Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 128 Harmonic Analysis Smooth and Non-smooth Palle E.T. Jorgensen with support from the 10.1090/cbms/128 Harmonic Analysis Smooth and Non-smooth First row, from left to right: Kasso Okoudjou, Palle Jorgensen, Daniel Alpay, and Marius Ionescu. (Credit: Connie S. Steffen, MATH, Iowa State University.) Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics Number 128 Harmonic Analysis Smooth and Non-smooth Palle E.T. Jorgensen Published for the Conference Board of the Mathematical Sciences by the with support from the CBMS Conference on “Harmonic Analysis: Smooth and Non-smooth” held at Iowa State University in Ames, Iowa, June 4–8, 2018. The author acknowledges support from the Conference Board of the Mathematical Sciences and NSF grant number 1743819. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2010 Mathematics Subject Classification. Primary 28A80, 81Q35, 11K70, 60J70, 42C40, 60G22, 37A45, 42B37. For additional information and updates on this book, visit www.ams.org/bookpages/cbms-128 Library of Congress Cataloging-in-Publication Data Names: Jørgensen, Palle E.T., 1947- author. | Conference Board of the Mathematical Sciences. | National Science Foundation (U.S.) Title: Harmonic analysis: smooth and non-smooth / Palle E.T. Jorgensen. Description: Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, [2018] | Series: CBMS regional conference series in mathematics; number 128 | “Support from the National Science Foundation.” | “NSF- CBMS Regional Conference in the Mathematical Sciences on Harmonic Analysis: Smooth and Non-smooth, held at Iowa State University, June 4–8, 2018.” | Includes bibliographical references and index. Identifiers: LCCN 2018030996 | ISBN 9781470448806 (alk. paper) Subjects: LCSH: Harmonic analysis. | AMS: Measure and integration – Classical measure theory –Fractals.msc| Quantum theory – General mathematical topics and methods in quantum theory – Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.. msc | Number theory – Probabilistic theory: distribution modulo 1; metric theory of algorithms – Harmonic analysis and almost periodicity. msc | Probability theory and stochastic processes – Markov processes – Applications of Brownian motions and diffusion theory (population genetics, ab- sorption problems, etc.). msc | Harmonic analysis on Euclidean spaces – Nontrigonometric harmonic analysis – Wavelets and other special systems. msc | Probability theory and sto- chastic processes – Stochastic processes – Fractional processes, including fractional Brownian motion. msc | Dynamical systems and ergodic theory – Ergodic theory – Relations with num- ber theory and harmonic analysis. msc | Harmonic analysis on Euclidean spaces – Harmonic analysis in several variables – Harmonic analysis and PDE. msc Classification: LCC QA403 .J67 2018 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2018030996 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 232221201918 Dedicated to the memory of Kiyosi Itˆo Contents Preface ix Acknowledgments xi Chapter 1. Introduction. Smooth vs the non-smooth categories 1 1.1. Preview 1 1.2. Historical context 3 1.3. Iterated function systems (IFS) 7 1.4. Frequency bands, filters, and representations of the Cuntz-algebras 13 1.5. Frames 16 1.6. Key themes in the book 17 Chapter 2. Spectral pair analysis for IFSs 23 2.1. The scale-4 Cantor measure, and its harmonic analysis 23 2.2. The middle third Cantor measure 27 2.3. Infinite Bernoulli convolutions 29 2.4. The scale-4 Cantor measure, and scaling by 5 in the spectrum 33 2.5. IFS measures and admissible harmonic analyses 37 2.6. Harmonic analysis of IFS systems with overlap 38 Chapter 3. Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures 45 3.1. Harmonic analysis in the smooth vs the non-smooth categories 45 3.2. Spectral pairs and the Fuglede conjecture 46 3.3. Spectral pairs 51 3.4. Spectral theory of multiple intervals 58 Chapter 4. Four kinds of harmonic analysis 69 4.1. Orthogonal Fourier expansions 69 4.2. Frame and related non-orthogonal Fourier expansions 73 4.3. Wavelet expansions 79 4.4. Harmonic analysis via reproducing kernel Hilbert spaces (RKHS) 98 Chapter 5. Harmonic analysis via representations of the Cuntz relations 135 5.1. From frequency band filters to signals and to wavelet expansions 136 5.2. Stochastic processes via representations of the Cuntz