Methods of the theory of generalized functions

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Methods of the theory of generalized functions / V.S. Vladimirov

資料種別:: 図書
出版情報:: London : Taylor & Francis, 2002
形態:: xiv, 311 p. ; 26 cm
シリーズ名:: Analytical methods and special functions ; v. 6 <BA41065805>
著者名:: Vladimirov, V. S. (Vasiliĭ Sergeevich) <DA00742385>
ISBN:: 9780415273565 [0415273560]
書誌ID:: BA58694532

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注記:

Includes bibliographical references (p. 303-308) and index

言語:

英語

目次情報:

Preface

Symbols and Definitions

Generalized Functions and their Properties / Chapter 1.：

Test and Generalized Functions / 1.：

Introduction / 1.1.：

The space of test functions D(O) / 1.2.：

The space of generalized functions D'(O) / 1.3.：

The completeness of the space of generalized functions D'(O) / 1.4.：

The support of a generalized function / 1.5.：

Regular generalized functions / 1.6.：

Measures / 1.7.：

Sochozki formulae / 1.8.：

Change of variables in generalized functions / 1.9.：

Multiplication of generalized functions / 1.10.：

Differentiation of Generalized Functions / 2.：

Derivatives of generalized functions / 2.1.：

The antiderivative (primitive) of a generalized function / 2.2.：

Examples / 2.3.：

The local structure of generalized functions / 2.4.：

Generalized functions with compact support / 2.5.：

Generalized functions with point support / 2.6.：

Generalized functions P([pi subscript v vertical bar]x[vertical bar superscript alpha-1]) / 2.7.：

Direct Product of Generalized Functions / 3.：

The definition of a direct product / 3.1.：

The properties of a direct product / 3.2.：

Some applications / 3.3.：

Generalized functions that are smooth with respect to some of the variables / 3.4.：

The Convolution of Generalized Functions / 4.：

The definition of convolution / 4.1.：

The properties of a convolution / 4.2.：

The existence of a convolution / 4.3.：

Cones in R[superscript n] / 4.4.：

Convolution algebras D'([Gamma]+) and D'([Gamma]) / 4.5.：

Mean functions of generalized functions / 4.6.：

Convolution as a continuous linear translation-invariant operator / 4.7.：

Tempered Generalized Functions / 4.9.：

The space S of test (rapidly decreasing) functions / 5.1.：

The space S' of tempered generalized functions / 5.2.：

Examples of tempered generalized functions and elementary operations in S' / 5.3.：

The structure of tempered generalized functions / 5.4.：

The direct product of tempered generalized functions / 5.5.：

The convolution of tempered generalized functions / 5.6.：

Homogeneous generalized functions / 5.7.：

Integral Transformations of Generalized Functions / Chapter 2.：

The Fourier Transform of Tempered Generalized Functions / 6.：

The Fourier transform of test functions in S / 6.1.：

The Fourier transform of tempered generalized functions / 6.2.：

Properties of the Fourier transform / 6.3.：

The Fourier transform of generalized functions with compact support / 6.4.：

The Fourier transform of a convolution / 6.5.：

The Mellin transform / 6.6.：

Fourier Series of Periodic Generalized Functions / 7.：

The definition and elementary properties of periodic generalized functions / 7.1.：

Fourier series of periodic generalized functions / 7.2.：

The convolution algebra D'[subscript T] / 7.3.：

Positive Definite Generalized Functions / 7.4.：

The definition and elementary properties of positive definite generalized functions / 8.1.：

The Bochner-Schwartz theorem / 8.2.：

The Laplace Transform of Tempered Generalized Functions / 8.3.：

Definition of the Laplace transform / 9.1.：

Properties of the Laplace transform / 9.2.：

The Cauchy Kernel and the Transforms of Cauchy-Bochner and Hilbert / 9.3.：

The space H[subscript s] / 10.1.：

The Cauchy kernel K[subscript C](z) / 10.2.：

The Cauchy-Bochner transform / 10.3.：

The Hilbert transform / 10.4.：

Holomorphic functions of the class H[superscript (s) subscript a] (C) / 10.5.：

The generalized Cauchy-Bochner representation / 10.6.：

Poisson Kernel and Poisson Transform / 11.：

The definition and properties of the Poisson kernel / 11.1.：

The Poisson transform and Poisson representation / 11.2.：

Boundary values of the Poisson integral / 11.3.：

Algebras of Holomorphic Functions / 12.：

The definition of the H[subscript +] (C) and H (C) algebras / 12.1.：

Isomorphism of the algebras S'(C* +) - H[subscript +] (C) and S'(C*) - H (C) / 12.2.：

