# Commutative and Noncommutative Harmonic Analysis and Applications

Share this page *Edited by *
*Azita Mayeli; Alex Iosevich; Palle E. T. Jorgensen; Gestur Ólafsson*

This volume contains the proceedings of the
AMS Special Session on Wavelet and Frame Theoretic Methods in Harmonic
Analysis and Partial Differential Equations, held September 22–23,
2012, at the Rochester Institute of Technology, Rochester, NY.

The book features new directions, results and ideas in commutative and
noncommutative abstract harmonic analysis, operator theory and
applications. The commutative part includes shift invariant spaces,
abelian group action on Euclidean space and frame theory; the
noncommutative part includes representation theory, continuous and
discrete wavelets related to four dimensional Euclidean space, frames
on symmetric spaces, \(C^*\)-algebras, projective
multiresolutions, and free probability algebras.

The scope of the book goes beyond traditional harmonic analysis,
dealing with Fourier tools, transforms, Fourier bases, and associated
function spaces. A number of papers take the step toward wavelet
analysis, and even more general tools for analysis/synthesis problems,
including papers on frames (over-complete bases) and their practical
applications to engineering, cosmology and astrophysics.

Other applications in this book include explicit families of wavelets
and frames, as they are used in signal processing, multiplexing, and
the study of Cosmic Microwave Background (CMB) radiation.

For the purpose of organization, we have divided the book into
three parts: noncommutative, commutative, and applications. The first
group of papers are devoted to problems in noncommutative harmonic
analysis, the second to topics in commutative harmonic analysis, and
the third to such applications as wavelet and frame theory and to some
real-world applications.

# Table of Contents

## Commutative and Noncommutative Harmonic Analysis and Applications

- Preface ix10 free
- Remembering Daryl Geller (1950 - 2011) xi12 free
- Publications xiii14 free
- List of speakers xvii18 free
- List of participants xix20 free
- Part I [.3cm] Noncommutative Harmonic Analysis 122 free
- Krein-Space Operators Induced by Dirichlet Characters 324
- 1. Introduction 324
- 2. Preliminaries 627
- 3. Free Probabilistic Model of 𝒜 Induced by Primes 1132
- 4. Dirichlet Character Subsets of 𝒜 1334
- 5. Free Distributional Data of Dirichlet Characters 1536
- 6. Representations of the Arithmetic Algebra 1637
- 7. Krein-Space Operators Induced by Dirichlet Characters on \frak{𝐾}_{𝑝} 2546
- 8. Krein-Space Operator Induced by Dirichlet Characters on ℂ_{𝔸ₒ}² 2849
- References 3253

- Noncommutative solenoids and their projective modules 3556
- Paley-Wiener-Schwartz nearly Parseval frames on noncompact symmetric spaces 5576
- 1. Introduction 5576
- 2. Harmonic analysis on Riemannian symmetric spaces of the noncompact type 5677
- 3. Paley-Wiener spaces 𝑃𝑊_{𝜔}(𝑋) 6182
- 4. Average sampling and almost Parseval frames in Paley-Wiener spaces on Riemannian manifolds 6384
- 5. Nearly Parseval Paley-Wiener frames on 𝑋=𝐺/𝐾 6990
- References 7091

- Projective multiresolution analyses over irrational rotation algebras 7394
- Part II [.3cm]Commutative Harmonic Analysis 87108
- Regularity of abelian linear actions 89110
- 𝑛-widths and approximation theory on compact Riemannian manifolds 111132
- A four dimensional continuous wavelet transform 123144
- Part III Applications 137158
- Dynamical Sampling in Shift-Invariant Spaces 139160
- High-Frequency Tail Index Estimation by Nearly Tight Frames 149170
- 1. Introduction 149170
- 2. Random fields and mexican needlets 152173
- 3. Mexican Needlet Whittle-like approximation to likelihood function 156177
- 4. Asymptotic Properties 158179
- 5. Narrow band estimates 171192
- 6. The Plug-in procedure 174195
- Appendix A. Auxiliary results: preliminaries 175196
- Appendix B. Auxiliary Results: Covariance terms 179200
- Appendix C. Auxiliary Results: Cumulants 183204
- References 184205

- Multiplexing and demultiplexing Frame Pairs 189210