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.

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+ and x-


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


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


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


3.6. The Generalized Functions (x + iO) and (x - iO)


3.7. Canonical Regularization


3.8. Regularization of Other Integrals


3.9. The Generalized Function r


3.10. Plane-Wave Expansion of r


3.10. Homogeneous Functions


4. Associate Functions


4.1. Definition


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


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


4.4. The Generalized Functions (x + iO) and (x - iO)


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


4.6. Expansion of r


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


1.2. The Space


1.3. The Case of Several Variables


1.4. Functionals on


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+ , x- | x | , and | x | sgn x


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


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


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 r


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 mu = f


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


1.3. The Generalized Function d(P)


1.4. Example: Derivation of Green's Theorem


1.5. The Differential Forms k( ) and the Generalized Functions d(k)( )


1.6. Recurrence Relations for the d(k)( )


1.7. Recurrence Relations for the d(k)(a )


1.8. Multiplet Layers


1.9. The Generalized Function d( 1,..., Pj,) and Its Derivatives


2. Generalized Functions Associated with Quadratic Forms


2.1. Definition of d1(k)( ) and d2(k)( )


2.2. The Generalized Function P+


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


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


2.5. Elementary Solutions of Linear Differential Equations


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


2.7. Generalized Functions Associated with Bessel Functions


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


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


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


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 r f, where f Is a Generalized Function on the Unit Sphere


4. Arbitrary Functions Raised to the Power


4.1. Reducible Singular Points


4.2. The General…