Preface |
Normed and Countably Normed Spaces / 1: |
Introduction / 1.1: |
Seminorms and Locally Convex Spaces / 1.2: |
Inductive Limits and Union of Multinormed Spaces / 1.3: |
The Test Function Space D / 1.4: |
The Test Function Space E / 1.5: |
The Test Function Space J / 1.6: |
The Test Function Spaces Z(a) and Z / 1.7: |
The Test Function Space W(I) / 1.8: |
Spaces of Generalized Functions / 1.9: |
Some special generalized function spaces / 1.9.1: |
Convergence of generalized functions / 1.9.2: |
Weak convergence and strong convergence / 1.9.3: |
A boundedness property / 1.9.4: |
Linear operators / 1.9.5: |
Operations on Generalized Functions / 1.10: |
Differentiation of generalized functions / 1.10.1: |
The Multiplication and Division of Generalized Functions / 1.11: |
Structures of Generalized Functions / 1.12: |
The Tensor Product of Distributions / 1.13: |
The Convolution of Generalized Functions / 1.14: |
Fundamental Solutions of Linear Differential Operators / 1.15: |
Problems |
Fourier Transforms of Distributions / 2: |
The Fourier Transform of Tempered Distributions / 2.1: |
The generalized Fourier transform / 2.2.1: |
Fourier transforms in L[superscript 2] / 2.2.2: |
Examples / 2.2.3: |
Properties of the Fourier Transform / 2.3: |
The Fourier transform of a convolution of generalized functions / 2.3.1: |
Applications / 2.4: |
Application of the Fourier transform to differential equations / 2.4.1: |
Application of the Fourier transform to a convolution equation / 2.4.2: |
Dual Integral Equations with Trigonometric Kernels / 2.5: |
Existence of Fundamental Solutions of a Certain Boundary Value Problem / 2.6: |
The Fourier Transform on D[prime] / 2.7: |
Distributional Boundary Values of Analytic Functions / 2.8: |
The Fourier Transform on E[prime] / 2.9: |
The Structure of Generalized Functions in Z[prime] / 2.10: |
The Cauchy Problem for the Two Dimensional Diffusion Equation / 2.11: |
Fourier Transforms of Ultradistributions / 3: |
Ultradifferentiable Functions / 3.1: |
The function M([rho]) / 3.2.1: |
The Space E(M[subscript p];[Omega]) / 3.3: |
The Space D(M[subscript p];[Omega]) / 3.4: |
Some Operations on Test Functions / 3.5: |
The Fourier Transform of Ultradifferentiable Functions / 3.6: |
Ultradistributions / 3.7: |
Convolution of ultradistributions / 3.7.1: |
Structure of ultradistributions / 3.7.2: |
The Fourier Transform of Ultradistributions / 3.8: |
Paley-Wiener theorem / 3.8.1: |
The Cauchy Integral Representation for Ultradistributions of Compact Support / 3.9: |
The Poisson Integral Representation / 3.10: |
Tempered Ultradistributions / 3.11: |
The Spaces D[subscript r](M[subscript p];R[superscript n]) and D[prime subscript r](M[subscript p];R[superscript n]) / 3.12: |
Laplace Transform / 4: |
The Test Function Space L[subscript a] and Its Dual L[prime subscript a] / 4.1: |
The Laplace Transform of Generalized Functions / 4.3: |
Analyticity / 4.3.1: |
Important properties / 4.3.3: |
Inversion / 4.4: |
Convolution / 4.5: |
An Operational Calculus / 4.6: |
A Cauchy Problem for the Diffusion Equation / 4.7: |
Laplace Transform via Fourier Transform / 4.8: |
Ordinary Linear Differential Equations with Constant Coefficients / 4.9: |
The Non-Homogeneous Heat Equation / 4.10: |
An Integro-Differential Equation / 4.11: |
Asymptotic Behaviour of the Laplace Transform of Functions / 4.12: |
Asymptotic Behaviour of the Laplace Transform of Generalized Functions / 4.13: |
Stieltjes Transform / 5: |
The Test Function Spaces S[subscript alpha](I) and S[subscript alpha](I) / 5.1: |
Preliminary Lemmas / 5.3: |
The Distributional Stieltjes Transform / 5.4: |
Asymptotic behaviour of F[superscript (m)](x) / 5.4.1: |
A Complex Inversion Theorem / 5.5: |
A Real Inversion Theorem / 5.6: |
Iteration of the Laplace Transform / 5.7: |
Abelian Theorems for the Stieltjes Transform of Functions / 5.8: |
Quasi-Asymptotic and Abelian Theorems / 5.9: |
Hilbert Transform / 6: |
The Schwartz Test Function Space D[subscript L[superscript p]] and Its Dual D[prime subscript L[superscript p]] / 6.1: |
The Hilbert transform on [characters not reproducible] / 6.2.1: |
