Spaces of Fundamental and Generalized Functions (eBook)

Front Cover 1
Spaces of Fundamental and Generalized Functions 4
Copyright Page 5
Table of Contents 10
Preface to the Russian Edition 6
CHAPTER I. LINEAR TOPOLOGICAL SPACES 12
1. Definition of a Linear Topological Space 
12 
2. Normed Spaces. Comparability and Compatibility of Norms 22
3. Countably Normed Spaces 26
4. Continuous Linear Functionals and the Conjugate Space 43
5. Topology in a Conjugate Space 52
6. Perfect Spaces 64
7. Continuous Linear Operators 71
8. Union of Countably Normed Spaces 77
Appendix 1: Elements, Functionals, Operators Depending on a Parameter 
81 
Appendix 1: Elements, Functionals, Operators Depending on a Parameter 
81 
Appendix 1: Elements, Functionals, Operators Depending on a Parameter 
81 
Appendix 2: Differentiable Abstract Functions 83
Appendix 3: Operators Depending on a Parameter 84
Appendix 4: Integration of Continuous Abstract Functions with Respect to the Parameter 86
CHAPTER II. FUNDAMENTAL AND GENERALIZED FUNCTION 88
1. Definition of Fundamental and Generalized Functions 88
2. Topology in the Spaces K{Mp} and Z{Mp} 97
3. Operations with Generalized Functions 109
4. Structure of Generalized Functions 120
CHAPTER III. FOURIER TRANSFORMATIONS OF FUNDAMENTAL AND GENERALIZED FUNCTIONS 133
1. Fourier Transformations of Fundamental Functions 133
2. Fourier Transforms of Generalized Functions 139
3. Convolution of Generalized Functions and Its Connection to Fourier Transforms 146
4. Fourier Transformation of Entire Analytic Functions 165
CHAPTER IV. SPACES OF TYPE S 177
1. Introduction 177
2. Various Modes of Defining Spaces of Type S 180
3. Topological Structure of Fundamental Spaces 187
4. Simplest Bounded Operations in Spaces of Type S 195
5. Differential Operators 204
6. Fourier Transformations 208
8. The Question of the Nontriviality of Spaces of Type S 236
Appendix 1: Generalization of Spaces of Type S 255
Appendix 2: Spaces of Type W 257
NOTES AND REFERENCES 264
BIBLIOGRAPHY 268
INDEX 270