Discrete Analogues in Harmonic Analysis

Ben Krause, King's College, London, UK.

Harmonic analytic preliminaries: Tools On oscillation and convergence The linear theory Discrete analogues in harmonic analyis: Radon transforms, I: Bourgain's maximal functions on $\ell^2(\mathbb{Z})$ Random pointwise ergodic theory An application to discrete Ramsey theory Bourgain's $\ell(\mathbb{Z})$=argument, revisited Discrete analogues in harmonic analysis: Radon transforms, II: Ionescu-Wainger theory Establishing Ionescu-Wainger theory The spherical maximal function The lacunary spherical maximal function Disctrete improving inequalities Discrete analogues in harmonic analysis: Maximally modulated singular integrals: Monomial ``Carleson'' operators Maximally modulated singular integrals: A theorem of Stein and Wainger Discrete analogues in harmonic analysis: An introduction to multilinear theory: Bilinear considerations Arithmetic Sobolev estimates, examples Conclusion and appendices: Further directions Remembering my collaboration with Stein and Bourgain-M. Mirek Introduction to additive combinatorics Oscillatory integrals and exponential sums Bibliography Index

"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