relations 141 5.3. Representations in a universal Hilbert space 165 Chapter 6. Positive definite functions and kernel analysis 175 6.1. Positive definite kernels and harmonic analysis in L2(μ)whenμ is a gap IFS fractal measure 175 vii viii CONTENTS 6.2. Positive definite kernels and harmonic analysis in L2 (μ)whenμ is a general singular measure in a finite interval 180 6.3. Positive definite kernels and the associated Gaussian processes 211 Chapter 7. Representations of Lie groups. Non-commutative harmonic analysis 219 7.1. Fundamental domains as non-commutative tiling constructions 220 7.2. Symmetry for unitary representations of Lie groups 226 7.3. Symmetry in physics and reflection positive constructions via unitary representations of Lie groups 241 Bibliography 245 Index 265 Preface Mathematics is an experimental science, and definitions do not come first, but later on. — Oliver Heaviside (1850–1925) Traditionally the notion of harmonic analysis has centered around analysis of trans- forms and expansions, and involving dual variables. The area of partial differential equations (PDE) has been a source of motivation and a key area of applications; dating back to the days of Fourier. Or rather, the applications might originate in such neighboring areas as signal processing, diffusion equations, and in more gen- eral applied inverse problems. The dual variables involved are typical notions of time vs frequency,orposition vs momentum (in quantum physics). As most stu- dents know, Fourier series and Fourier transforms have been a mainstay in analysis courses we teach. In the case of Fourier series, and Fourier transforms, we refer to the variables involved as dual variables. If a function in an x-domain admits a Fourier expansion, the associated transform will be a function in the associated dual variable, often denoted λ,inwhatistofollow. Now the x-domain may involve a suitable subset Ω in Rd. The aim of Fourier harmonic analysis is orthogonal L2 Fourier expansions, at least initially. Alterna- tively, the x-domain may refer to a prescribed measure,sayμ with compact support in Rd. These settings are familiar, at least in the classical case, which we shall here refer to as the “smooth case.” Now if the measure μ might be fractal,sayaCantor measure, an iterated function system (IFS) measure, e.g., a Sierpinski construction, then it is not at all clear that the familiar setting of Fourier duality will yield useful orthogonal L2 decompositions. Take for example the Cantor IFS constructions, arising from scaling by 3, and one gap; or the corresponding IFS measure resulting from scaling by 4, but now with two gaps. In a paper in 1998, Jorgensen and Ped- ersen showed that the first of these Cantor measures does not admit any orthogonal L2 Fourier series, while the second does. In the two decades that followed, a rich theory of harmonic analysis for fractal settings has ensued. If a set Ω, or a measure μ, admits an L2 spectrum, we shall talk about spectral pairs (Ω, Λ), or (μ, Γ), where the second set in the pair will be called a spectrum.If thefirstvariablearisesasafractal in the small, we will see that associated spectra will arise as fractals in the large; in some cases as lacunary Fourier expansions, series with large gaps, or lacunae, between the non-zero coefficients in expansions. We will focus on Fourier series with similar gaps between non-zero coefficients, gaps being a power of a certain scale number. There is a slight ambiguity in modern usage of the term lacunary series. When needed, clarification will be offered in the notes. ix xPREFACE In addition to harmonic analyses via Fourier duality, there are also multireso- lution wavelet approaches; work by Dutkay and Jorgensen. In the notes, both will be developed systematically, and it will be demonstrated that the wavelet tools are more flexible but perhaps not as precise for certain fractal applications. A third tool for our fractal harmonic analysis will be L2 spaces derived from appropriate Gaussian processes and their analysis. In our development of some of these duality approaches, or multiresolution analysis constructions, we shall have occasion to rely on certain non-commutative harmonic analyses. They too will be developed from scratch (as needed) in the notes. The present book is based on a series of 10 lectures delivered in June 2018 at Iowa State University.