
Discrete Analogues in Harmonic Analysis
Bourgain, Stein, and Beyond
Ben Krause(Author)
American Mathematical Society (Publisher)
Will be published approx. on 30. January 2023
Book
Paperback/Softback
542 pages
978-1-4704-7174-3 (ISBN)
Description
Reviews / Votes
"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
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Weight
363 gr
ISBN-13
978-1-4704-7174-3 (9781470471743)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Person
Ben Krause, King's College, London, UK.
Content
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
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