The Hilbert Transform of Generalized Functions in [characters not reproducible] / 6.3: |
Approximate Hilbert Transform / 6.4: |
Distributional Representation of Analytic Functions / 6.5: |
The Hilbert Problem for Generalized Functions / 6.6: |
Existence and Uniqueness of the Solution to a Dirichlet Problem / 6.7: |
Hilbert Transform via Fourier Transform / 6.8: |
The Hilbert Transform of Ultradistributions / 6.9: |
Modified Hilbert Transforms / 6.10: |
The Spaces H[subscript alpha characters not reproducible] and K[subscript alpha characters not reproducible] / 6.11: |
Distributional Modified Hilbert Transforms / 6.12: |
Finite Hilbert Transform / 6.13: |
Mellin and Watson Transforms / 7: |
The Test Function Spaces M[subscript a,b] and M(w, z) and Their Duals / 7.1: |
The Mellin Transform / 7.3: |
The Space T([lambda],[mu]) and the Mellin Transform / 7.4: |
The Watson Transform on T([lambda,mu]) / 7.7: |
Examples and Applications / 7.8: |
Product Convolutions and Fractional Integral Operators on T([lambda],[mu]) / 7.9: |
A Dual Distributional Equation of Titchmarsh Type / 7.10: |
Hankel Transforms of Distributions / 8: |
The Test Function Space H[subscript mu] and its Dual H[prime subscript mu] / 8.1: |
Operations on H[subscript mu] and H[prime subscript mu] / 8.2.1: |
Hankel Transforms on H[subscript mu] and H[prime subscript mu] / 8.3: |
The n-dimensional Distributional Hankel Transform / 8.4: |
Fractional Integrals and Hankel Transforms / 8.5: |
Dual Integral Equations of Titchmarsh Type / 8.7: |
Existence / 8.7.1: |
Regularity / 8.7.2: |
Uniqueness / 8.7.3: |
An Extension of H[subscript mu] by Kernel Method / 8.8: |
The Hankel transform of generalized functions in G[prime subscript mu, alpha](I) / 8.8.1: |
Inversion and Uniqueness / 8.9: |
Axisymmetric Dirichlet Problem for a Thick Plate / 8.10: |
Hankel Transforms of Ultradistributions / 9: |
Test Function Spaces and Their Duals / 9.1: |
Some Operations on Test Function Spaces / 9.3: |
Operation of multiplication by x / 9.3.1: |
Multiplication by an infinitely differentiable function / 9.3.2: |
The differential operator D / 9.3.3: |
Some other differential and integral operators / 9.3.4: |
Operations in dual spaces / 9.3.5: |
The Hankel Transform of Test Functions / 9.4: |
The Generalized Hankel Transform of Ultradistributions / 9.5: |
The Hankel Transform of Arbitrary Order / 9.6: |
Some Operational Formulae / 9.7: |
The Hankel Transform of Ultradistributions of Rapid Growth / 9.8: |
The test function spaces [characters not reproducible] / 9.8.1: |
The Hankel transform of arbitrary order / 9.8.2: |
A Dirichlet Problem in Cylindrical Coordinates / 9.9: |
Kontorovich-Lebedev Transform / 10: |
The Test Function Space K(I) and Its Dual / 10.1: |
The Kontorovich-Lebedev Transform of Generalized Functions / 10.3: |
The Function G[subscript N](t, x) / 10.4: |
Dirichlet's Problem for a Wedge with a Distributional Boundary Condition / 10.5: |
The Hankel-Form of the Kontorovich-Lebedev Transform / 10.8: |
Generalized Mehler-Fock Transform / 11: |
The Test Function Space M[superscript alpha subscript beta](I) and Its Dual / 11.1: |
The Distributional Generalized Mehler-Fock Transform / 11.3: |
The Function F[subscript N](t, x) / 11.4: |
A Dirichlet Problem with a Distributional Boundary Condition / 11.5: |
Certain Dual Integral Equations / 11.8: |
Eigenfunction Expansion of Generalized Functions / 12: |
The Test Function Space N(I) / 12.1: |
The Sturm-Liouville Expansion of Genralized Functions in N[prime](I) / 12.3: |
Special Cases / 12.4: |
Expansion in Fourier series / 12.4.1: |
Expansion in a series of Jacobi polynomials / 12.4.2: |
Expansion in a series of Legendre polynomials / 12.4.3: |
Expansion in series of spherical harmonics / 12.4.4: |
Expansion in series of Bessel functions / 12.4.5: |
Dirichlet Problem for the Interior of a Unit Sphere / 12.5: |
Temperature in a Long Cylinder / 12.7: |
A Mathematical Model of Volcanoes / 12.8: |
Bibliography |
Index of Symbols |
Author Index |
Subject Index |
Preface |
Normed and Countably Normed Spaces / 1: |
Introduction / 1.1: |