Properties and Operations

Properties and Operations

1st Edition - January 1, 1964

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  • Authors: I. M. Gel'fand, G. E. Shilov
  • eBook ISBN: 9781483261591

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Generalized Functions, Volume 1: Properties and Operations provides a systematic development of the theory of generalized functions and problems in analysis connected with it. This book focuses on the concept of a convenient link that connects many aspects of analysis, functional analysis, theory of differential equations, representation theory of locally compact Lie groups, and theory of probability and statistics. This volume is essentially devoted to algorithmic questions of the theory, covering many applications of generalized functions to various problems of analysis. The topics discussed include the local properties of generalized functions, differentiation as a continuous operation, and Fourier Transforms of test functions. The wave equation in space of odd dimension, derivation of Green's theorem, and reducible singular points are also described. This publication is a good reference for mathematicians, researchers, and students concerned with generalized functions.

Table of Contents

  • Translator's Note

    Foreword to the First Russian Edition

    Foreword to the Second Russian Edition

    Chapter I Definition and Simplest Properties of Generalized Functions

    1. Test Functions and Generalized Functions

    1.1. Introductory Remarks

    1.2. Test Functions

    1.3. Generalized Functions

    1.4. Local Properties of Generalized Functions

    1.5. Addition and Multiplication by a Number and by a Function

    1.6. Translations, Rotations, and Other Linear Transformations on the Independent Variables

    1.7. Regularization of Divergent Integrals

    1.8. Convergence of Generalized Function Sequences

    1.9. Complex Test Functions and Generalized Functions

    1.10. Other Test-Function Spaces

    2. Differentiation and Integration of Generalized Functions

    2.1. Fundamental Definitions

    2.2. Examples for the Case of a Single Variable

    2.3. Examples for the Case of Several Variables

    2.4. Differentiation as a Continuous Operation

    2.5. Delta-Convergent Sequences

    2.6. Differential Equations for Generalized Function

    2.7. Differentiation in S

    3. Regularization of Functions with Algebraic Singularities

    3.1. Statement of the Problem

    3.2. The Generalized Functions x+λ and x-λ

    3.3. Even and Odd Combinations of x+λ and x-λ

    3.4. Indefinite Integrals of x+λ , x-λ | x |λ sgn x

    3.5. Normalization of x+λ , x-λ | x |λ sgn x

    3.6. The Generalized Functions (x + iO)λ and (x — iO)λ

    3.7. Canonical Regularization

    3.8. Regularization of Other Integrals

    3.9. The Generalized Function rλ

    3.10. Plane-Wave Expansion of rλ

    3.10. Homogeneous Functions

    4. Associate Functions

    4.1. Definition

    4.2. Taylor's and Laurent Series for x+λ and x-λ

    4.3. Expansion of | x |λ and | x |λ sgn x

    4.4. The Generalized Functions (x + iO)λ and (x — iO)λ

    4.5. Taylor's Series for (x + iO)λ and (x — iO)λ

    4.6. Expansion of rλ

    5. Convolutions of Generalized Functions

    5.1. Direct Product of Generalized Functions

    5.2. Convolutions of Generalized Functions

    5.3. Newtonian Gravitational Potential and Elementary Solutions of Differential Equations

    5.4. Poisson's Integral and Elementary Solutions of Cauchy's Problem

    5.5. Integrals and Derivatives of Higher Orders

    6. Elementary Solutions of Differential Equations with Constant Coefficients

    6.1. Elementary Solutions of Elliptic Equations

    6.2. Elementary Solutions of Regular Homogeneous Equations

    6.3. Elementary Solutions of Cauchy's Problem

    Appendix 1. Local Properties of Generalized Functions

    A1.1. Test Functions as Averages of Continuous Functions

    A1.2. Partition of Unity

    A1.3. Local Properties of Generalized Functions

    A1.4. Differentiation as a Local Operation

    Appendix 2. Generalized Functions Depending on a Parameter

    A2.1. Continuous Functions

    A2.2. Differentiable Functions

    A2.3. Analytic Functions

    Chapter II Fourier Transforms of Generalized Functions

    1. Fourier Transforms of Test Functions

    1.1. Fourier Transforms of Functions in Κ

    1.2. The Space Ζ

    1.3. The Case of Several Variables

    1.4. Functionals on Ζ

    1.5. Analytic Functionals

    1.6. Fourier Transforms of Functions in S

    2. Fourier Transforms of Generalized Functions. A Single Variable

    2.1. Definition

    2.2. Examples

    2.3. Fourier Transforms x+λ , x-λ | x |λ, and | x |λ sgn x

