Distributions and the Boundary Values of Analytic Functions

Distributions and the Boundary Values of Analytic Functions focuses on the tools and techniques of distribution theory and the distributional boundary behavior of analytic functions and their applications.

The publication first offers information on distributions, including spaces of testing functions, distributions of finite order, convolution and regularization, and testing functions of rapid decay and distributions of slow growth. The text then examines Laplace transform, as well as Laplace transforms of distributions with arbitrary support.

The manuscript ponders on distributional boundary values of analytic functions, including causal and passive operators, analytic continuation and uniqueness, boundary value theorems and generalized Hilbert transforms, and representation theorems for half-plane holomorphic functions with S' boundary behavior.

The publication is a valuable source of data for researchers interested in distributions and the boundary values of analytic functions.

Fore word

Preface

Chapter I. Distributions

1. Introduction

1.1 Preliminaries

1.2 Some Spaces of Testing Functions

1.3 Schwartz Distributions

1.4 Distributions of Finite Order

1.5 Convolution and Regularization

1.6 Sobolev Spaces

1.7 Testing Functions of Rapid Decay and Distributions of Slow Growth

1.8 Fourier Transforms

Chapter II. The Laplace Transform

2.1 Laplace Transforms of Distributions with Arbitrary Support

2.2 Laplace Transforms of Distributions in D'

2.3 Laplace Transforms of Distribution in E'

Chapter III. Distributional Boundary Values of Analytic Functions

3. Introduction

3.1 Causal Operators

3.2 Representation Theorems for Half Plane Holomorphic Functions with S' Boundary Behavior

3.3 Boundary Value Theorems and Generalized Hubert Transforms

3.4 Analytic Continuation and Uniqueness

3.5 Passive Operators

3.6 Notes and Remarks

Appendix I. Representation of Positive-Real Matrices

Appendix II. Supplementary Remarks

Bibliography

Index