# Generalized Functions

## 1st Edition

### Applications of Harmonic Analysis

**Authors:**I. M. Gel'fand N. Ya. Vilenkin

**eBook ISBN:**9781483262246

**Imprint:**Academic Press

**Published Date:**1st January 1964

**Page Count:**398

## Description

Generalized Functions, Volume 4: Applications of Harmonic Analysis is devoted to two general topics—developments in the theory of linear topological spaces and construction of harmonic analysis in n-dimensional Euclidean and infinite-dimensional spaces.

This volume specifically discusses the bilinear functionals on countably normed spaces, Hilbert-Schmidt operators, and spectral analysis of operators in rigged Hilbert spaces. The general form of positive generalized functions on the space S, continuous positive-definite functions, and conditionally positive generalized functions are also deliberated. This publication likewise considers the mean of a generalized random process, multidimensional generalized random fields, simplest properties of cylinder sets, and definition of Gaussian measures.

This book is beneficial to students, specialists, and researchers aiming to acquire knowledge of functional analysis.

## Table of Contents

Translator's Note

Foreword

Chapter I The Kernel Theorem. Nuclear Spaces. Rigged Hilbert Space

1. Bilinear Functionals on Countably Normed Spaces. The Kernel Theorem

1.1. Convex Functionals

1.2. Bilinear Functionals

1.3. The Structure of Bilinear Functionals on Specific Spaces (the Kernel Theorem)

Appendix. The Spaces K, S, and 2

2. Operators of Hilbert-Schmidt Type and Nuclear Operators

2.1. Completely Continuous Operators

2.2. Hilbert-Schmidt Operators

2.3. Nuclear Operators

2.4. The Trace Norm

2.5. The Trace Norm and the Decomposition of an Operator into a Sum of Operators of Rank 1

3. Nuclear Spaces. The Abstract Kernel Theorem

3.1. Countably Hilbert Spaces

3.2. Nuclear Spaces

3.3. A Criterion for the Nuclearity of a Space

3.4. Properties of Nuclear Spaces

3.5. Bilinear Functionals on Nuclear Spaces

3.6. Examples of Nuclear Spaces

3.7. The Metric Order of Sets in Nuclear Spaces

3.8. The Functional Dimension of Linear Topological Spaces

4. Rigged Hilbert Spaces. Spectral Analysis of Self-Adjoint and Unitary Operators

4.1. Generalized Eigenvectors

4.2. Rigged Hilbert Spaces

4.3. The Realization of a Hilbert Space as a Space of Functions, and Rigged Hilbert Spaces

4.4. Direct Integrals of Hilbert Spaces, and Rigged Hilbert Spaces

4.5. The Spectral Analysis of Operators in Rigged Hilbert Spaces

Appendix. The Spectral Analysis of Self-Adjoint and Unitary Operators in Hilbert Space

