Sinossi
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.
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Contenuti
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
Schrödinger'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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A Guide to Distribution Theory and Fourier Transforms
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