Generalized Functions, Volume 4: Applications of Harmonic Analysis

✍ Scribed by I. M. Gelfand, N. Ya Vilenkin

• Publisher: American Mathematical Society
• Year: 2016
• Tongue: English
• Leaves: 399
• Series: AMS Chelsea Publishing
• Category: Library

✦ Synopsis

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gelfand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. The main goal of Volume 4 is to develop the functional analysis setup for the universe of generalized functions. The main notion introduced in this volume is the notion of rigged Hilbert space (also known as the equipped Hilbert space, or Gelfand triple). Such space is, in fact, a triple of topological vector spaces $E \subset H \subset E'$, where $H$ is a Hilbert space, $E'$ is dual to $E$, and inclusions $E\subset H$ and $H\subset E'$ are nuclear operators. The book is devoted to various applications of this notion, such as the theory of positive definite generalized functions, the theory of generalized stochastic processes, and the study of measures on linear topological spaces.

✦ Table of Contents

Cover
Title page
Translator's Note
Foreword
Contents
Chapter I The Kernel Theorem. Nuclear Spaces. Rigged Hubert 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, 5, 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 Hubert 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 Hubert Spaces. Spectral Analysis of Self-Adjoint and Unitary Operators
4.1. Generalized Eigenvectors
4.2. Rigged Hubert Spaces
4.3. The Realization of a Hubert Space as a Space of Functions, and Rigged Hubert Spaces
4.4. Direct Integrals of Hubert Spaces, and Rigged Hubert Spaces
4.5. The Spectral Analysis of Operators in Rigged Hubert Spaces
Appendix. The Spectral Analysis of Self-Adjoint and Unitary Operators in Hubert Space
1. The Abstract Theorem on Spectral Decomposition
2. Cyclic Operators
3. The Decomposition of a Hubert Space into a Direct Integral Corresponding to a Given Self-Adjoint Operator
Chapter II Positive and Positive-Definite Generalized Functions
1. Introduction
1.1. 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 Z
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
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 S
5.3. Evenly Positive-Definite Generalized Functions on S12
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 Z
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(int [f(phit)]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-Order 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 Hubert Spaces
2.3. Cylinder Sets Measures in the Adjoint Spaces of Nuclear Countably Hubert 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 Hubert 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 Hubert 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