, these approximations converge. Iterating instead on the Fourier side using Equation (5.2) leads to the following theorem, which is responsible for the construction of many of the most useful MRA wavelets, those that are both well-localized and smooth. Theorem 5.1 Let h and g be periodic C 1 functions that satisfy the orthonormality conditions (5.4). Suppose in addition that h(ω) = 0 for all ω ∈ [− 14 , 14 ) and that √ |h(0)| = 2. If

5.1 Classical Filters

(ω) = φ

57

∞  1 √ h(2−j ω) 2 j =1

and

ω ω 1 ( ), (ω) = √ g( )φ ψ 2 2 2

(5.5)

then the function φ is a scaling function for an MRA and the function ψ is an orthonormal wavelet for dilation by 2 in L2 (R).  

Proof See [23, 25]

To find filters h and g that satisfy the hypothesis of this theorem, first note that if h is any periodic function that satisfies the first of the filter equations (5.4), then taking 1 g(ω) = e2π iω h(ω + ) 2

(5.6)

will satisfy the other two as well. Thus, the challenge in building wavelets from classical filters is to find low-pass filters h that lead to scaling functions and wavelets with desirable properties. I. Daubechies used powers of the trigonometric identity sin2 (ω) + cos2 (ω) = 1, which fit naturally into the first of the orthonormality conditions (5.4), to find low-pass filters that yield C r wavelets with compact support [17]. A good introductory description of this construction appears in [31]. Several authors in the late 1990s and early 2000s smoothed the filters associated with wavelet sets in order to construct well-localized approximations for MRA MSF wavelets. In [21], Hernández, Wang, and Weiss did this for a large class of MSF wavelets for dilation by 2 in L2 (R). Calogero ([14]) similarly smoothed the filter associated with the wavelet set for the quincunx matrix shown in the center graph of Figure 4.1. Later, Bownik and Speegle used this technique to show that every expansive integer 2 × 2 matrix A has an orthonormal wavelet {ψ1 , . . . , ψ| det A|−1 } whose Fourier transforms are compactly supported and smooth [11]. We will use generalized filters to carry out a similar process for non-MRA wavelet sets in Section 5.4. of Theorem 5.1 that the low-pass filter must satisfy |h(0)| = √ The requirements 2 and h ∈ C 1 are natural restrictions to make the infinite product converge. The condition that h = 0 on [− 14 , 14 ) appears less natural, but is needed to prove L2 convergence of the infinite product in order to ensure the orthonormality of integer translates of φ. A famous example due to A. Cohen [15] showed that removing this condition can in fact lead to functions φ and ψ whose translates are not orthonormal. Cohen’s example took h(ω) = √1 (1 + e−6π iω ) and g(ω) = √1 (1 − e−6π iω ), 2

2

resulting in stretched versions of the Haar scaling function φ = 13 1[0,3) and wavelet ψ = 13 (1[0, 3 ) − 1[ 3 ,3) ). While this stretched version of the Haar wavelet is not itself 2 2 an orthonormal wavelet, it is a Parseval wavelet. We will explore this phenomenon in the next section.

58

5 Generalized Filters

5.2 Extended Uses of Classical Filters The first extensions of the use of classical filters came from generalizing Cohen’s example, to produce Parseval wavelets when omitting the nonvanishing condition on h in the classical construction. In 1990, W. Lawton used a cascade algorithm in the time domain to show that other examples like Cohen’s also produce Parseval wavelets. Specifically, he generalized Daubechies’ construction by starting with a filter h that is a trigonometric polynomial, but does not necessarily satisfy h = 0 on [− 14 , 14 ), and then proving that the function ψ that then results from the procedure described above is a compactly supported Parseval wavelet [22]. In the mid-1990s, O. Bratteli and P. E. T. Jorgensen extended Lawton’s result, requiring that the filters be only Lipschitz continuous rather than trigonometric polynomials, and including dilation by any integer d > 1. Their generalization starts with filter equations associated with an orthogonal wavelet MRA, even though the translates of the resulting φ and ψ do not necessarily turn out to be orthogonal. As in Corollary 2.1, the hypotheses thus require d − 1 wavelet functions. They require orthonormality conditions that are like (5.4), but modified in an obvious way for dilation by d. The proof of this result analyzes partial isometries that satisfy relations similar to those defining the Cuntz algebra, a method we will see again in Section 5.4. Theorem 5.2 Suppose h, g1 √ , . . . gd−1 are periodic Lipschitz continuous functions on R, which satisfy |h(0)| = d and the filter equations d−1 

|h(ω +

k=0 d−1 

gi (ω +

k=0 d−1  k=0

h(ω +

k 2 )| = d d

k k )gj (ω + ) = dδi,j d d k k )gi (ω + ) = 0. d d

(5.7)

for i = 1, . . . d − 1.