The Paley-Wiener-Schwartz theorem and its generalizations / 12.3.：

The space H[subscript a](C) is the projective limit of the spaces H[subscript a'](C') / 12.4.：

The Schwartz representation / 12.5.：

A generalization of the Phragmen-Lindelof theorem / 12.6.：

Equations in Convolution Algebras / 13.：

Divisors of unity in the H[subscript +] (C) and H(C) algebras / 13.1.：

On division by a polynomial in the H(C) algebra / 13.2.：

Estimates for holomorphic functions with nonnegative imaginary part in T[subscript C] / 13.3.：

Divisors of unity in the algebra W(C) / 13.4.：

Example / 13.5.：

Tauberian Theorems for Generalized Functions / 14.：

Preliminary results / 14.1.：

General Tauberian theorem / 14.2.：

One-dimensional Tauberian theorems / 14.3.：

Tauberian and Abelian theorems for nonnegative measures / 14.4.：

Tauberian theorems for holomorphic functions of bounded argument / 14.5.：

Some Applications in Mathematical Physics / Chapter 3.：

Differential Operators with Constant Coefficients / 15.：

Fundamental solutions in D' / 15.1.：

Tempered fundamental solutions / 15.2.：

A descent method / 15.3.：

A comparison of differential operators / 15.4.：

Elliptic and hypoelliptic operators / 15.6.：

Hyperbolic operators / 15.7.：

The sweeping principle / 15.8.：

The Cauchy Problem / 16.：

The generalized Cauchy problem for a hyperbolic equation / 16.1.：

Wave potential / 16.2.：

Surface wave potentials / 16.3.：

The Cauchy problem for the wave equation / 16.4.：

A statement of the generalized Cauchy problem for the heat equation / 16.5.：

Heat potential / 16.6.：

Solution of the Cauchy problem for the heat equation / 16.7.：

Holomorphic Functions with Nonnegative Imaginary Part in T[subscript C] / 17.：

Preliminary remarks / 17.1.：

Properties of functions of the class P[subscript +](T[superscript C]) / 17.2.：

Estimates of the growth of functions of the class H[subscript +](T[superscript C]) / 17.3.：

Smoothness of the spectral function / 17.4.：

Indicator of growth of functions of the class P[subscript +](T[superscript C]) / 17.5.：

An integral representation of functions of the class H[subscript +](T[superscript C]) / 17.6.：

Holomorphic Functions with Nonnegative Imaginary Part in T[superscript n] / 18.：

Lemmas / 18.1.：

Functions of the classes H[subscript +](T[superscript 1]) and P[subscript +](T[superscript 1]) / 18.2.：

Functions of the class P[subscript +](T[superscript n]) / 18.3.：

Functions of the class H[subscript +](T[superscript n]) / 18.4.：

Positive Real Matrix Functions in T[superscript C] / 19.：

Positive real functions in T[superscript C] / 19.1.：

Positive real matrix functions in T[superscript C] / 19.2.：

Linear Passive Systems / 20.：

Corollaries to the condition of passivity / 20.1.：

The necessary and sufficient conditions for passivity / 20.3.：

Multidimensional dispersion relations / 20.4.：

The fundamental solution and the Cauchy problem / 20.5.：

What differential and difference operators are passive operators? / 20.6.：

Quasiasymptotics of the solutions of equations in convolutions / 20.7.：

Abstract Scattering Operator / 21.：

The definition and properties of an abstract scattering matrix / 21.1.：

A description of abstract scattering matrices / 21.2.：

The relationship between passive operators and scattering operators / 21.3.：

Bibliography

Index

Preface

Symbols and Definitions

Generalized Functions and their Properties / Chapter 1.：

Test and Generalized Functions / 1.：

Introduction / 1.1.：

The space of test functions D(O) / 1.2.：

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