| TABLE OF CONTENTS
DISTRIBUTION THEORY AND TRANSFORM ANALYSIS |
| TABLE OF CONTENTS Chapter 1. Distributions: Their Definitions and Basic Properties 1.1 Introduction 1.2 The Space D of Testing Functions 1.3 Distributions 1.4 Pseudofunctions, Hadamard's Finite Part, and Cauchy's Principal Value 1.5 Testing Functions and Distributions of Several Variables 1.6 Equality of Distributions over Open Sets 1.7 Some Operations on Distributions 1.8 Distributions as Local Phenomena Chapter 2. The Calculus of Distributions 2.1 Introduction 2.2 Convergence of a Sequence of Distributions (Convergence in the Space D') 2.3 Some Special Cases of Convergence in D' 2.4 The Differentiation of Distributions 2.5 Hadamard's Finite Part and Some Pseudofunctions Generated by It 2.6 The Primitives of Distributions Defined over R^{1}. 2.7 Continuity and Differentiability with Respect to a Parameter upon Which the Testing Functions Depend 2.8 Distributions That Depend upon a Parameter and Integration with Respect to That Parameter Chapter 3. Further Properties of Distributions 3.1 Introduction 3.2 A Characterization of the Delta Functional and Its Derivatives 3.3 A Local-Boundedness Property of Distributions 3.4 Locally Every Distribution Is a Finite-Order Derivative of a Continuous Function 3.5 Only Finite Linear Combinations of the Delta Functional and Its Derivatives Are Concentrated on a Point Chapter 4. Distributions of Slow Growth 4.1 Introduction 4.2 The Space S of Testing Functions of Rapid Descent 4.3 The Space S' of Distributions of Slow Growth 4.4 A Boundedness Property for Distributions of Slow Growth 4.5 A Differentiability Property for the Application of a Distribution in S_{tau} to a Testing Function in S_{t,tau} Chapter 5. Convolution 5.1 Introduction 5.2 The Direct Product of Distributions 5.3 The Support, Commutativity, and Associativity of the Direct Product 5.4 The Convolution of Distributions 5.5 Some Operations on the Convolution Process 5.6 The Continuity of the Convolution Process 5.7 The Convolution of a Distribution in S' with a Testing Function in S 5.8 Convolution Operators Chapter 6. Convolution Equations 6.1 Introduction 6.2 Convolution Algebras 6.3 An Application to Ordinary Linear Differential Equations with Constant Coefficients 6.4 Mikusinski's Operational Calculus Chapter 7. The Fourier Transformation 7.1 Introduction 7.2 The Ordinary Fourier Transformation 7.3 The Fourier Transforms of Testing Functions of Rapid Descent 7.4 The Fourier Transforms of Distributions of Slow Growth 7.5 The Fourier Transformation of Convolutions of Distributions Having Bounded Supports 7.6 The Space Z of Testing Functions Whose Fourier Transforms Are in D 7.7 The Space Z' of Ultradistributions 7.8 The Fourier Transforms of Arbitrary Distributions 7.9 The Fourier Transformation of the Convolution of Two Distributions One of Which Has a Bounded Support 7.10 The General Solution of a Homogeneous Linear Differential Equation with Constant Coefficients Chapter 8. The Laplace Transformation 8.1 Introduction 8.2 The Laplace Transforms of Ordinary Right-Sided Functions 8.3 The Laplace Transforms of Right-Sided Distributions 8.4 The Inversion of the Laplace Transformation for Right-Sided Distributions 8.5 The Laplace Transformation of Convolutions of Right-Sided Distributions 8.6 Some Abelian Theorems of the Initial-Value Type 8.7 Some Abelian Theorems of the Final-Value Type 8.8 The Laplace Transforms of Left-Sided Distributions 8.9 The Laplace Transforms of Distributions Having, in General, Unbounded Supports Chapter 9. The Solution of Differential and Difference Equations by Transform Analysis 9.1 Introduction 9.2 The Use of the Laplace Transformation in Solving Convolution Equations in the Algebra D'_{R} 9.3 Ordinary Linear Differential Equations with Constant Coefficients 9.4 Ordinary Linear Integrodifferential Equations with Constant Coefficients 9.5 Ordinary Linear Difference Equations with Constant Coefficients: The Continuous-Variable Case 9.6 Ordinary Linear Difference Equations with Constant Coefficients: The Discrete-Variable Case 9.7 Ordinary Linear Differential Equations with Polynomial Coefficients Chapter 10. Passive Systems 10.1 Introduction 10.2 One-Ports Having Convolution Representations 10.3 Causality and Passivity 10.4 The Positive-Reality of the Immittance Function 10.5 A Representation for the Unit-Impulse Response Corresponding to a Positive-Real Immittance Function 10.5 The Realizability of Every Positive-Real Function Chapter 11. Periodic Distributions 11.1 Introduction 11.2 The Space P_{T} of Periodic Testing Functions 11.3 The Space P'_{T} of Periodic Distributions 11.4 T-Convolution 11.5 The T-Convolution Algebra P'_{T} 11.6 The Fourier Series 11.7 The Finite Fourier Transformation and the Solution of T-Convolution Equations 11.8 Some Applications to Differential and Difference Equations Appendix A. The Axioms for a Linear Space Appendix B. Tables of Formulas for the Right-Sided Laplace Transformation Appendix C. Glossary of Symbols Bibliography Index |