Define (ω) = φ

∞  1 √ h(d −j ω) d j =1

and

ω 1 ω ( ) l (ω) = √ gl ( )φ ψ d d d

(5.8)

for ω ∈ R and l = 1, . . . , d − 1. Then φ ∈ L2 (R), and {ψ1 , . . . ψd−1 } is a Parseval wavelet for dilation by d in L2 (R). The set of functions {ψ1 , . . . ψd−1 } is an orthonormal wavelet if and only if  k∈Z

(ω(t + k)|2 = 1 |φ

(5.9)

5.2 Extended Uses of Classical Filters

59

Proof See [12] (Section 12) and [13] (Theorem 2.2.1)

 

Other researchers in the late 1990s used versions of the classical filter construction to build wavelets from given scaling functions in GMRAs for dilation by 2 in L2 (R). For example, given a function φ whose translates form a frame for V0 in an FMRA, Benedetto and Li defined a low-pass filter h using Equation (5.2), and then derived conditions on a high-pass filter g that are sufficient to produce an associated  (ω + k)|2 , frame wavelet ([7, 8]). These conditions involve the function k∈Z |φ which we know by Lemmas 2.1 and 3.2 is the same as 1S1 = m in the FMRA case, providing a tie to the generalized filter equations for GMRAs that we will see in the next section. Paluszy´nski, Šiki´c, Weiss, and Xiao in 2001 also used what are essentially classical filters for dilation by 2 in L2 (R) to build Parseval wavelets ([27, 28]). They called periodic functions h that satisfy the first orthonormality equation in (5.4) generalized filters, and then defined pseudo-scaling functions to be φ’s that satisfy Equation (5.2) with such an h. They classified filters as low pass if lim

n→∞

∞ 

|h(2−j ω)| = 1

for a.e. ω ∈ R,

(5.10)

j =n

and were able to show that for such filters, (5.3) and (5.6) always result in a Parseval wavelet. They also established that all orthonormal MRA wavelets can be obtained from this construction. Ron and Shen developed a key tool called the Unitary Extension Principle (UEP), which gives sufficient conditions on filters in a GMRA in Rd , under which a collection {ψ1 , . . . ψL } ⊂ V1 determines a Parseval wavelet ([30]). The UEP starts (ω) = with a function φ whose translates generate V0 , and which satisfies limω→0 φ (0) = 1, and φ (ω) = O(|ω|−ρ ) for ρ > d as ω → ∞. Let h be a filter determined φ 2 by φ as in Equation (5.2), and let filters {g1 , . . . , gL } be determined by {ψ1 , . . . ψL } as in Equation (5.8). Then the UEP states that {ψ1 , . . . , ψL } is a Parseval wavelet if the following (L + 1) × 2 matrix has orthonormal columns for a.e. ω ∈ [0, 12 ): ⎛

h(ω) ⎜ g1 (ω) ⎜ ⎜ ⎝

⎞ h(ω + 12 ) g1 (ω + 12 ) ⎟ ⎟ ⎟. .. ⎠ .

(5.11)

gL (ω) gL (ω + 12 ) The authors generalized the UEP to allow any finite number of generators {φ1 , . . . φc } for V0 and any expansive integer matrix dilation. We will see in the next section that the UEP condition is a consequence of the generalized filter equations for GMRAs.

