Generalized Functions, Volume 3: Theory of Differential Equations 1470426617, 9781470426613
The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, altho
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English Pages 222 [233] Year 2016
Table of contents :
Cover
Title page
Translator's Note
Preface to the Russian Edition
Contents
Chapter I Spaces of Type W
1. Definitions
2. Bounded Operators in Spaces of Type W
3. Fourier Transforms
4. The Case of Several Variables
Chapter II Uniqueness Classes for the Cauchy Problem
1. Introduction
2. The Cauchy Problem in a Topological Vector Space
3. The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method
4. The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms
5. Examples
6. The Connection between the Reduced Order of a System and Its Characteristic Roots
7. A Theorem of the Phragmen-Lindelof Type
Appendix I. Convolution Equations
Appendix 2. Equations with Coefficients Which Depend on ?
Appendix 3. Systems with Elliptic Operators
Chapter III Correctness Classes for the Cauchy Problem
1. Introduction
2. Parabolic Systems
3. Hyperbolic Systems
4. Systems Which Are Petrovskii-Correct
5. On the Solutions of Incorrect Systems
Chapter IV Generalized Eigenfunction Expansions
1. Introduction
2. Differentiation of Functionals of Strongly Bounded Variation
3. Differentiation of Functionals of Weakly Bounded Variation
4. Existence and Completeness Theorems for the System of Eigenfunctionals
5. Generalized Eigenfunctions of Self-Adjoint Operators
6. The Structure of the Generalized Eigenfunctions
7. Dynamical Systems
Notes and References
Bibliography
Index
Cover
Title page
Translator's Note
Preface to the Russian Edition
Contents
Chapter I Spaces of Type W
1. Definitions
2. Bounded Operators in Spaces of Type W
3. Fourier Transforms
4. The Case of Several Variables
Chapter II Uniqueness Classes for the Cauchy Problem
1. Introduction
2. The Cauchy Problem in a Topological Vector Space
3. The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method
4. The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms
5. Examples
6. The Connection between the Reduced Order of a System and Its Characteristic Roots
7. A Theorem of the Phragmen-Lindelof Type
Appendix I. Convolution Equations
Appendix 2. Equations with Coefficients Which Depend on ?
Appendix 3. Systems with Elliptic Operators
Chapter III Correctness Classes for the Cauchy Problem
1. Introduction
2. Parabolic Systems
3. Hyperbolic Systems
4. Systems Which Are Petrovskii-Correct
5. On the Solutions of Incorrect Systems
Chapter IV Generalized Eigenfunction Expansions
1. Introduction
2. Differentiation of Functionals of Strongly Bounded Variation
3. Differentiation of Functionals of Weakly Bounded Variation
4. Existence and Completeness Theorems for the System of Eigenfunctionals
5. Generalized Eigenfunctions of Self-Adjoint Operators
6. The Structure of the Generalized Eigenfunctions
7. Dynamical Systems
Notes and References
Bibliography
Index
- Author / Uploaded
- I. M. Gelfand
- G. E. Shilov
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