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Softcover ISBN:  9781470471743 
Product Code:  GSM/224.S 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Sale Price:  $55.25 
eBook ISBN:  9781470471750 
EPUB ISBN:  9781470476601 
Product Code:  GSM/224.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Sale Price:  $55.25 
Softcover ISBN:  9781470471743 
eBook ISBN:  9781470471750 
Product Code:  GSM/224.S.B 
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Book DetailsGraduate Studies in MathematicsVolume: 224; 2022; 563 ppMSC: Primary 11; 37; 42
Reviews and Endorsements
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 selfcontained 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
ReadershipGraduate students and researchers interested in discrete harmonic analysis and pointwise ergodic theory.

Table of Contents

Chapters

Introduction

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

IonescuWainger theory

Establishing IonescuWainger 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


Additional Material

Reviews

The style is much like what we see in Tao's books, the author's Ph.D. advisor: extremely rich in heuristics, and interspersed with exercises complementing the text.
Faruk Temur (Izmir Institute of Technology) MathSciNet Mathematical Reviews Clippings


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Reviews and Endorsements
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 selfcontained 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
Graduate students and researchers interested in discrete harmonic analysis and pointwise ergodic theory.

Chapters

Introduction

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

IonescuWainger theory

Establishing IonescuWainger 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

The style is much like what we see in Tao's books, the author's Ph.D. advisor: extremely rich in heuristics, and interspersed with exercises complementing the text.
Faruk Temur (Izmir Institute of Technology) MathSciNet Mathematical Reviews Clippings