60

5 Generalized Filters

5.3 Generalized Filters in GMRAs In this section and the next, we will present the theory of filters derived from the representation theory approach to GMRAs. While some authors have labeled the filters used in the last section as generalized filters, we will reserve that term for the filters described below using the multiplicity function of a GMRA. Just as with classical filters, we first define filters within a multiresolution structure, and then see under what conditions the process can be reversed. The ideas and proofs for the beginning of this process use calculations involving Fourier coefficients that are much the same in the general case as in L2 (RN ). Thus, we begin this section in the context of a general Hilbert space. At the end of this section, we specialize to L2 (RN ), and continue with that context in Section 5.4. The material in this section was first developed in L2 (RN ) in [2, 3, 16]; for the general Hilbert space context, see [5, 19, 29] and [6]. Let {Vj } be a GMRA in a Hilbert space H with affine structure given by group Γ and related operator δ such that δ −1 Γ δ has index d in Γ . We need to make the restrictions that Γ is abelian and that the core measure class (see Theorem 2.3) is absolutely continuous with respect to Haar measure. We will also assume that the multiplicity function m is finite a.e.. Recall from Theorems 2.4 and 2.5 that any GMRA with the first two restrictions has generalized scaling vectors {φi }i∈I , where φi = J −1 (1Si ), and a semiorthogonal Parseval wavelet {ψk }k∈K , with ψk = J−1 (1S˜k ). Here I and K are countable (possibly finite) index sets, and the nonunique operators J and J come from applying the theory of spectral multiplicity to the representation of Γ on V0 and W0 , respectively. (See Theorem 2.3 for details.) Recall that the sets Si and S˜k are related to the multiplicity functions for these representations by Si = {ω ∈ Γ : m(ω) ≥ i} and S˜k = {ω ∈ Γ : m (ω) ≥ k}. Analogously to the MRAsetting in Section 5.1, we use that fact that V−1 ⊂ V0  to conclude that δ −1 (φi ) = j ∈I γ ∈Γ aj,γ γ (φj ). Applying the map J , we obtain  2 functions hi ∈ L (Sj ) such that J (δ −1 φi ) = hi =



hi,j .

(5.12)

j ∈I

  Similarly applying the map J to δ −1 (ψk ) = j ∈I γ ∈Γ bj,γ γ (φj ), which follows  2 from the containment W−1 ⊂ V0 , we obtain functions gk ∈ L (Sj ) such that J (δ −1 ψk ) = gk =



gk,j .

(5.13)

j ∈I

Recall the map α defined on Γ in Section 2.1 by α(γ ) = δ −1 γ δ, and the induced map α ∗ on Γ given by α ∗ (ω) = ω ◦ α. The index d of δ −1 Γ δ in Γ is the number of elements in α ∗−1 ω = {ω0 , ω1 , . . . ωd−1 }. Using this notation, we have the following generalized filter equations:

5.3 Generalized Filters in GMRAs

61

Theorem 5.3 The functions hi,j and gk,j , defined as in Equations (5.12) and (5.13), are supported on Sj ⊂ Γ, and satisfy d−1 

hi,j (ωl )hi ,j (ωl ) = dδi,i 1Si (ω),

(5.14)

gk,j (ωl )gk ,j (ωl ) = dδk,k 1S˜k (ω),

(5.15)

j ∈I l=0 d−1  j ∈I l=0

and

d−1 

hi,j (ωl )gk,j (ωl ) = 0

(5.16)

j ∈I l=0

for a.e. ω ∈ Γ. Proof The equations follow from a straightforward comparison of the Fourier coefficients of the two sides as L1 functions on Γ. See [5] (Theorem 10), or [2] for details.    2 We call functions hi , gk ∈ L (Sj ) that satisfy Theorem 5.3 generalized lowand high-pass filters, respectively. As with classical filters, we wish to reverse the process just described by starting with generalized filters and building GMRAs and wavelets. For our first result in this direction, we start with a GMRA and its associated generalized scaling functions and low-pass filter, and use these ingredients to build a wavelet. The first step is to construct a generalized highpass filter {gk } from the low-pass filter {hi }. It was shown in [2] that this is always possible, using a procedure that depends on linear algebra arguments first developed in [16], which we summarize below. Fix an ω ∈ Γ, and again write ω0 , ω1 , . . . ωd−1 for points in α ∗−1 (ω).  the d  fj ∈ L2 (Sj ), we have Let lj = m(ωj ), and note that for any function f = − →ω that fj (ωk ) = 0 for lk < j . Thus, we can define a vector f whose components include all the nonzero values of the fj at the points ωk as follows: − →ω f =(f1 (ω0 ), . . . , fl0 (ω0 ), f1 (ω1 ), . . . , fl1 (ω1 ), . . . , f1 (ωd−1 ), . . . , fld−1 (ωd−1 )). − → m(ω) , and we use the ordinary The consistency equation shows that f ω ∈ Cm(ω)+ inner product and norm there. Because the vectors include all the nonzero values of the fj at ωk , the left-hand sides of the generalized filter equations in Theorem 5.3 → − → →ω − → − → → g ωk , − g ωk , and h ωi , − g k , can then be written as the inner products h ωi , h ωi , − respectively. By the first of these, the rewritten Equation (5.14), we see that, for√each fixed ω, the low-pass filter {hi } determines m(ω) orthogonal vectors of norm d in m(ω) . To produce the high-pass filter, we must measurably construct m Cm(ω)+ (ω) √ → m(ω) , each of norm more orthogonal vectors − g ωk ∈ Cm(ω)+ d. That such vectors exist follows from the obvious dimension argument. To assure measurability, we

