Introduction to Hyperfunctions and Their Integral Transforms

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ISBN: 9783034604079 | 3034604076
Cover: Hardcover
Copyright: 5/15/2010

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This textbook presents an introduction to generalized functions through Sato's hyperfunctions, i.e. based on complex variables theory. Laplace transforms, Fourier transforms, Hilbert transforms, Mellin tranforms and Hankel transforms of hyperfunctions and ordinary functions are then treated, and some applications mainly to integral equations are presented. The book is written from an applied and computational point of view and contains many concrete examples.

Preface	p. ix
Introduction to Hyperfunctions	p. 1
Generalized Functions	p. 1
The Concept of a Hyperfunction	p. 2
Properties of Hyperfunctions	p. 12
Linear Substitution	p. 13
Hyperfunctions of the Type f (ø(x))	p. 15
Differentiation	p. 18
The Shift Operator as a Differential Operator	p. 24
Parity, Complex Conjugate and Realness	p. 25
The Equation ø(x)f(x) = h(x)	p. 28
Finite Part Hyperfunctions	p. 33
Integrals	p. 37
Integrals with respect to the Independent Variable	p. 37
Integrals with respect to a Parameter	p. 43
More Familiar Hyperfunctions	p. 44
Unit-Step, Delta Impulses, Sign, Characteristic Hyperfunctions	p. 44
Integral Powers	p. 45
Non-integral Powers	p. 49
Logarithms	p. 51
Upper and Lower Hyperfunctions	p. 56
The Normalized Power x^¿₊/¿(¿ + 1)	p. 58
Hyperfunctions Concentrated at One Point	p. 61
Analytic Properties	p. 63
Sequences, Series, Limits	p. 63
Cauchy-type Integrals	p. 71
Projections of Functions	p. 76
Functions Satisfying the Hölder Condition	p. 77
Projection Theorems	p. 78
Convergence Factors	p. 87
Homologous and Standard Hyperfunctions	p. 89
Projections of Hyperfunctions	p. 92
Holomorphic and Meromorphic Hyperfunctions	p. 92
Standard Defining Functions	p. 95
Micro-analytic Hyperfunctions	p. 111
Support, Singular Support and Singular Spectrum	p. 111
Product of Hyperfunctions	p. 114
Product of Upper or Lower Hyperfunctions	p. 114
Products in the Case of Disjoint Singular Supports	p. 116
The Integral of a Product	p. 120
Hadamard's Finite Part of an Integral	p. 126
Periodic Hyperfunctions and Their Fourier Series	p. 128
Convolutions of Hyperfunctions	p. 137
Definition and Existence of the Convolution	p. 137
Sufficient Conditions for the Existence of Convolutions	p. 141
Operational Properties	p. 145
Principal Value Convolution	p. 150
Integral Equations I	p. 152
Laplace Transforms	p. 155
Loop Integrals	p. 155
The Two-Sided Laplace Transform	p. 159
The Classical Laplace Transform	p. 159
Laplace Transforms of Hyperfunctions	p. 162
Transforms of some Familiar Hyperfunctions	p. 171
Dirac Impulses and their Derivatives	p. 171
Non-negative Integral Powers	p. 173
Negative Integral Powers	p. 174
Non-integral Powers	p. 175
Powers with Logarithms	p. 175
Exponential Integrals	p. 177
Transforms of Finite Part Hyperfunctions	p. 182
Operational Properties	p. 188
Linearity	p. 188
Image Translation Rule	p. 189
The Multiplication or Image Differentiation Rule	p. 193
Similarity Rule	p. 193
Differentiation Rule	p. 195
Integration Rule	p. 198
Original Translation Rule	p. 201
Linear Substitution Rules	p. 201
Inverse Laplace Transforms and Convolutions	p. 204
Inverse Laplace Transforms	p. 204
The Convolution Rule	p. 216
Fractional Integrals and Derivatives	p. 223
