Discrete Analogues in Harmonic Analysis: Bourgain, Stein, and Beyond (Graduate Studies in Mathematics)

This timely book explores certain modern topics and connections at the interface of harmonic ysis, 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 ysis. 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 ysis is a rapidly developing field of mathematics that fuses together clical Fourier ysis, probability theory, ergodic theory, ytic 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 ysis holistically. The presentation is highly accessible and suitable for students with an introductory graduate knowledge of ysis, 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