62

5 Generalized Filters

construct these vectors piecewise on the sets Pl0 ,l1 ,...,ld−1 ,l˜ = {ω ∈ Γ : m(ωk ) = ˜ using Gram-Schmidt or a cross product technique developed by (ω) = l}, lk and m Courter [2]. The following theorem from [2] assures that this high-pass filter will always yield Parseval wavelets. Theorem 5.4 Let {Vj } be a GMRA in the Hilbert space H whose multiplicity function is finite a.e., with generalized scaling functions {φi = J −1 (1Si )}, and generalized low-pass filter {hi,j } given by Equation (5.12). If {gk,j } are functions supported on Sj which satisfy Equations (5.15) and (5.16), and we set ψk = δ(J −1 (⊕gk,j )),

(5.17)

then {ψk } is a Parseval wavelet for H. Proof It will suffice to show that the functions {γ (ψk ) : k ∈ K, γ ∈ Γ } ∪ {γ (φi ) : i ∈ I, γ ∈ Γ } form a Parseval frame for V1 . Applying the unitary map δ −1 , we −1 φ )} form a can equivalently show that the functions {α(γ )(δ −1 ψk )} and {α(γ i )(δ ∞ Parseval frame for V0 . To do this, we let T = {hi }i∈I ∪{gk }k∈K ⊂ j =1 L2 (Sj ), and  show that {α(γ )(τ ) : τ ∈ T , γ ∈ Γ } is a Parseval frame for L2 (Sj ). Accordingly, let f = ⊕fj be an arbitrary element of L2 (Sj ), write e for the identity of Γ , and write c (ρ) for the γ th Fourier coefficient of a function ρ on Γ. If we let ρ(ω) ˜ = d−1 γ ρ(ω ), then a straightforward calculation shows that c ( ρ) ˜ = dc (ρ). k γ α(γ ) k=0 Using this, we have: 

| f, α(γ )(τ )  L2 (Sj ) |2 =

τ,γ



|cα(γ ) (

τ,γ



fj τ j )|2

j

1  − → →ω 2 = 2 |cγ ( f ω , − τ )| d τ,γ 1  − → →ω 2  f ω , − τ )L2 (Γ) d2 τ  1 − → =  f ω 2Cm(ω)+m(ω) dω d Γ ⎛ ⎞ d−1 1 ⎝  = ce |fj (ωk )|2 ⎠ d =

j

k=0

⎛ ⎞  = cα(e) ⎝ |fj |2 ⎠ j

=

f 2

L2 (Sj )

.  

5.3 Generalized Filters in GMRAs

63

Corollary 5.1 If m  ≡ L, then {ψ1 , . . . ψL } defined as in Theorem 5.4 is an orthonormal wavelet for H. Proof If m  ≡ L, we have  Sk = Γ for 1 ≤ k ≤ L and  Sk = ∅ if k > L. Thus, → g ωk form an orthonormal basis for CL . A similar for almost all ω, the vectors √1 − d calculation to the proof of Theorem 5.4 shows that ⎛ ⎞  1 → → α(γ )gk , α(γ )gk =cα(γ −γ ) ⎝ gk,j g k ,j ⎠ = cγ −γ ( − g ωk , − g ωk )=δk,k δγ ,γ , d j

