Integral Transforms of Generalized Functions and Their Applications

by Pathak,Ram Shankar

• ISBN13: 9789056995546
• ISBN10: 9056995545
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-12-01
• Publisher: G & B Pub

Summary

For those who have a background in advanced calculus, elementary topology and functional analysis - from applied mathematicians and engineers to physicists - researchers and graduate students alike - this work provides a comprehensive analysis of the many important integral transforms and renders particular attention to all the technical aspects of the subject. The author presents the last two decades of research and includes important results from other works.

Author Biography

Ram Shankar Pathak is a professor of mathematics at Banaras Hindu University in Varanasi, India.

Table of Contents

Preface	p. xv
Normed and Countably Normed Spaces	p. 1
Introduction	p. 1
Seminorms and Locally Convex Spaces	p. 1
Inductive Limits and Union of Multinormed Spaces	p. 4
The Test Function Space D	p. 4
The Test Function Space E	p. 5
The Test Function Space J	p. 5
The Test Function Spaces Z(a) and Z	p. 6
The Test Function Space W(I)	p. 7
Spaces of Generalized Functions	p. 8
Some special generalized function spaces	p. 9
Convergence of generalized functions	p. 13
Weak convergence and strong convergence	p. 14
A boundedness property	p. 15
Linear operators	p. 16
Operations on Generalized Functions	p. 17
Differentiation of generalized functions	p. 18
The Multiplication and Division of Generalized Functions	p. 20
Structures of Generalized Functions	p. 22
The Tensor Product of Distributions	p. 24
The Convolution of Generalized Functions	p. 26
Fundamental Solutions of Linear Differential Operators	p. 27
Problems	p. 29
Fourier Transforms of Distributions	p. 33
Introduction	p. 33
The Fourier Transform of Tempered Distributions	p. 34
The generalized Fourier transform	p. 36
Fourier transforms in L[superscript 2]	p. 36
Examples	p. 37
Properties of the Fourier Transform	p. 39
The Fourier transform of a convolution of generalized functions	p. 40
Applications	p. 41
Application of the Fourier transform to differential equations	p. 41
Application of the Fourier transform to a convolution equation	p. 42
Dual Integral Equations with Trigonometric Kernels	p. 42
Existence of Fundamental Solutions of a Certain Boundary Value Problem	p. 50
The Fourier Transform on D[prime]	p. 54
Distributional Boundary Values of Analytic Functions	p. 58
The Fourier Transform on E[prime]	p. 62
The Structure of Generalized Functions in Z[prime]	p. 66
The Cauchy Problem for the Two Dimensional Diffusion Equation	p. 67
Problems	p. 70
Fourier Transforms of Ultradistributions	p. 73
Introduction	p. 73
Ultradifferentiable Functions	p. 74
The function M([rho])	p. 75
The Space E(M[subscript p];[Omega])	p. 77
The Space D(M[subscript p];[Omega])	p. 78
Some Operations on Test Functions	p. 80
The Fourier Transform of Ultradifferentiable Functions	p. 82
Ultradistributions	p. 84
Convolution of ultradistributions	p. 85
Structure of ultradistributions	p. 88
The Fourier Transform of Ultradistributions	p. 91
Paley-Wiener theorem	p. 92
The Cauchy Integral Representation for Ultradistributions of Compact Support	p. 95
The Poisson Integral Representation	p. 103
Tempered Ultradistributions	p. 106
The Spaces D[subscript r](M[subscript p];R[superscript n]) and D[prime subscript r](M[subscript p];R[superscript n])	p. 111
Problems	p. 114
Laplace Transform	p. 119
Introduction	p. 119
The Test Function Space L[subscript a] and Its Dual L[prime subscript a]	p. 120
The Laplace Transform of Generalized Functions	p. 121
Examples	p. 122
Analyticity	p. 122
Important properties	p. 123
Inversion	p. 124
Convolution	p. 130
An Operational Calculus	p. 133
A Cauchy Problem for the Diffusion Equation	p. 135
Laplace Transform via Fourier Transform	p. 136
Ordinary Linear Differential Equations with Constant Coefficients	p. 137
The Non-Homogeneous Heat Equation	p. 139
An Integro-Differential Equation	p. 140
Asymptotic Behaviour of the Laplace Transform of Functions	p. 141
Asymptotic Behaviour of the Laplace Transform of Generalized Functions	p. 144
Problems	p. 147
Stieltjes Transform	p. 151
Introduction	p. 151
The Test Function Spaces S[subscript alpha](I) and S[subscript alpha](I)	p. 152
Preliminary Lemmas	p. 154
The Distributional Stieltjes Transform	p. 159
Asymptotic behaviour of F[superscript (m)](x)	p. 161
A Complex Inversion Theorem	p. 162
A Real Inversion Theorem	p. 166
Iteration of the Laplace Transform	p. 172
Abelian Theorems for the Stieltjes Transform of Functions	p. 174
Quasi-Asymptotic and Abelian Theorems	p. 178
Problems	p. 182
Hilbert Transform	p. 185
Introduction	p. 185
