Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions 1470426595, 9781470426590
The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, altho
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English Pages 261 [273] Year 2016
Table of contents :
Cover
Title page
Preface to the Russian Edition
Contents
Chapter I Linear Topological Spaces
1. Definition of a Linear Topological Space
2. Normed Spaces. Comparability and Compatibility of Norms
3. Countably Normed Spaces
4. Continuous Linear Functionals and the Conjugate Space
5. Topology in a Conjugate Space
6. Perfect Spaces
7. Continuous Linear Operators
8. Union of Countably Normed Spaces
Appendix 1. Elements, Functionals, Operators Depending on a Parameter
Appendix 2. Differentiable Abstract Functions
Appendix 3. Operators Depending on a Parameter
Appendix 4. Integration of Continuous Abstract Functions with Respect to the Parameter
Chapter II Fundamental and Generalized Functions
1. Definition of Fundamental and Generalized Functions
2. Topology in the Spaces K{MV) and Z{MV)
3. Operations with Generalized Functions
4. Structure of Generalized Functions
Chapter III Fourier Transformations of Fundamental and Generalized Functions
1. Fourier Transformations of Fundamental Functions
2. Fourier Transforms of Generalized Functions
3. Convolution of Generalized Functions and Its Connection to Fourier Transforms
4. Fourier Transformation of Entire Analytic Functions
Chapter IV Spaces of Type S
1. Introduction
2. Various Modes of Defining Spaces of Type S
3. Topological Structure of Fundamental Spaces
4. Simplest Bounded Operations in Spaces of Type S
5. Differential Operators
6. Fourier Transformations
7. Entire Analytic Functions as Elements or Multipliers in Spaces of Type S
8. The Question of the Nontriviality of Spaces of Type S
9. The Case of Several Independent Variables
Appendix 1. Generalization of Spaces of Type S
Appendix 2. Spaces of Type W
Notes and References
Bibliography
Index
Cover
Title page
Preface to the Russian Edition
Contents
Chapter I Linear Topological Spaces
1. Definition of a Linear Topological Space
2. Normed Spaces. Comparability and Compatibility of Norms
3. Countably Normed Spaces
4. Continuous Linear Functionals and the Conjugate Space
5. Topology in a Conjugate Space
6. Perfect Spaces
7. Continuous Linear Operators
8. Union of Countably Normed Spaces
Appendix 1. Elements, Functionals, Operators Depending on a Parameter
Appendix 2. Differentiable Abstract Functions
Appendix 3. Operators Depending on a Parameter
Appendix 4. Integration of Continuous Abstract Functions with Respect to the Parameter
Chapter II Fundamental and Generalized Functions
1. Definition of Fundamental and Generalized Functions
2. Topology in the Spaces K{MV) and Z{MV)
3. Operations with Generalized Functions
4. Structure of Generalized Functions
Chapter III Fourier Transformations of Fundamental and Generalized Functions
1. Fourier Transformations of Fundamental Functions
2. Fourier Transforms of Generalized Functions
3. Convolution of Generalized Functions and Its Connection to Fourier Transforms
4. Fourier Transformation of Entire Analytic Functions
Chapter IV Spaces of Type S
1. Introduction
2. Various Modes of Defining Spaces of Type S
3. Topological Structure of Fundamental Spaces
4. Simplest Bounded Operations in Spaces of Type S
5. Differential Operators
6. Fourier Transformations
7. Entire Analytic Functions as Elements or Multipliers in Spaces of Type S
8. The Question of the Nontriviality of Spaces of Type S
9. The Case of Several Independent Variables
Appendix 1. Generalization of Spaces of Type S
Appendix 2. Spaces of Type W
Notes and References
Bibliography
Index
- Author / Uploaded
- I. M. Gelfand
- G. E. Shilov
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