so that the functions α(γ )gk form an orthonormal basis for their span in  ∞ 2 j =1 L (Sj ). Therefore, the functions γ (ψk ) for γ ∈ Γ and 1 ≤ k ≤ L form an orthonormal basis for W0 .   We prove one final theorem about filters in an abstract GMRA, before restricting our attention to GMRAs in L2 (RN ). The following theorem from [3], taken together with Theorem 5.4, gives a generalization of the Unitary Extension Principle of Ron and Shen.  Theorem 5.5 Let {hi } ∪ {gk } ⊂ L2 ( Sj . Then {hi } and {gk } satisfy the generalized filter equations (5.14), (5.15), and (5.16) if and only if they satisfy  i∈I

hi,j (ωl )hi,j (ωl ) +



gk,j (ωl )gk,j (ωl ) = dδj,j δl,l 1Sj (ωl ).

(5.18)

k∈K

Proof It follows from the discussion preceding Theorem 5.4 that Equations (5.14), (5.15), and (5.16) are equivalent to the condition that the vectors − → → m(ω) for a.e. ω. Forming √1 h ω and √1 − g ωk form an orthonormal basis for Cm(ω)+ i d d a matrix with these vectors as rows produces a (m(ω) + m (ω)) × (m(ω) + m (ω)) matrix, whose rows are orthonormal if and only if its columns are orthonormal. Equation (5.18) is equivalent to the orthonormality of the columns.   Remark 5.1 In [10], Bownik and Rzeszotnik gave a similar generalization of the Unitary Extension Principle to GMRAs for the case of L2 (RN ). We now focus on the use of these theorems in the context of L2 (RN ), with Γ = ZN acting by translation, and δ = δA , dilation by an expansive integer matrix A. Examples in other Hilbert spaces and with other groups Γ will appear in Chapters 6, 7, and 8. Note that in the L2 (RN ) case, α(k) = Ak, and α ∗ acts on [− 12 , 12 )N by α ∗ (ω) = A∗ ω (mod 1), where A∗ is the transpose of the dilation matrix A. The index d of δ −1 Γ δ in Γ , which is the number of elements in α ∗−1 ω, is in this case equal to | det A|. The following result from [2] makes it easier to relate the theorems in this section to those in Section 5.2 by showing that prior knowledge of the operator J is not required in order to determine that a set of vectors {φi } are generalized scaling vectors for a GMRA in L2 (RN ). Note the similarity to the well-known MRA result (see, e.g., [20], p. 382).

64

5 Generalized Filters

Proposition 5.1 Let m be an integrable multiplicity function for a GMRA in L2 (RN ) (see Theorem 3.3), and let {hi,j } be functions supported on Sj that satisfy the filter equation (5.14). Suppose that {φi } is a collection of functions in L2 (RN ) that satisfy the following three conditions: (i) For almost all ω ∈ [− 12 , 12 )N , 

i (ω + k)φ j (ω + k) = δi,j 1Si (ω). φ

k∈ZN

(ii) For each i, for almost all ξ ∈ RN , 1  i (A∗ (ξ )) = √ j (ξ ). φ hi,j (ξ )φ d j (iii) For almost all ξ ∈ RN , lim sup j →∞



i (A∗−j (ξ ))|2 ≥ 1. |φ

i

Then the φi ’s are generalized scaling functions of the form φi = J −1 (1Si ) for a GMRA {Vj } whose multiplicity function is the given m.  

Proof See [2].

Note that condition (i) is the same as what appears in Lemma 2.1. Also note that in the context of L2 (RN ), knowing that φj = J −1 (1Sj ) makes Equations (5.12) and (5.13) equivalent to the more familiar forms given by condition (ii) and by 1  k (A∗ (ξ )) = √ j (ξ ). ψ gk,j (ξ )φ d j

(5.19)

We see this since the fact that J intertwines translation with multiplication by   exponentials implies that J −1 ( fj ) = j γ ∈ZN cj,γ γ (φj ), where the cj,γ are the Fourier coefficients of the function fj . Thus, applying J −1 to both sides of Equation (5.12), and then taking the Fourier transform, yields condition (ii). Similarly, applying the map δ −1 to Equation (5.17) in Theorem 5.4 results in Equation (5.19). In both the equation in condition (ii) and Equation (5.17), the filters act as periodic functions on RN . Here is an example of the ideas and results in this section in the context of L2 (RN ). Example 5.1 Let m = 1[− 1 , 1 )2 + 21 3 3

