Applied hyperfunction theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English Japanese |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1992
|
| Schriftenreihe: | Mathematics and its applications / Japanese series
8 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Japan. übers. |
| Beschreibung: | XIX, 438 S. graph. Darst. |
| ISBN: | 0792315073 |
Internformat
MARC
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| 035 | |a (DE-599)BVBBV009021147 | ||
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| 084 | |a SK 750 |0 (DE-625)143254: |2 rvk | ||
| 100 | 1 | |a Imai, Isao |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Ōyō chōkansūron |
| 245 | 1 | 0 | |a Applied hyperfunction theory |c by Isao Imai |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1992 | |
| 300 | |a XIX, 438 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications / Japanese series |v 8 | |
| 500 | |a Aus dem Japan. übers. | ||
| 650 | 7 | |a Functies (wiskunde) |2 gtt | |
| 650 | 7 | |a Functionaalanalyse |2 gtt | |
| 650 | 7 | |a Hyperfonctions |2 ram | |
| 650 | 4 | |a Hyperfunctions | |
| 650 | 0 | 7 | |a Hyperfunktion |0 (DE-588)4161056-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hyperfunktion |0 (DE-588)4161056-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 810 | 2 | |a Japanese series |t Mathematics and its applications |v 8 |w (DE-604)BV000613779 |9 8 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005965617&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-005965617 | ||
Datensatz im Suchindex
| _version_ | 1804123371553685504 |
|---|---|
| adam_text | Contents
Series Editor s Preface v
Preface xvii
Chapter 1. INTRODUCTION 1
§1 What is a hyperfunction? 1
§2 Sato s hyperfunction 2
§3 Aim 3
§4 Complex velocity and analytic function 4
§5 Distribution of vortices and hyperfunctions 6
§6 Ordinary functions and hyperfunctions 8
Chapter 2. OPERATIONS ON HYPERFUNCTIONS 11
§1 Definition of hyperfunctions 11
§2 Linear combinations 13
§3 Product of a hyperfunction and an analytic function 14
§4 Reinterpretation of ordinary functions as hyperfunctions 15
§5 Differentiation of hyperfunctions 19
§6 Definite integrals of hyperfunctions 22
§7 Summary 23
viii
Chapter 3. BASIC HYPERFUNCTIONS 25
§1 Preliminary 25
§2 Hyperfunction with generating function F(—z) 25
§3 Even hyperfunctions and odd hyperfunctions 27
§4 Hyperfunction with generating function F(z) 29
§5 Real hyperfunctions and imaginary hyperfunctions 30
§6 Single valued analytic functions reinterpreted as hyperfunctions 33
§7 Cauchy s principal value 34
§6 Hyperfunction of the form f(ax + b) 36
§9 Formal product of a hyperfunction and a single valued analytic func¬
tion 37
§10 H(z), l(z), sgnz 40
§11 x~m, x mE(x), x msgnx 43
§12 log x , log |x| H(z), log x sgnz 45
§13or mlog|x|, x mlog|a;|H(x), x~m log |x| sgnz 46
§14 x m{ og x )n, z m(log|z|)nH(z), z m(log|z|)nsgnz 47
§15 x a, |z|aH(z), |z|asgnz 47
§16 Equation j (x) ¦ f(x) = h{x) 49
§17 Summary 51
Chapter 4. HYPERFUNCTIONS DEPENDING ON PARAMETERS 53
§1 Preliminary 53
§2 Hyperfunction depending on a parameter 53
§3 Operations on parameter dependent hyperfunctions 56