The Schwartz Test Function Space D[subscript L[superscript p]] and Its Dual D[prime subscript L[superscript p]]	p. 186
The Hilbert transform on [characters not reproducible]	p. 187
The Hilbert Transform of Generalized Functions in [characters not reproducible]	p. 188
Approximate Hilbert Transform	p. 190
Distributional Representation of Analytic Functions	p. 193
The Hilbert Problem for Generalized Functions	p. 196
Existence and Uniqueness of the Solution to a Dirichlet Problem	p. 197
Hilbert Transform via Fourier Transform	p. 199
The Hilbert Transform of Ultradistributions	p. 202
Modified Hilbert Transforms	p. 203
The Spaces H[subscript alpha characters not reproducible] and K[subscript alpha characters not reproducible]	p. 205
Distributional Modified Hilbert Transforms	p. 206
Finite Hilbert Transform	p. 207
Problems	p. 209
Mellin and Watson Transforms	p. 211
Introduction	p. 211
The Test Function Spaces M[subscript a,b] and M(w, z) and Their Duals	p. 213
The Mellin Transform	p. 214
Convolution	p. 217
An Operational Calculus	p. 219
The Space T([lambda],[mu]) and the Mellin Transform	p. 219
The Watson Transform on T([lambda,mu])	p. 223
Examples and Applications	p. 226
Product Convolutions and Fractional Integral Operators on T([lambda],[mu])	p. 229
A Dual Distributional Equation of Titchmarsh Type	p. 234
Problems	p. 235
Hankel Transforms of Distributions	p. 239
Introduction	p. 239
The Test Function Space H[subscript mu] and its Dual H[prime subscript mu]	p. 240
Operations on H[subscript mu] and H[prime subscript mu]	p. 242
Hankel Transforms on H[subscript mu] and H[prime subscript mu]	p. 244
The n-dimensional Distributional Hankel Transform	p. 246
Existence of Fundamental Solutions of a Certain Boundary Value Problem	p. 248
Fractional Integrals and Hankel Transforms	p. 253
Dual Integral Equations of Titchmarsh Type	p. 258
Existence	p. 260
Regularity	p. 263
Uniqueness	p. 264
An Extension of H[subscript mu] by Kernel Method	p. 264
The Hankel transform of generalized functions in G[prime subscript mu, alpha](I)	p. 266
Inversion and Uniqueness	p. 270
An Operational Calculus	p. 274
Axisymmetric Dirichlet Problem for a Thick Plate	p. 277
Problems	p. 279
Hankel Transforms of Ultradistributions	p. 283
Introduction	p. 283
Test Function Spaces and Their Duals	p. 283
Some Operations on Test Function Spaces	p. 289
Operation of multiplication by x	p. 290
Multiplication by an infinitely differentiable function	p. 291
The differential operator D	p. 291
Some other differential and integral operators	p. 292
Operations in dual spaces	p. 294
The Hankel Transform of Test Functions	p. 294
The Generalized Hankel Transform of Ultradistributions	p. 296
The Hankel Transform of Arbitrary Order	p. 297
Some Operational Formulae	p. 300
The Hankel Transform of Ultradistributions of Rapid Growth	p. 303
The test function spaces [characters not reproducible]	p. 304
The test function spaces [characters not reproducible]	p. 305
The Hankel transform of arbitrary order	p. 308
An Operational Calculus	p. 313
A Dirichlet Problem in Cylindrical Coordinates	p. 314
Problems	p. 317
Kontorovich-Lebedev Transform	p. 319
Introduction	p. 319
The Test Function Space K(I) and Its Dual	p. 321
The Kontorovich-Lebedev Transform of Generalized Functions	p. 322
The Function G[subscript N](t, x)	p. 323
Inversion	p. 329
An Operational Calculus	p. 335
Dirichlet's Problem for a Wedge with a Distributional Boundary Condition	p. 337
The Hankel-Form of the Kontorovich-Lebedev Transform	p. 339
Problems	p. 340
Generalized Mehler-Fock Transform	p. 343
Introduction	p. 343
The Test Function Space M[superscript alpha subscript beta](I) and Its Dual	p. 345
The Distributional Generalized Mehler-Fock Transform	p. 346
The Function F[subscript N](t, x)	p. 348
Inversion	p. 353
An Operational Calculus	p. 358
A Dirichlet Problem with a Distributional Boundary Condition	p. 360
Certain Dual Integral Equations	p. 364
Problems	p. 365
Eigenfunction Expansion of Generalized Functions	p. 367
Introduction	p. 367
The Test Function Space N(I)	p. 368
The Sturm-Liouville Expansion of Genralized Functions in N[prime](I)	p. 369
Special Cases	p. 370
Expansion in Fourier series	p. 371
Expansion in a series of Jacobi polynomials	p. 371
Expansion in a series of Legendre polynomials	p. 371
Expansion in series of spherical harmonics	p. 373
Expansion in series of Bessel functions	p. 373
An Operational Calculus	p. 375
Dirichlet Problem for the Interior of a Unit Sphere	p. 376
Temperature in a Long Cylinder	p. 378
A Mathematical Model of Volcanoes	p. 382
Problems	p. 384
Bibliography	p. 387
Index of Symbols	p. 401
Author Index	p. 405
Subject Index	p. 409
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