[− 12 ,− 13 )∪[ 31 , 12 )

2

be the multiplicity function

for dilation by 2 in L2 (R2 ) introduced in Example 4.1, so that S1 = [− 13 , 13 )2 ∪

5.3 Generalized Filters in GMRAs

65



2  2 [− 12 , − 13 ) ∪ [ 13 , 12 ) and S2 = [− 12 , − 13 ) ∪ [ 13 , 12 ) . Here we build a 2-wavelet set for a different GMRA with the same multiplicity function but different filters. Note that both this wavelet and that of Example 4.1 can be built more simply using wavelet set techniques. We include the filter construction here to make the vector techniques discussed in this section concrete. First, we use Proposition 5.1 to show that {φ1 , φ2 } given by 1 = 1 1 1 2 + 1 1 2 2 φ [− , ) [ , ) 3 3

3 3

2 = 1 2 1 2 φ [− ,− )

and

3

3

(5.20)

are generalized scaling vectors for a GMRA with the given multiplicity function. Conditions (i) and (iii) are easily checked. Condition (ii) can be satisfied by taking   h1,1 = 2 1[− 1 , 1 )2 + 1[ 1 , 1 )2 6 6

h2,1 = 21[− 1 ,− 1 )2 3

6

6 3

h1,2 = 0 h2,2 = 0

Note that, as before, we use functions on [− 12 , 12 )2 to define the filters, which are periodic functions on R2 . To find a high-pass filter from the low-pass filter we use the vector method described before Theorem 5.4. Here the points {ωl } = { ω2 , ω2 + ( 12 , 0), ω2 + − → (0, 12 ), ω2 + ( 12 , 12 )}. Recall that the vectors h ωi are defined on Si , and only contain those components hi,j (ωl ) that are allowed to be nonzero under the restriction ωl ∈ Sj . Thus the vectors determined by these low-pass filters are, for ω ∈ [− 13 , 13 )2 ,      ω 1 1 1 1 ω ω − →ω , h1,1 + ( , ) , h1,2 +( , ) = (2, 0, 0). h 1 = h1,1 2 2 2 2 2 2 2      − → For ω ∈ S1 \[ − 13 , 13 )2 = S2 , we have h ωi = hi,1 ω2 , hi,1 ω2 + ( 12 , 0) ,     hi,1 ω2 + (0, 12 ) , hi,1 ω2 + ( 12 , 12 ) , so ⎧ ⎪ ⎪ (2, 0, 0, 0) − →ω ⎨ (0, 2, 0, 0) h1 = ⎪ (0, 0, 0, 2) ⎪ ⎩ (0, 0, 2, 0) ⎧ ⎪ ⎪ (0, 0, 0, 2) − →ω ⎨ (0, 0, 2, 0) h2 = ⎪ (2, 0, 0, 0) ⎪ ⎩ (0, 2, 0, 0)

for ω ∈ [ 13 , 12 )2 for ω ∈ [− 12 , − 13 ) × [ 13 , 12 ) for ω ∈ [− 12 , − 13 )2 for ω ∈ [ 13 , 12 ) × [− 12 , − 13 ) for ω ∈ [ 13 , 12 )2 for ω ∈ [− 12 , − 13 ) × [ 13 , 12 ) . for ω ∈ [− 12 , − 13 )2 for ω ∈ [ 13 , 12 ) × [− 12 , − 13 )

The vectors associated with g1 and g2 , which are both defined on all of T2 , must also only contain those components gi,j (ωl ) that are allowed to be nonzero under

