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A.H. Zemanian


Chapter 1. Vector-Valued Functions

1.1 Introduction
1.2 Notations and Terminology
1.3 Linear Spaces
1.4 Sequential-Convergence Spaces
1.5 Seminorms and Multinorms
1.6 Multinormed Spaces
1.7 Countable-Union Spaces
1.8 Duals of Countably Multinormed Spaces
1.9 Duals of Countable-Union Spaces
1.10 Operators and Adjoint Operators

Chapter 2. Distributions and Generalized Functions

2.1 Introduction
2.2 The Spaces D_{K}(I) and D(I), and Their Duals; Distributions
2.3 The Space E(I) and Its Dual; Distributions of Compact Support
2.4 Generalized Functions
2.5 Linear Partial Differential Operators Acting on Generalized Functions
2.6 Generalized Functions That Depend Upon a Parameter and Parametric Differentiation
2.7 Generalized Functions That Are Concentrated on Compact Sets

Chapter 3. The Two-Sided Laplace Transformation

3.1 Introduction
3.2 The Testing Function Spaces L_{a,b} and L(w,z) and Their Duals
3.3 The Two-Sided Laplace Transformation
3.4 Operation-Transform Formulas
3.5 Inversion and Uniqueness
3.6 Characterization of Laplace Transforms and an Operational Calculus
3.7 Convolution
3.8 The Laplace Transformation of Convolution
3.9 The Cauchy Problem for the Wave Equation in One-Dimensional Space
3.10 The Right-Sided Laplace Transformation
3.11 The n-Dimensional Laplace Transformaation
3.12 The Inhomogeneous Wave Equation in One-Dimensional Space

Chapter 4. The Mellin Transformaion

4.1 Introduction
4.2 The Testing-Function Spaces M_{a,b} and M(w,z) and Their Duals
4.3 The Mellin Transformation
4.4 Operation-Transform Formulas
4.5 An Operational Calculus for Euler Differential Equations
4.6 Mellin-Type Convolution
4.7 Dirichlet's Problem for a Wedge with a Generalized-Function Boundary Condition

Chapter 5. The Hankel Transformation

5.1 Introduction
5.2 The Testing-Function Space H_{mu} and Its Dual
5.3 Some Operations on H_{mu} and H'_{mu}
5.4 The Conventional Hankel Transformation on H_{mu}
5.5 The Generalized Hankel Transformation
5.6 The Hankel Transformation on E'(I)
5.7 An Operational Calculus
5.8 A Dirichlet Problem in Cylindrical Coordinates
5.9 A Cauchy Problem for Cylindrical Waves
5.10 Hankel Transforms of Arbitrary Order
5.11 Hankel Transforms of Certain Generalized Functions of Arbitrary Growth

Chapter 6. The K Transformation

6.1 Introduction
6.2 Some Classical Results
6.3 The Testing-Function Space K_{mu,a} and Its Dual
6.4 The K Transformation
6.5 The Analyticity of a K Transform
6.6 Inversion
6.7 Characterization of K Transforms
6.8 An Operational Calculus
6.9 Applications to Certain Time-varying Electrical Networks

Chapter 7. The Weierstrass Transformation

7.1 Introduction
7.2 The Testing-Function Spaces W_{a,b} and W(w,z) and Their Duals
7.3 The Weierstrass Transformation
7.4 Another Inversion Formula
7.5 The Cauchy Problem for the Heat Equation for One-Dimensional Flow

Chapter 8. The Convolution Transformation

8.1 Introduction
8.2 Convolution Kernels
8.3 The Convolution Transformation
8.4 Inversion
8.5 The One-Sided Laplace Transformation
8.6 The Stieltjes Transformation

Chapter 9. Transformations Arising from Orthonormal Series Expansions

9.1 Introduction
9.2 The Space L_{2}(I)
9.3 The Testing Function Space A
9.4 The Generalized-Function Space A'
9.5 Orthonormal Series Expansions and Generalized Integral Transformations
9.6 Characterizations of the Generalized Functions in A' and Their Transforms
9.7 An Operational Calculus
9.8 Particular Cases
9.9 Application of the Finite Fourier transformation:
A Dirichlet Problem for a Semi-Infinite Channel
9.10 Applications of the Laguerre and Jacobi Transformations:
The Time Domain Synthesis of Signals
9.11 Application of the Legendre Transformation:
A Dirichlet Problem for the Interior of the Unit Sphere
9.12 Application of the Finite Hankel Transformation of the First Form:
A Dirichlet Problem for a Semi-Infinite Cylinder
9.13 Application of the Finite Hankel Transformation of the Second Form:
Heat Flow in an Infinite Cylinder with a Radiation Condition


Index of Symbols