A guide to distribution theory and Fourier transforms

著者

書誌事項

A guide to distribution theory and Fourier transforms

Robert S. Strichartz

（Studies in advanced mathematics）

CRC Press, c1994

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注記

Includes bibliographical references (p. 207) and index

内容説明・目次

内容説明

This book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudo-differential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book. in applied analysis and mathematical physics.

目次

What are Distributions? Generalized functions and test functions Examples of distributions What good are distributions? Problems The Calculus of Distributions Functions as distributions Operations on distributions Adjoint identities Consistency of derivatives Distributional solutions of differential equations Problems Fourier Transforms From Fourier series to Fourier integrals The Schwartz class S Properties of the Fourier transform on S The Fourier inversion formula on S The Fourier transform of a Gaussian Problems Fourier Transforms of Tempered Distributions The definitions Examples Convolutions with tempered distributions Problems Solving Partial Differential Equations The Laplace equation The heat equation The wave equation Schroedinger's equation and quantum mechanics Problems The Structure of Distributions The support of a distribution Structure theorems Distributions with point support Positive distributions Continuity of distribution Approximation by test functions Local theory of distributions Problems Fourier Analysis The Riemann-Lebesgue lemma Paley-Wiener theorems The Poisson summation formula Probability measures and positive definite functions The Heisenberg uncertainty principle Hermite functions Radial Fourier transforms and Bessel functions Haar functions and wavelets Problems Sobolev Theory and Microlocal Analysis Sobolev inequalities Sobolev spaces Elliptic partial differential equations (constant coefficients) Pseudodifferential operators Hyperbolic operators The wave front set Microlocal analysis of singularities Problems Suggestions for Further Reading Index

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