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Product Identifiers
PublisherCambridge University Press
ISBN-101108499708
ISBN-139781108499705
eBay Product ID (ePID)23038626950
Product Key Features
Number of Pages345 Pages
Publication NameFourier Restriction, Decoupling and Applications
LanguageEnglish
SubjectGeneral, Mathematical Analysis
Publication Year2020
TypeTextbook
AuthorCiprian Demeter
Subject AreaMathematics
SeriesCambridge Studies in Advanced Mathematics Ser.
FormatHardcover
Dimensions
Item Height0.9 in
Item Weight21.2 Oz
Item Length9.2 in
Item Width6.2 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN2019-040373
Dewey Edition23
ReviewsI think that the book Fourier restriction, decoupling and applications, by Ciprian Demeter, will be a really valuable addition to the literature, and I endorse it strongly. Restriction theory is an exciting area of Fourier analysis. It is organized around some deep central questions, raised by Stein more than 50 years ago, about the connection between geometry and the Fourier transform. These central questions are still open and look very difficult, but there has been fundamental progress in our understanding over that time, and especially in the last decade. The questions are interesting in their own right, but they also have a number of applications. Early work on restriction theory led to the Strichartz estimates, which are a ubiquitous tool in dispersive PDE, and later developments have also had impact in PDE. - for instance giving the sharp Strichartz estimates for periodic solutions of the Schrodinger equation. Recently, ideas coming from restriction theory have had a significant impact in number theory: for instance, they led to a proof of Vinogradov's mean value conjecture from the 1930s, which in turn gives improved estimates for Waring's problem. Ideas from restriction theory also play a role in the best current estimates for the Lindelhof hypothesis and the Gauss circle problem. The author is one of the main contributors to the recent developments that we mentioned above. This book discusses the major recent developments in the field. He has worked hard to explain them clearly and reduce the technical details. For instance, he presents relatively simple cases of each development first, and then describes how to build to more general cases. He pauses to discuss examples and heuristics, or to look back at the end of a proof and reflect on the strategy. He is also careful and rigorous. And he provides plenty of exercises. I work in this area myself and I have a number of students learning the field. In the past, I have used lecture notes fr, The topic of decoupling is now a major active area of research in both harmonic analysis and analytic number theory. There are a number of survey articles and lecture notes already on these topics, but this book - by one of the leading contributors to the field - is more comprehensive than any of these, being almost completely self-contained and detailing a number of older results as well as the more recent ones. It also has a number of exercises and insightful commentary. This text is an excellent resource, both for students and for existing researchers in the field. Terence Tao, UCLA
Series Volume NumberSeries Number 184
IllustratedYes
Dewey Decimal515.2433
Table Of ContentBackground and notation; 1. Linear restriction theory; 2. Wave packets; 3. Bilinear restriction theory; 4. Parabolic rescaling and a bilinear-to-linear reduction; 5. Kakeya and square function estimates; 6. Multilinear Kakeya and restriction inequalities; 7. The Bourgain-Guth method; 8. The polynomial method; 9. An introduction to decoupling; 10. Decoupling for the elliptic paraboloid; 11. Decoupling for the moment curve; 12. Decouplings for other manifolds; 13. Applications of decoupling; References; Index.
SynopsisRecent years have seen a flurry of advances in Fourier restriction theory, with significant applications in number theory, PDEs and geometric analysis. This timely text for specialists and graduate students in analysis, written by a leader in the field, brings the reader to the forefront of these exciting developments., The last fifteen years have seen a flurry of exciting developments in Fourier restriction theory, leading to significant new applications in diverse fields. This timely text brings the reader from the classical results to state-of-the-art advances in multilinear restriction theory, the Bourgain-Guth induction on scales and the polynomial method. Also discussed in the second part are decoupling for curved manifolds and a wide variety of applications in geometric analysis, PDEs (Strichartz estimates on tori, local smoothing for the wave equation) and number theory (exponential sum estimates and the proof of the Main Conjecture for Vinogradov's Mean Value Theorem). More than 100 exercises in the text help reinforce these important but often difficult ideas, making it suitable for graduate students as well as specialists. Written by an author at the forefront of the modern theory, this book will be of interest to everybody working in harmonic analysis.
LC Classification NumberQA403.5.D46 2019