    2.4. Fourier Transforms of In x^ and Similar Generalized Functions

    2.5. Fourier Transform of the Generalized Function (ax2 + bx + c)+λ

    2.6. Fourier Transforms of Analytic Functionals

    3. Fourier Transforms of Generalized Functions. Several Variables

    3.1. Definitions

    3.2. Fourier Transform of the Direct Product

    3.3. Fourier Transform of rλ

    3.4. Fourier Transform of Generalized Function with Bounded Support

    3.5. The Fourier Transform as the Limit of a Sequence of Functions

    4. Fourier Transforms and Differential Equations

    4.1. Introductory Remarks

    4.2. The Iterated Laplace Equation Δmu = ƒ

    4.3. The Wave Equation in Space of Odd Dimension

    4.4. The Relation between the Elementary Solution of an Equation and the Corresponding Cauchy Problem

    4.5. Classical Operational Calculus

    Chapter III Particular Types of Generalized Functions

    1. Generalized Functions Concentrated on Smooth Manifolds of Lower Dimension

    1.1. Introductory Remarks on Differential Forms

    1.2. The Form ω

    1.3. The Generalized Function δ(P)

    1.4. Example: Derivation of Green's Theorem

    1.5. The Differential Forms ωk(ȹ) and the Generalized Functions δ(k)(Ρ)

    1.6. Recurrence Relations for the δ(k)(Ρ)

    1.7. Recurrence Relations for the δ(k)(aΡ)

    1.8. Multiplet Layers

    1.9. The Generalized Function δ(Ρ1,..., Pj,) and Its Derivatives

    2. Generalized Functions Associated with Quadratic Forms

    2.1. Definition of δ1(k)(Ρ) and δ2(k)(Ρ)

    2.2. The Generalized Function P+λ

    2.3. The Generalized Function Pλ Associated with a Quadratic Form with Complex Coefficients

    2.4. The Generalized Functions (P + iO)λ and (P - iO)λ

    2.5. Elementary Solutions of Linear Differential Equations

    2.6. Fourier Transforms of (P + iO)λ and (P - iO)λ

    2.7. Generalized Functions Associated with Bessel Functions

    2.8. Fourier Transforms of (c2 + P + iO)λ and (c2 + P - iO)λ

    2.9. Fourier Transforms of (c2 + P)+λ and (c2 + P)-λ

    2.10. Fourier Transforms of (c2 + Ρ)+λ/Γ(λ + 1) and(c2 + P)-λ/Γ(λ + 1) for Integral λ

    3. Homogeneous Functions

    3.1. Introduction

    3.2. Positive Homogeneous Functions of Several Independent Variables

    3.3. Generalized Homogeneous Functions of Degree —n

    3.4. Generalized Homogeneous Functions of Degree —n — m

    3.5. Generalized Functions of the Form rλƒ, where ƒ Is a Generalized Function on the Unit Sphere

    4. Arbitrary Functions Raised to the Power λ

    4.1. Reducible Singular Points

    4.2. The Generalized Function Gλ when G = 0 Consists Entirely of First-Order Points

    4.3. The Generalized Function Gλ when G = 0 Has No Points of Order Higher Than Two

    4.4. The Generalized Function Gλ in General

    4.5. Integrals of an Infinitely Differentiable Function over a Surface Given by G = c

    Summary of Fundamental Definitions and Equations of Volume I

    Table of Fourier Transforms

    Appendix A. Proof of the Completeness of the Generalized-Function Space

    Appendix B. Generalized Functions of Complex Variables

    Bl. Generalized Functions of a Single Complex Variable

    Β1.1. The Variables z and z

    Β1.2. Homogeneous Functions of a Complex Variable

    B1.3. The Homogeneous Generalized Functions zλzμ

    Β1.4. The Generalized Functions z-k-1 and Its Derivatives

    B1.5. Associated Homogeneous Functions

    Β1.6. Uniqueness Theorem for Homogeneous Generalized Functions

    Β1.7. Fourier Transforms of Test Functions and of Generalized Functions

    Β1.8. The Generalized Functions ƒλ(z) ƒμ(z), Where ƒ( z ) is a Meromorphic Function

    B2. Generalized Functions of m Complex Variables

    B2.1. The Generalized Functions δ(P) and δ(k,l)(Ρ)

    B2.2. The Generalized Functions GλCμ

    B2.3. Homogeneous Generalized Functions

    B2.4. Associated Homogeneous Functions

    B2.5. The Residue of a Homogeneous Function

    B2.6. Homogeneous Generalized Functions of Degree (—m,—m)

    B2.7. The Generalized Function ΡλΡμ Where Ρ Is a Nondegenerate Quadratic Form

    B2.8. Elementary Solutions of Linear Differential Equations in the Complex Domain

    B2.9. The Generalized Function GλCμ (General Case)

    B2.10 Generalized Functions Corresponding to Meromorphic Functions of m Complex Variables

    Notes and References to the Literature



    Index of Particular Generalized Functions

Product details

  • No. of pages: 442
  • Language: English
  • Copyright: © Academic Press 1964
  • Published: January 1, 1964
  • Imprint: Academic Press
  • eBook ISBN: 9781483261591

About the Authors

I. M. Gel'fand

G. E. Shilov

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