Discrete analogues in harmonic analysis : Bourgain, Stein, and beyond

This timely book explores certain modern topics and connections at the interface of harmonic analysis, ergodic theory, number theory, and additive combinatorics. The main ideas were pioneered by Bourgain and Stein, motivated by questions involving averages over polynomial sequences, but the subject has grown significantly over the last 30 years, through the work of many researchers, and has steadily become one of the most dynamic areas of modern harmonic analysis. The author has succeeded admirably in choosing and presenting a large number of ideas in a mostly self-contained and exciting monograph that reflects his interesting personal perspective and expertise into these topics. --Alexandru Ionescu, Princeton University Discrete harmonic analysis is a rapidly developing field of mathematics that fuses together classical Fourier analysis, probability theory, ergodic theory, analytic number theory, and additive combinatorics in new and interesting ways. While one can find good treatments of each of these individual ingredients from other sources, to my knowledge this is the first text that treats the subject of discrete harmonic analysis holistically. The presentation is highly accessible and suitable for students with an introductory graduate knowledge of analysis, with many of the basic techniques explained first in simple contexts and with informal intuitions before being applied to more complicated problems; it will be a useful resource for practitioners in this field of all levels. --Terence Tao, University of California, Los Angeles List of Symbols Asymptotic Notation Miscellaneous Symbols Functions and Operators in the Euclidean Setting Functions and Operators in the Discrete Setting Fourier Multipliers in the Discrete Setting Foreword Acknowledgements Introduction 1. Discrete Analogues in Harmonic Analysis in Action 2. A Brief History of Discrete Analogues in Harmonic Analysis 3. Where We’re Going 4. Part One: Harmonic Analytic Preliminaries 5. Part Two: Discrete Analogues in Harmonic Analysis: Radon Transforms I 6. Part Three: Discrete Analogues in Harmonic Analysis: Radon Transforms II 7. Part Four: Discrete Analogues in Harmonic Analysis: Maximally Modulated Singular Integrals 8. Part Five: Discrete Analogues in Harmonic Analysis: An Introduction to Multilinear Theory 9. Part Six: Conclusion and Appendices Part 1. Harmonic Analytic Preliminaries Chapter 1. Tools 1. Exploiting Invariance 2. Interpolation of L^{p}-Spaces 3. The Hardy-Littlewood Maximal Function 4. Continuous Operators on Infinite Dimensional Vector Spaces 5. The Fourier Transform on l2(Z) 6. The Euclidean Fourier Transform Chapter 2. On Oscillation and Convergence Chapter 3. The Linear Theory 1. The Pointwise Ergodic Theorem 2. Birkhoff’s Theorem 3. Introduction to Variation 4. The Proof of Lépingle’s Inequality Part 2. Discrete Analogues in Harmonic Analysis: Radon Transforms, I Chapter 4. Bourgain’s Maximal Functions on l2(Z) 1. Number Theoretic Approximations 2. The Multi-Frequency Maximal Theory: Preliminaries 3. Controling a Maximal Function on L2 4. Proving the Multifrequency Maximal Theory 5. Oscillation and Convergence Chapter 5. Random Pointwise Ergodic Theory 1. Probabilistic Preliminaries 2. The L^{p}(X)-Theory, 1