66

5 Generalized Filters

the restriction ωl ∈ Sj . They must be orthogonal to the hi vectors and of norm 2 for almost all ω. We can take, for ω ∈ [− 13 , 13 )2 ,      ω ω ω 1 1 1 1 − → ω , g1,1 + ( , ) , g1,2 +( , ) = (0, 2, 0) g 1 = g1,1 2 2 2 2 2 2 2      ω ω ω 1 1 1 1 − → ω g 2 = g2,1 , g2,1 + ( , ) , g2,2 +( , ) = (0, 0, 2). 2 2 2 2 2 2 2        → For ω ∈ S2 , we have − g ωi = gi,1 ω2 , gi,1 ω2 + ( 12 , 0) , gi,1 ω2 + (0, 12 ) ,   gi,1 ω2 + ( 12 , 12 ) , and can take ⎧ (0, 0, 2, 0) ⎪ ⎪ ⎨ (0, 0, 0, 2) − → g ω1 = ⎪ (0, 2, 0, 0) ⎪ ⎩ (2, 0, 0, 0) ⎧ (0, 2, 0, 0) ⎪ ⎪ ⎨ (2, 0, 0, 0) − → ω g2 = ⎪ (0, 0, 2, 0) ⎪ ⎩ (0, 0, 0, 2)

for ω ∈ [ 13 , 12 )2 for ω ∈ [− 12 , − 13 ) × [ 13 , 12 ) . for ω ∈ [− 12 , − 13 )2 for ω ∈ [ 13 , 12 ) × [− 12 , − 13 ) for ω ∈ [ 13 , 12 )2 for ω ∈ [− 12 , − 13 ) × [ 13 , 12 ) . for ω ∈ [− 12 , − 13 )2 1 1 1 1 for ω ∈ [ 3 , 2 ) × [− 2 , − 3 )

Finally, for ω ∈ T2 \ S1 , we can take  

 ω ω 1 (0, 2) − → ω , g1,1 +( , 0) = g 1 = g1,1 (2, 0) 2 2 2  

 ω ω 1 (2, 0) − → , g2,1 +( , 0) = g ω2 = g2,1 (0, 2) 2 2 2 

  ω ω 1 (2, 0) − → , g1,1 +(0, ) = g ω1 = g1,1 (0, 2) 2 2 2  

 ω ω 1 (0, 2) − → , g2,1 +(0, ) = g ω2 = g2,1 (2, 0) 2 2 2

for ω ∈ [− 12 , − 13 ) × [− 13 , 13 ) , for ω ∈ [ 13 , 12 ) × [− 13 , 13 ) for ω ∈ [− 12 , − 13 ) × [− 13 , 13 ) , for ω ∈ [ 13 , 12 ) × [− 13 , 13 ) for ω ∈ [− 13 , 13 ) × [− 12 , − 13 ) , for ω ∈ [− 13 , 13 ) × [ 13 , 12 ) for ω ∈ [− 13 , 13 ) × [− 12 , − 13 ) . for ω ∈ [− 13 , 13 ) × [ 13 , 12 )

These vectors result in the high-pass filter g1,1 = 21

S2

g2,1 = 21

 "  " 1 1 [− 16 , 16 )×[− 13 ,− 16 ) [ 6 , 3 )×[− 31 , 16 )

[− 13 , 16 )×[ 16 , 13 )

 "

[− 13 ,− 16 )×[− 16 , 16 )



g1,2 = 0 g2,2 = 21S2 ,

5.3 Generalized Filters in GMRAs

67

which by Equation (5.19) gives the Parseval wavelet {ψ1 , ψ2 } with Fourier transforms   1 ξ ξ ξ ξ    g1,1 ( )φ1 ( ) + g1,2 ( )φ2 ( ) ψ1 (ξ ) = 2 2 2 2 2 = 1 2 (ξ ) = ψ

[ 23 , 43 )2

 "

[ 13 , 23 )×[− 23 , 13 )

 "

[− 13 , 13 )×[− 23 ,− 13 )



  1 ξ ξ ξ ξ 1 ( ) + g2,2 ( )φ 2 ( ) g2,1 ( )φ 2 2 2 2 2

= 1

[− 43 ,− 23 )2

 " .  " [− 23 ,− 13 )×[− 31 , 13 ) [− 23 , 13 )×[ 31 , 23 )

(When performing the multiplications above, recall that the filters are periodic 1 and φ 2 are not.) Since m satisfies the consistency equation, this functions, while φ Parseval wavelet is actually an orthonormal wavelet. The graph of the associated 2-wavelet set is shown in Figure 5.1.

