Introduction to hyperfunctions and their integral transforms

Cover......Page 1
Introduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach......Page 4
9783034604079......Page 5
Contents......Page 6
Preface......Page 10
1.1 Generalized Functions......Page 14
1.2 The Concept of a Hyperfunction......Page 15
1.3 Properties of Hyperfunctions......Page 25
1.3.1 Linear Substitution......Page 26
1.3.2 Hyperfunctions of the Type f(φ(x))......Page 28
1.3.3 Differentiation......Page 31
1.3.4 The Shift Operator as a Differential Operator......Page 37
1.3.5 Parity, Complex Conjugate and Realness......Page 38
1.3.6 The Equation φ(x)f(x) = h(x)......Page 41
1.4 Finite Part Hyperfunctions......Page 46
1.5.1 Integrals with respect to the Independent Variable......Page 50
1.5.2 Integrals with respect to a Parameter......Page 56
1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyperfunctions......Page 57
1.6.2 Integral Powers......Page 58
1.6.3 Non-integral Powers......Page 62
1.6.4 Logarithms......Page 64
1.6.5 Upper and Lower Hyperfunctions......Page 69
1.6.6 The Normalized Power x^α_+/Γ(α + 1)......Page 71
1.6.7 Hyperfunctions Concentrated at One Point......Page 74
2.1 Sequences, Series, Limits......Page 76
2.2 Cauchy-type Integrals......Page 84
2.3 Projections of Functions......Page 89
2.3.1 Functions Satisfying the H¨older Condition......Page 90
2.3.2 Projection Theorems......Page 91
2.3.3 Convergence Factors......Page 100
2.3.4 Homologous and Standard Hyperfunctions......Page 102
2.4.1 Holomorphic and Meromorphic Hyperfunctions......Page 105
2.4.2 Standard Defining Functions......Page 108
Sato’s Decomposition Theorems......Page 112
Perfect Hyperfunctions and Their Standard Defining Functions......Page 113
Extension to Non-Perfect Hyperfunctions......Page 117
Strong Defining Functions......Page 118
2.4.4 Support, Singular Support and Singular Spectrum......Page 124
2.5.1 Product of Upper or Lower Hyperfunctions......Page 127
2.5.2 Products in the Case of Disjoint Singular Supports......Page 129
2.5.3 The Integral of a Product......Page 133
2.5.4 Hadamard’s Finite Part of an Integral......Page 139
2.6 Periodic Hyperfunctions and Their Fourier Series......Page 141
2.7.1 Definition and Existence of the Convolution......Page 150
2.7.2 Sufficient Conditions for the Existence of Convolutions......Page 154
2.7.3 Operational Properties......Page 158
2.7.4 Principal Value Convolution......Page 163
2.8 Integral Equations I......Page 165
3.1 Loop Integrals......Page 168
3.2.1 The Classical Laplace Transform......Page 172
3.3 Laplace Transforms of Hyperfunctions......Page 175
3.4.1 Dirac Impulses and their Derivatives......Page 184
3.4.2 Non-negative Integral Powers......Page 186
3.4.3 Negative Integral Powers......Page 187
3.4.5 Powers with Logarithms......Page 188
3.4.6 Exponential Integrals......Page 190
3.4.7 Transforms of Finite Part Hyperfunctions......Page 195
3.5.1 Linearity......Page 201
3.5.2 Image Translation Rule......Page 202
3.5.4 Similarity Rule......Page 206
3.5.5 Differentiation Rule......Page 208
3.5.6 Integration Rule......Page 211
3.5.8 Linear Substitution Rules......Page 214
3.6.1 Inverse Laplace Transforms......Page 217
The Inversion Formulas......Page 225
Laplace hyperfunctions......Page 228
3.6.2 The Convolution Rule......Page 229
3.6.3 Fractional Integrals and Derivatives......Page 236
3.7 Right-sided Laplace Transforms......Page 237
3.8.1 Volterra Integral Equations of Convolution Type......Page 240
3.8.2 Convolution Integral Equations over an Infinite Range......Page 246
4.1.1 Basic Definitions......Page 254
4.1.2 Connection to Laplace Transformation......Page 257
4.2 Fourier Transforms of Some Familiar Hyperfunctions......Page 259
4.3 Inverse Fourier Transforms......Page 264
4.3.1 Reciprocity......Page 267
4.4 Operational Properties......Page 268
4.4.1 Linear Substitution Rule......Page 269
4.4.3 Complex Conjugation and Realness......Page 270
4.4.4 Differentiation and Multiplication Rule......Page 271
4.4.5 Convolution Rules......Page 273
4.5 Further Examples......Page 278
4.6 Poisson’s Summation Formula......Page 280
4.7.1 Integral Equations III......Page 283
4.7.2 Heat Equation and Weierstrass Transformation......Page 286
5 Hilbert Transforms......Page 288
5.1.1 Definition and Basic Properties......Page 289
5.1.2 Operational Properties......Page 303
5.1.3 Using Fourier Transforms......Page 306
5.2 Analytic Signals and Conjugate Hyperfunctions......Page 309
5.3 Integral Equations IV......Page 313
6.1 The Classical Mellin Transformation......Page 322
6.2 Mellin Transforms of Hyperfunctions......Page 326
6.3.2 Scale Changes......Page 328
6.3.3 Multiplication by (log x)^n......Page 329
6.3.5 Reflection......Page 330
6.3.6 Differentiation Rules......Page 331
6.3.7 Integration Rules......Page 332
6.4 Inverse Mellin Transformation......Page 334
6.5 \mathcal{M}-Convolutions......Page 336
6.5.1 Reciprocal Integral Transforms......Page 338
6.5.2 Transform of a Product and Parseval’s Formula......Page 339
6.6.1 Dirichlet’s Problem in a Wedge-shaped Domain......Page 341
6.6.2 Euler’s Differential Equation......Page 343
6.6.3 Integral Equations V......Page 345
6.6.4 Summation of Series......Page 346
7.1.1 Genesis of the Hankel Transform......Page 350
7.1.2 Cylinder Functions......Page 353
7.1.3 Lommel’s Integral......Page 359
7.1.4 MacRobert’s Proof......Page 361
7.1.5 Some Hankel Transforms of Ordinary Functions......Page 362
The Similarity and Division Rule......Page 365
Differentiation Rules......Page 366
7.2.1 Basic Definitions......Page 369
7.2.2 Transforms of some Familiar Hyperfunctions......Page 371
The Similarity Rule......Page 376
The Division Rule......Page 377
The Differentiation Rule......Page 378
7.3 Applications......Page 380
A.1.1 Flow Fields and Holomorphic Functions......Page 386
A.1.2 Polya fields and Defining Functions......Page 388
A.2.1 Functions of Exponential Type......Page 390
A.2.2 Laplace Hyperfunctions and their Transforms......Page 399
A.3.1 Interchanging Infinite Series with Improper Integrals......Page 402
A.3.3 Defining Holomorphic Functions by Series and Integrals......Page 404
Operational Rules for the Laplace Transformation......Page 408
Some Laplace Transforms of Hyperfunctions......Page 409
Some Fourier Transforms of Hyperfunctions......Page 411
Operational Rules for the Hilbert Transformation......Page 413
Some Hilbert Transforms of Hyperfunctions......Page 414
Operational Rules for the Mellin Transformation......Page 415
Some Mellin Transforms of Hyperfunctions......Page 416
Operational Rules for the Hankel Transformation......Page 417
Some Hankel Transforms of order ν of Hyperfunctions......Page 418
Bibliography......Page 420
List of Symbols......Page 424
Index......Page 426