Right-sided Laplace Transforms	p. 224
Integral Equations II	p. 227
Volterra Integral Equations of Convolution Type	p. 227
Convolution Integral Equations over an Infinite Range	p. 233
Fourier Transforms	p. 241
Fourier Transforms of Hyperfunctions	p. 241
Basic Definitions	p. 241
Connection to Laplace Transformation	p. 244
Fourier Transforms of Some Familiar Hyperfunctions	p. 246
Inverse Fourier Transforms	p. 251
Reciprocity	p. 254
Operational Properties	p. 255
Linear Substitution Rule	p. 256
Shift-Rules	p. 257
Complex Conjugation and Realness	p. 257
Differentiation and Multiplication Rule	p. 258
Convolution Rules	p. 260
Further Examples	p. 265
Poisson's Summation Formula	p. 267
Application to Integral and Differential Equations	p. 270
Integral Equations III	p. 270
Heat Equation and Weierstrass Transformation	p. 273
Hubert Transforms	p. 275
Hilbert Transforms of Hyperfunctions	p. 276
Definition and Basic Properties	p. 276
Operational Properties	p. 290
Using Fourier Transforms	p. 293
Analytic Signals and Conjugate Hyperfunctions	p. 296
Integral Equations IV	p. 300
Mellin Transforms	p. 309
The Classical Mellin Transformation	p. 309
Mellin Transforms of Hyperfunctions	p. 313
Operational Properties	p. 315
Linearity	p. 315
Scale Changes	p. 315
Multiplication by (log x)ⁿ	p. 316
Multiplication by x^¿, ¿ C	p. 317
Reflection	p. 317
Differentiation Rules	p. 318
Integration Rules	p. 319
Inverse Mellin Transformation	p. 321
M-Convolutions	p. 323
Reciprocal Integral Transforms	p. 325
Transform of a Product and Parseval's Formula	p. 326
Applications	p. 328
Dirichlet's Problem in a Wedge-shaped Domain	p. 328
Euler's Differential Equation	p. 330
Integral Equations V	p. 332
Summation of Series	p. 333
Hankel Transforms	p. 337
Hankel Transforms of Ordinary Functions	p. 337
Genesis of the Hankel Transform	p. 337
Cylinder Functions	p. 340
Lommel's Integral	p. 346
MacRobert's Proof	p. 348
Some Hankel Transforms of Ordinary Functions	p. 349
Operational Properties	p. 352
Hankel Transforms of Hyperfunctions	p. 356
Basic Definitions	p. 356
Transforms of some Familiar Hyperfunctions	p. 358
Operational Properties	p. 363
Applications	p. 367
Complements	p. 373
Physical Interpretation of Hyperfunctions	p. 373
Flow Fields and Holomorphic Functions	p. 373
Pólya fields and Defining Functions	p. 375
Laplace Transforms in the Complex Plane	p. 377
Functions of Exponential Type	p. 377
Laplace Hyperfunctions and their Transforms	p. 386
Some Basic Theorems of Function Theory	p. 389
Interchanging Infinite Series with Improper Integrals	p. 389
Reversing the Order of Integration	p. 391
Defining Holomorphic Functions by Series and Integrals	p. 391
Tables	p. 395
Convolution Properties of Hyperfunctions	p. 395
Operational Rules for the Laplace Transformation	p. 395
Some Laplace Transforms of Hyperfunctions	p. 396
Operational Rules for the Fourier Transformation	p. 398
Some Fourier Transforms of Hyperfunctions	p. 398
Operational Rules for the Hilbert Transformation	p. 400
Some Hilbert Transforms of Hyperfunctions	p. 401
Operational Rules for the Mellin Transformation	p. 402
Some Mellin Transforms of Hyperfunctions	p. 403
Operational Rules for the Hankel Transformation	p. 404
Some Hankel Transforms of order ¿ of Hyperfunctions	p. 405
Bibliography	p. 407
List of Symbols	p. 411
Index	p. 413
Table of Contents provided by Ingram. All Rights Reserved.

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