Fourier Integrals in Classical Analysis is an advanced treatment of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author in particular studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions.
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Editorial Reviews
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"The writing is agile and somewhat colloquial, giving a refreshing informal tone to the presentation of quite arduous topics. The goals of the collection in which this book has been published state that '[works] have to be rigorous, definite, and of lasting value to mathematicians working in the relevant disciplines'. I believe that the book under consideration generously fulfills these goals." Josefina Alvarez, Mathematical Reviews
"...provides an excellent introduction to oscillatory integral operators and a detailed treatment of some of the most recent developments....rewards the reader with a thorough account of some of the last decade's most important developments in Fourier analysis, many of them due to its author. It belongs on the bookshelf of anyone seriously interested in the subject." Allan Greenleaf, Bulletin of the American Mathematical Society
"...provides an excellent introduction to oscillatory integral operators and a detailed treatment of some of the most recent developments....rewards the reader with a thorough account of some of the last decade's most important developments in Fourier analysis, many of them due to its author. It belongs on the bookshelf of anyone seriously interested in the subject." Allan Greenleaf, Bulletin of the American Mathematical Society
Book Description
Fourier Integrals in Classical Analysis is an advanced monograph concerned with modern treatments of central problems in harmonic analysis. This self-contained book starts with a rapid review of important topics in Fourier analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equations and their counterparts in classical analysis. In particular, basic problems in classical analysis, such as estimates for maximal functions and eigenfunctions, are attacked using modern microlocal techniques.
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Inside This Book
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First Sentence:
The purpose of this chapter and the next is to present the background material that will be needed. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
circular maximal theorem, local smoothing estimates, oscillatory integral operators, phase function theorem, small conic neighborhood, spectral projection operators, smoothing theorem, canonical graph, orthogonality arguments, maximal theorems, canonical relation, curvature hypothesis, wave front set, small enough support, oscillatory integrals, maximal estimates, smoothing error, fixed compact set, restriction theorem, symplectic basis, cone condition, homogeneity relations, conormal bundle, maximal operator, multiplier theorem New!
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The purpose of this chapter and the next is to present the background material that will be needed. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
circular maximal theorem, local smoothing estimates, oscillatory integral operators, phase function theorem, small conic neighborhood, spectral projection operators, smoothing theorem, canonical graph, orthogonality arguments, maximal theorems, canonical relation, curvature hypothesis, wave front set, small enough support, oscillatory integrals, maximal estimates, smoothing error, fixed compact set, restriction theorem, symplectic basis, cone condition, homogeneity relations, conormal bundle, maximal operator, multiplier theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Back Cover | Surprise Me!
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This book cites 5
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6
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- Harmonic Analysis by Elias M. Stein in Back Matter
- Partial Differential Equations (Applied Mathematical Sciences) (v. 1) by Fritz John in Back Matter
- Trigonometric Series (Cambridge Mathematical Library) by A. Zygmund in Back Matter
- Beijing Lectures in Harmonic Analysis. (AM-112) (Annals of Mathematics Studies) by Elias M. Stein in Back Matter
- Lectures on Differential Geometry by Sholomo Sternberg in Back Matter
- Topics in Analysis and Its Applications: Selected Theses by Ronald R. Coifman on page 59, and page 96
- Geometric Methods in Inverse Problems and PDE Control (The IMA Volumes in Mathematics and its Applications) by Chrisopher B. Croke on page 262
- Nonlinear Wave Equations (Chapman & Hall/CRC Pure and Applied Mathematics) by Satyanad Kichenassamy on page 44
- Beyond Wavelets, Volume 10 (Studies in Computational Mathematics) by Grant Welland on page 60
- Advances in Imaging and Electron Physics, Volume 111 by Peter W. Hawkes on page 326
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