Fourier Integrals in Classical Analysis - Christopher Sogge - 1993
"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
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.
Introduction to Fourer Analysis on Euclidean Spaces - Elias Weiss - Guido Stein - 1971
The authors present a unified treatment of basic topics that arise in Fourier analysis.
Their intention is to illustrate the role played by the structure of Euclidean spaces,
particularly the action of translations, dilatations, and rotations, and to motivate the
study of harmonic analysis on more general spaces having an analogous structure, e.g.,
symmetric spaces.
Singular Integrals And Differentiability Properties Of Functions - Elias Stein - 1971
Singular integrals are among the most interesting and important objects of study in
analysis, one of the three main branches of mathematics. They deal with real and complex
numbers and their functions. In this book, Princeton professor Elias Stein, a leading
mathematical innovator as well as a gifted expositor, produced what has been called the
most influential mathematics text in the last thirty-five years. One reason for its
success as a text is its almost legendary presentation: Stein takes arcane material,
previously understood only by specialists, and makes it accessible even to beginning
graduate students. Readers have reflected that when you read this book, not only do you
see that the greats of the past have done exciting work, but you also feel inspired that
you can master the subject and contribute to it yourself.
Singular integrals were known to only a few specialists when Stein's book was first
published. Over time, however, the book has inspired a whole generation of researchers
to apply its methods to a broad range of problems in many disciplines, including
engineering, biology, and finance.
Stein has received numerous awards for his research, including the Wolf Prize of Israel,
the Steele Prize, and the National Medal of Science. He has published eight books with
Princeton, including Real Analysis in 2005.
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