1.0

0.5

-1.0

-0.5

0.5

1.0

-0.5

-1.0

Fig. 5.1 The two wavelet set W = W1 ∪ W2 , with W1 and W2 shown in different shades of gray

68

5 Generalized Filters

5.4 Building GMRAs and Parseval Wavelets from Filters in L2 (RN ) Example 5.1 shows the limitations of building Parseval wavelets from given GMRAs. The easily obtainable scaling functions are often characteristic functions, leading to Parseval wavelets with Fourier transforms that are not smooth. In this section, we see how to build the GMRAs and scaling functions themselves from filters, allowing for the construction of smooth and well-localized Parseval wavelets. We restrict to the case of H = RN , Γ = ZN acting by translation, and δ = δA , dilation by an expansive integer matrix A, whose determinant has absolute value d. We also assume that the multiplicity function is bounded by some positive integer c, which implies that the multiplicity function m  is also bounded by a positive integer c. ˜ We will use filters in this context to build GMRAs and their scaling functions and wavelets by starting with just a multiplicity function. We will discuss how to build abstract GMRAs using filters in Chapter 8. Just as when building MRAs from classical filters, we will construct generalized scaling vectors and Parseval wavelets using an infinite product of dilates of the generalized low-pass filters. Here the low-pass filter is a matrix of functions H = (hi,j )i,j ≤c on TN that have supp(hi,j ) ⊂ Sj and that satisfy the generalized filter Equation (5.14). Some narrow conditions that force the infinite matrix product to converge and produce Parseval wavelets were presented in [2]. A more general result in [3] simplified these conditions to minimal requirements analogous to those of Theorem 5.2. Theorem 5.6 Let H = (hi,j ) and G = (gi,j ) be matrices of functions on TN which satisfy the filter equations of Theorem 5.3 and have supp(hi,j ) ⊂ Sj and Lipschitz continuous functions in a supp(gi,j ) ⊂ Sj . Suppose that the hi,j are √ neighborhood of 0, and that hi,j (0) = δi,1 δj,1 d. Then: # √1 H (A∗−q (ξ )) converges almost everywhere (1) The infinite product P = ∞ q=1 d

on RN , with components Pi,j that are square-integrable functions on RN and satisfy Pi,j = 0 for j > 1. (2) If V0 is defined to be the closed linear span of the integer translates of the i = Pi,1 for 1 ≤ i ≤ c, then {Vj = δ j V0 } is a GMRA functions φi defined by φ 2 N in L (R ). (3) If ψk ∈ L2 (RN ) for 1 ≤ k ≤ c˜ is defined by c 1  k (ξ ) = √ j (A∗−1 (ξ )), ψ gk,j (A∗−1 (ξ ))φ d j =1

(5.21)

then {ψ1 , ψ2 . . . ψc˜ } is a Parseval wavelet on L2 (RN ). Proof See [3].

 

5.4 Building GMRAs and Parseval Wavelets from Filters in L2 (RN )

69

The proof in [3] that the functions {ψk } given (5.21) form a Parseval  by Equation wavelet uses two auxiliary operators, SH : ci=1 L2 (Si ) → ci=1 L2 (Si ) and SG : c˜ c 2  2 k=1 L (Sk ) → i=1 L (Si ), which are defined from the filters by SH (f )(ω) = H t (ω)f (A∗ (ω))

and

SG (f )(ω) = Gt (ω)f (A∗ (ω)),

(5.22)

where H t and Gt stand for the transposes of H and G. These operators are known as Ruelle operators. The generalized filter equations of Theorems 5.3 and 5.5 translate to show that they satisfy the relations ∗ SH = I, SH

∗ SG SG = I

∗ SH SG = 0, and ∗ ∗ SH SH + SG SG = I,

(5.23)

which are similar to the relations defining the Cuntz algebra O2 . The first of these conditions shows that SG and SH are isometries. The Ruelleoperators are used in the proof of Theorem 5.6 to construct a Parseval frame for ci=1 L2 (Si ), which is in turn used to show that the {ψk } are a Parseval wavelet for L2 (RN ). We will use these operators again in Chapter 8 to build GMRAs and wavelets using direct limits. The statement of Theorem 5.6 is relatively straightforward, but several limitations need to be kept in mind. While the Parseval wavelet functions are in the set V1 = δV0 of the constructed GMRA, it may not be the case that Vj is the closure of the span of {δ i (γ (ψk ))}i