Generalized Functions (eBook)

Front Cover 1
Applications of Harmonic Analysis 4
Copyright Page 5
Table of Contents 10
Translator's Note 6
Foreword 8
CHAPTER I. THE KERNEL THEOREM. NUCLEAR SPACES. RIGGED HILBERT SPACE 16
1. Bilinear Functionals on Countably Normed Spaces. The Kernel Theorem 17
Appendix. The Spaces K, S, and . 35
2. Operators of Hilbert-Schmidt Type and Nuclear Operators 41
3. Nuclear Spaces. The Abstract Kernel Theorem 71
4. Rigged Hilbert Spaces. Spectral Analysis of Self-Adjoint and Unitary Operators 118
Appendix. The Spectral Analysis of Self-Adjoint and Unitary Operators in Hilbert Space 142
CHAPTER II. Positive and Positive-Definite Generalized Functions 150
1. Introduction 150
2. Positive Generalized Functions 157
3. Positive-Definite Generalized Functions. Bochner's Theorem 166
4. Conditionally Positive-Definite Generalized Functions 190
Appendix 209
5. Evenly Positive-Definite Generalized Functions 211
6. Evenly Positive-Definite Generalized Functions on the Space of Functions of One Variable with Bounded Supports 231
7. Multiplicatively Positive Linear Functionals on Topological Algebras with Involutions 244
CHAPTER III. Generalized Random Processes 252
1. Basic Concepts Connected with Generalized Random Processes 252
2. Moments of Generalized Random Processes. Gaussian Processes. Characteristic Functionals 261
3. Stationary Generalized Random Processes. Generalized Random Processes with Stationary A?th-Order Increments 277
4. Generalized Random Processes with Independent Values at Every Point 288
5. Generalized Random Fields 304
CHAPTER IV. MEASURES IN LINEAR TOPOLOGICAL SPACES 318
1. Basic Definitions 318
2. The Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Nuclear Spaces 327
3. Gaussian Measures in Linear Topological Spaces 350
4. Fourier Transforms of Measures in Linear Topological Spaces 360
5. Quasi-Invariant Measures in Linear Topological Spaces 365
NOTES AND REFERENCES TO THE LITERATURE 386
BIBLIOGRAPHY 392
SUBJECT INDEX 396