**Contemporary Mathematics**

Volume: 464;
2008;
264 pp;
Softcover

MSC: Primary 42; 44;
Secondary 43; 51; 52; 65

Print ISBN: 978-0-8218-4327-7

Product Code: CONM/464

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-0-8218-8143-9
Product Code: CONM/464.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

# Radon Transforms, Geometry, and Wavelets

Share this page *Edited by *
*Gestur Ólafsson; Eric L. Grinberg; David Larson; Palle E. T. Jorgensen; Peter R. Massopust; Eric Todd Quinto; Boris Rubin*

This volume is based on two special sessions held at the AMS Annual Meeting
in New Orleans in January 2007, and a satellite workshop held in Baton Rouge
on January 4-5, 2007. It consists of invited expositions that together
represent a broad spectrum of fields, stressing surprising interactions and
connections between areas that are normally thought of as disparate. The
main topics are geometry and integral transforms. On the one side are
harmonic analysis, symmetric spaces, representation theory (the groups
include continuous and discrete, finite and infinite, compact and
non-compact), operator theory, PDE, and mathematical probability. Moving in
the applied direction we encounter wavelets, fractals, and engineering
topics such as frames and signal and image processing.

The subjects covered in this book form a unified whole, and they stand at
the crossroads of pure and applied mathematics. The articles cover a broad
range in harmonic analysis, with the main themes related to integral
geometry, the Radon transform, wavelets and frame theory. These themes can
loosely be grouped together as follows:

- Frame Theory and Applications
- Harmonic Analysis and Function Spaces
- Harmonic Analysis and Number Theory
- Integral Geometry and Radon Transforms
- Multiresolution Analysis, Wavelets, and Applications

#### Readership

Graduate students and research mathematicians interested in harmonic analysis, integral geometry and applications.

# Table of Contents

## Radon Transforms, Geometry, and Wavelets

- Contents v6 free
- Preface vii8 free
- Combined List of Speakers and Titles of Their Talks xi12 free
- Composite Wavelet Transforms: Applications and Perspectives 116 free
- 1. Introduction. 116
- 2. Composite wavelet transforms for dilated kernels. 217
- 3. Wavelet transforms associated to one-parametric semigroups and inversion of potentials. 520
- 4. Wavelet transforms with the generalized translation operator. 924
- 5. Beta-semigroups. 1126
- 6. Parabolic wavelet transforms. 1429
- 7. Some applications to inversion of the k-plane Radon transform. 1631
- 8. Higher-rank composite wavelet transforms and open problems. 1732
- References 2338

- Complex Sigma-Delta Quantization Algorithms for Finite Frames 2742
- Decomposition and Admissibility for the Quasiregular Representation for Generalized Oscillator Groups 5166
- Fourier Series on Fractals: A Parallel with Wavelet Theory 7590
- 1. Introduction 7691
- 2. Multiresolution wavelets and Iterated Function Systems 7994
- 2.1. The attractor 8095
- 2.2. Multiresolutions 8196
- 2.3. The scaling equation, the invariance equation 8297
- 2.4. Orthogonality of the translates of the scaling function; orthogonality of the exponentials 8499
- 2.5. The quadrature mirror filter condition and Hadamard triples 85100
- 2.6. Lawton's theorem and transfer operators 87102
- 2.7. Cohen's orthogonality condition and the Laba-Wang theorem 88103
- 2.8. Super-wavelets and the completion of Λ0 89104

- 3. Open problems 91106
- 4. Appendix, with some of Jorgensen's recollections of conversations with Irving Segal, Marshall Stone, Bent Fuglede, and David Shale. 93108
- References 99114

- A Computational Complexity Paradigm for Tomography 103118
- Invariant Differential Operators on Matrix Motion Groups and Applications to the Matrix Radon Transform 107122
- Sums and Products in Finite Fields: an Integral Geometric Viewpoint 129144
- Frame Potential and Finite Abelian Groups 137152
- Robustness of Fusion Frames under Erasures of Subspaces and of Local Frame Vectors 149164
- Smooth, Well-localized Parseval Wavelets Based on Wavelet Sets in R2 161176
- Explicit Schemes in Seismic Migration and Isotropic Multiscale Representations 177192
- Invariant Functions on Grassmanians 201216
- Harmonic Analysis Related to Schrödinger Operators 213228
- A Discrete Helgason-Fourier Transform for Sobolev and Besov Functions on Noncompact Symmetric Spaces 231246
- Helgason's Support Theorem and Spherical Radon Transforms 249264