**Contemporary Mathematics**

Volume: 444;
2007;
228 pp;
Softcover

MSC: Primary 37; 42; 47; 60; 65;

Print ISBN: 978-0-8218-4235-5

Product Code: CONM/444

List Price: $75.00

Individual Member Price: $60.00

**Electronic ISBN: 978-0-8218-8123-1
Product Code: CONM/444.E**

List Price: $75.00

Individual Member Price: $60.00

# Topics in Harmonic Analysis and Ergodic Theory

Share this page *Edited by *
*Joseph M. Rosenblatt; Alexander M. Stokolos; Ahmed I. Zayed*

There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to problems in number theory.

Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2–4, 2005). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness theorems for multi-dimensional trigonometric series, and Bellman and \(s\)-functions.

In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on multi-dimensional trigonometric series.

#### Table of Contents

# Table of Contents

## Topics in Harmonic Analysis and Ergodic Theory

- Contents v6 free
- Preface vii8 free
- List of Participants ix10 free
- Topics in Ergodic Theory and Harmonic Analysis: An Overview 114 free
- The mathematical work of Roger Jones 922
- The Central Limit Theorem for Random Walks on Orbits of Probability Preserving Transformations 3144
- Probability, Ergodic Theory, and Low-Pass Filters 5366
- (1) Introduction. An overview. Basic notation 5467
- (2) Two simple examples: the Haar function and the stretched Haar function. Correcting defective filters 5770
- (3) An outline of the probability argument: Low-pass filters as transition probabilities and a zero-one principle 5972
- (4) The Paul Lévy Borel-Cantelli Lemma and the convergence/divergence of an infinite product 6477
- (5) Doeblin's coupling for low-pass filters 6578
- (6) The state space and the path space. Basic probability theory for this application 6578
- (7) Coding R1 into the state space: The signed magnitude representation versus the two's complement representation 6679
- (8) The construction of a stationary Markov process. P-invariant measures, martingales, and harmonic functions 6780
- (9) The crux of the problem: Invariant sets. Cycles and perfect sets. Forbidden zeros 7184
- (10) The asymptotic behavior of paths from an initial point. Recurrent and transient points. Attractors and inaccessible sets. Examples 7588
- (11) The probabilistic description of low-pass filters (Theorem 11.1) 7790
- (12) The polynomial case: Daubechies' filters and the Pascal-Fermat correspondence. Cohen's necessary and sufficient conditions. A zero-one principle (Theorem 12.1) 7992
- (13) Analytic conditions for low-pass filters. A class of examples from subshifts of finite type (Theorem 13.1) 8295
- (14) Concluding remarks 8699
- (15) References 8699

- Ergodic Theory on Borel Foliations by Rn and Zn 89102
- Short review of the work of Professor J. Marshall Ash 115128
- Uniqueness questions for multiple trigonometric series 129142
- 1. Introduction 130143
- 2. Some Cantor-Lebesgue Type Theorems 134147
- 3. A Uniqueness Theorem for Unrestrictedly Rectangular Convergence 138151
- 4. A Uniqueness Theorem for Spherical Convergence 142155
- 5. Sets of Uniqueness under Spherical Summation 148161
- 6. Questions about Square and Restricted Rectangular Uniqueness 152165
- 7. Orthogonal Trigonometric Polynomials 159172
- References 163176

- Smooth interpolation of functions on Rn 167180
- Problems in interpolation theory related to the almost everywhere convergence of Fourier series 175188
- Lectures on Nehari's Theorem on the Polydisk 185198
- The s-function and the exponential integral 215228

#### Readership

Research mathematicians interested in harmonic analysis, ergodic theory, and their interaction.