1. The Abstract Theorem on Spectral Decomposition

2. Cyclic Operators

3. The Decomposition of a Hilbert Space into a Direct Integral Corresponding to a Given Self-Adjoint Operator

Chapter II Positive and Positive-Definite Generalized Functions

1. Introduction

l.l. Positivity and Positive Definiteness

2. Positive Generalized Functions

2.1. Positive Generalized Functions on the Space of Infinitely Differentiable Functions Having Bounded Supports

2.2. The General Form of Positive Generalized Functions on the Space S

2.3. Positive Generalized Functions on Some Other Spaces

2.4. Multiplicatively Positive Generalized Functions

3. Positive-Definite Generalized Functions. Bochner's Theorem

3.1. Positive-Definite Generalized Functions on S

3.2. Continuous Positive-Definite Functions

3.3. Positive-Definite Generalized Functions on K

3.4. Positive-Definite Generalized Functions on Ζ

3.5. Translation-Invariant Positive-Definite Hermitean Bilinear Functionals

3.6. Examples of Positive and Positive-Definite Generalized Functions

4. Conditionally Positive-Definite Generalized Functions

4.1. Basic Definitions

4.2. Conditionally Positive Generalized Functions (Case of One Variable)

4.3. Conditionally Positive Generalized Functions (Case of Several Variables)

4.4. Conditionally Positive-Definite Generalized Functions on Κ

4.5. Bilinear Functionals Connected with Conditionally Positive-Definite Generalized Functions

Appendix

5. Evenly Positive-Definite Generalized Functions

5.1. Preliminary Remarks

5.2. Evenly Positive-Definite Generalized Functions on S1/2 1/2

5.3. Evenly Positive-Definite Generalized Functions on S1/2 1/2

5.4. Positive-Definite Generalized Functions and Groups of Linear Transformations

6. Evenly Positive-Definite Generalized Functions on the Space of Functions of One Variable with Bounded Supports

6.1. Positive and Multiplicatively Positive Generalized Functions

6.2. A Theorem on the Extension of Positive Linear Functionals

6.3. Even Positive Generalized Functions on Ζ

6.4. An Example of the Nonuniqueness of the Positive Measure Corresponding to a Positive Functional on Z+

7. Multiplicatively Positive Linear Functionals on Topological Algebras with Involutions

7.1. Topological Algebras with Involutions

7.2. The Algebra of Polynomials in Two Variables

Chapter III Generalized Random Processes

1. Basic Concepts Connected with Generalized Random Processes

1.1. Random Variables

1.2. Generalized Random Processes

1.3. Examples of Generalized Random Processes

1.4. Operations on Generalized Random Processes

2. Moments of Generalized Random Processes. Gaussian Processes. Characteristic Functionals

2.1. The Mean of a Generalized Random Process

2.2. Gaussian Processes

2.3. The Existence of Gaussian Processes with Given Means and Correlation Functionals

2.4. Derivatives of Generalized Gaussian Processes

2.5. Examples of Gaussian Generalized Random Processes

2.6. The Characteristic Functional of a Generalized Random Process

3. Stationary Generalized Random Processes. Generalized Random Processes with Stationary nth-Order Increments

3.1. Stationary Processes

3.2. The Correlation Functional of a Stationary Process

3.3. Processes with Stationary Increments

3.4. The Fourier Transform of a Stationary Generalized Random Process

4. Generalized Random Processes with Independent Values at Every Point

4.1. Processes with Independent Values

4.2. A Condition for the Positive Definiteness of the Functional exp(∫ƒ[ȹ(t)]dt)

4.3. Processes with Independent Values and Conditionally Positive-Definite Functions

4.4. A Connection between Processes with Independent Values at Every Point and Infinitely Divisible Distribution Laws

4.5. Processes Connected with Functionals of the nth Order

4.6. Processes of Generalized Poisson Type

4.7. Correlation Functionals and Moments of Processes with Independent Values at Every Point

4.8. Gaussian Processes with Independent Values at Every Point

5. Generalized Random Fields

5.1. Basic Definitions

5.2. Homogeneous Random Fields and Fields with Homogeneous sth-0rder Increments

5.3. Isotropic Homogeneous Generalized Random Fields

5.4. Generalized Random Fields with Homogeneous and Isotropic sth-Order Increments

5.5. Multidimensional Generalized Random Fields

5.6. Isotropic and Vectorial Multidimensional Random Fields

Chapter IV Measures In Linear Topological Spaces

1. Basic Definitions

1.1. Cylinder sets

1.2. Simplest Properties of Cylinder Sets

1.3. Cylinder Set Measures

1.4. The Continuity Condition for Cylinder Set Measures

1.5. Induced Cylinder Set Measures

2. The Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Nuclear Spaces

2.1. The Additivity of Cylinder Set measures

2.2. A Condition for the Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Countably Hilbert Spaces

2.3. Cylinder Sets Measures in the Adjoint Spaces of Nuclear Countably Hilbert Spaces

2.4. The Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Union Spaces of Nuclear Spaces

2.5. A Condition for the Countable Additivity of Measures on the Cylinder Sets in a Hilbert Space

3. Gaussian Measures in Linear Topological Spaces

3.1. Definition of Gaussian Measures

3.2. A Condition for the Countable Additivity of Gaussian Measures in the Conjugate Spaces of Countably Hilbert Spaces

4. Fourier Transforms of Measures in Linear Topological Spaces

4.1. Definition of the Fourier Transform of a Measure

4.2. Positive-Definite Functionals on Linear Topological Spaces

5. Quasi-Invariant Measures in Linear Topological Spaces

5.1. Invariant and Quasi-Invariant Measures in Finite-Dimensional Spaces

5.2. Quasi-Invariant Measures in Linear Topological Spaces

5.3. Quasi-Invariant Measures in Complete Metric Spaces

5.4. Nuclear Lie Groups and Their Unitary Representations. The Commutation Relations of the Quantum Theory of Fields

Notes and References to the Literature

Bibliography

Subject Index

## Details

- No. of pages:
- 398

- Language:
- English

- Copyright:
- © Academic Press 1964

- Published:
- 1st January 1964

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483262246