Generalized Functions in Mathematical Physics : Main Ideas and Concepts

Main Ideas and Concepts

By: A S Demidov

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Hardcover | 1 April 2001

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This important book gives an interconnected presentation of some basic ideas, concepts, results of the theory of generalised functions (first of all, in the framework of the theory of distributions) and equations of mathematical physics. A part of the material is given according to the scheme: definition -- theorem -- proof. This scheme is convenient for presenting results in clear and concentrated form. However, it seems reasonable to give a student the possibility not only to study a priori given definitions and proofs of theorems, but also to discover them while considering the problems involved. A series of sections serve this purpose. Moreover, a part of the material is given as exercises and problems.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Science Physics Mathematical Physics

Preface	p. ix
Notation	p. xiii
Introduction to problems of mathematical physics	p. 1
Temperature at a point? No! In volumes contracting to the point	p. 1
The notion of [delta]-sequence and [delta]-function	p. 3
Some spaces of smooth functions. Partition of unity	p. 6
Examples of [delta]-sequences	p. 9
On the Laplace equation	p. 10
On the heat equation	p. 18
The Ostrogradsky-Gauss formula. The Green formulae and the Green function	p. 25
The Lebesgue integral	p. 28
The spaces L[superscript p] and L[superscript p subscript loc]	p. 35
Functions of L[superscript 1 subscript loc] as linear functional on C[superscript [infinity] subscript 0]	p. 40
Simplest hyperbolic equations. Generalized Sobolev solutions	p. 41
The spaces D[superscript b], D[superscript #] and D'. Elements of the distribution theory (generalized function in the sense of L. Schwartz)	p. 55
The space D[superscript b] of the Sobolev derivatives	p. 55
The space D[superscript #] of generalized functions	p. 59
The problem of regularization	p. 61
Generalized functions with a point support. The Borel theorem	p. 63
The space D' of generalized functions (distributions by L. Schwartz)	p. 66
The spaces H[superscript 3]. Pseudodifferential operators	p. 75
The Fourier series and the Fourier transform. The spaces S and S'	p. 75
The Fourier-Laplace transform. The Paley-Wiener theorem	p. 86
Fundamental solutions. Convolution	p. 90
On spaces H[superscript s]	p. 93
On pseudodifferential operators (PDO)	p. 97
On elliptic problems	p. 102
Addendum: A new approach to the theory of generalized functions	p. 117
Bibliography	p. 125
Index	p. 129
Table of Contents provided by Syndetics. All Rights Reserved.

Generalized Functions in Mathematical Physics : Main Ideas and Concepts

Main Ideas and Concepts

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