§4 Convergence of a sequence of functions and convergence of a sequence
of hyperfunctions 60
ix
§5 1*1* log W, x a log x H(x), |x|a log |*| sgna: 61
§6 |x|a(log |s|)n, |x|a(]og|x|)»H(x), |x|°(log|x|)Bsgnx 64
§7 x m( og x )n, x m(log|x|) H(x), ^ m(log|a;|) sgnx 65
§8 Power type hyperfunctions 67
§9 Finite part of a divergent integral 71
§10 Calculation of pf integrals 75
§11 Summary 81
Chapter 5. FOURIER TRANSFORMATION 83
§1 Preliminary 83
§2 Definition of Fourier transformations 83
§3 Theorems about Fourier transformation 88
§4 Inverse Fourier transformations 93
§5 Examples of calculations of Fourier transforms 97
§6 Summary 99
Chapter 6. FOURIER TRANSFORMATION OF POWER TYPE HY¬
PERFUNCTIONS 101
§1 Preliminary 101
§2 Tza 101
§3 T x a E(x), T x a, ^|x|a sgn x 104
§4 ^|x|a(log|x|) H(x), T x a( og x )n, ^|x|a(log|x|)nsgnx 105
§5 jF^Clog |x|)n H(x), JFx* (log |x|)n, JFx (log |x|) sgnx 107
§6 Tx m{ og x )nU.{x), ^x m(log|x|)n, JFx l(log|x|) sgnx 109
§7 Table of Fourier transforms of power type hyperfunctions 111
§8 Polygamma functions 111
X
§9 Examples of application 112
§10 Summary 114
Chapter 7. UPPER (LOWER) TYPE HYPERFUNCTIONS 115
§1 Preliminary 115
§2 Left (right) hyperfunctions and upper (lower) hyperfunctions 115
§3 Properties of upper (lower) type hyperfunctions 117
§4 Calculation of upper (lower) type hyperfunctions 121
§5 Fourier transforms of upper (lower) type hyperfunctions 124
§6 Upper (lower) power type hyperfunctions and their Fourier trans¬
forms 126
§7 Examples of application 128
§8 Summary 131
Chapter 8. FOURIER TRANSFORMS EXISTENCE AND REGULAR¬
ITY 133
§1 Preliminary 133
§2 eksa type functions and hyperfunctions 134
§3 Sufficient conditions for the existence of Fourier transforms 136
§4 Regularity of G(Q = f{ j){z)l+(z)} on the £ axis 137
§5 Examples of application of the theorems 141
§6 Summary 143
Chapter 9. FOURIER TRANSFORM ASYMPTOTIC BEHAVIOUR 147
§1 Preliminary 147
§2 Riemann Lebesgue theorem 147
§3 Reduction of hyperfunctions to ordinary functions 148
§4 Asymptotic behaviour of T{(p(z)l+(z)} 149
xi
§5 Generalization of Riemann Lebesgue theorem 152
§6 Formal product of tp( x ) and a power type hyperfunction 154
§7 Asymptotic behaviour of T[cj {x) ± 158
§8 Upper (lower) hyperfunctions and left (right) hyperfunctions 160
§9 From ff (x) to Tf{x) 162
§10 Summary 163
Chapter 10. PERIODIC HYPERFUNCTIONS and FOURIER SERIES
FOURIER SERIES 165
§1 Preliminary 165
§2 Standard generating functions 165
§3 Hyperfunction with standard generating function 169
§4 Periodic hyperfunction 172
§5 Fourier series of hyperfunctions 176
§6 Fourier transforms of periodic hyperfunctions 179
§7 Row of 5 functions and step functions 182
§8 Calculation of Fourier series 187
§9 Fourier series of upper (lower) hyperfunctions 193
§10 Behaviour of Fourier coefficients c,j for n —* oo 197
§11 Right (left) step function and right (left) 5 function row 199
§12 Summary 204
Chapter 11. ANALYTIC CONTINUATION and PROJECTION OF
HYPERFUNCTIONS 205
§1 Preliminary 205
§2 Projection 206
§3 Analytic continuation 207
xii
§4 Generalised ^ function 212
§5 Standard hyperfunction 213
§6 Extension of the definition of projection 214
§7 Theorems about projection 218
§8 Finite part of divergent integral 222
§9 Summary 223
Chapter 12. PRODUCT OF HYPERFUNCTIONS 225
§1 Preliminary 225
§2 Product of hyperfunctions 225
§3 Definite integral of the product of hyperfunctions 229
§4 Generating function of product of hyperfunctions 233
§5 pf £(x a)a{b xY dx 235
§6 Product of two upper (lower) type hyperfunctions 239
§7 Functions of upper (lower) type hyperfunctions 241
§8 Classification of products of hyperfunctions 246
§9 Summary 246
Chapter 13. CONVOLUTION OF HYPERFUNCTIONS 249
§1 Preliminary 249
§2 Convolution ordinary function and hyperfunction 250
§3 Convolution hyperfunction and hyperfunction 251
§4 Definition of convolutions 253
§5 Basic convolutions 260
§6 Basic properties of convolution 262
§7 Calculation of convolutions 264
xiii
§8 Sufficient conditions for the existence of convolution 267
§9 Convolution of two right (left) hyperfunctions 269
§10 Convolution of upper (lower) type hyperfunctions 271
§11 Fourier transforms of convolutions 275
§12 Fourier transforms of products 277
§13 Values of hyperfunctions and infinite principal value integrals 278
§14 Parseval s theorem 280
§15 Summary 282
Chapter 14. CONVOLUTION OF PERIODIC HYPERFUNCTIONS 283
§1 Preliminary 283
§2 Extension of definition of convolution 283
§3 Poisson s summation formula 287
§4 Convolution of periodic hyperfunctions 293
§5 Fourier expansions 296
§6 Convolution of two periodic hyperfunctions 299
§7 Summary 301
Chapter 15. HILBERT TRANSFORMS and CONJUGATE HYPER¬
FUNCTIONS 303
§1 Preliminary 303
§2 Hilbert transforms 303
§3 Properties of conjugate hyperfunctions 307
§4 Conjugate Fourier series 310
§5 Calculation of Hilbert transforms 312
§6 Formulae of Hilbert transforms 314
xiv
§7 Standard type generating function 322
§8 Hilbert type transforms 324
§9 Summary 326
Chapter 16. POISSON SCHWARZ INTEGRAL FORMULAE 327
§1 Preliminary 327
§2 The Poisson Schwarz integral formula for a half plane 327
§3 Poisson—Schwarz integral formula for a circle 329
§4 Generalization of the Poisson Schwarz integral formula 331
§5 Riemann—Hilbert problem 337
§6 Integral equations related to Hilbert transforms 339
§7 Ho f(t)(t x) 1 dt = g{x), oo x oo 340
%8f~f(t)(t x) ldt = g(x),x 0 341
§9 £ f(t)(t x) 1 dt = g(x), a x b 343
§10 Hf(x) = a(x)f(x) + /?(*) 348
§11 /ixf(t)log t x dt = g{x), x 352
§12 Summary 355
Chapter 17. INTEGRAL EQUATIONS 357
§1 Preliminary 357
§2 Classification of integral equations 357
§3 Solution of convolution equations 358
§4 Examples of application 360
§5 Alternative method 362
§6 Volterra integral equations 363
§7 Abel s integral equation 365
XV
§8 J^f(t)e ^ ^dt^g(x),x 0 368
§9 ft f{t) x t adt = g(x), 0 x 1 369
§10 Homogeneous equations 373
§11 Integrals of hyperfunctions 376
§11 Summary 379
Chapter 18. LAPLACE TRANSFORMS 381
§1 Preliminary 381
§2 Laplace transform 381
§3 Various properties of Laplace transforms 385
§4 Sine transform and cosine transform 388
§5 Summary 390
EPILOGUE 393
REFERENCES 395
APPENDICES 397
Appendix A. Symbols 399
Appendix B. Functions, hyperfunctions and generating functions 401
Appendix C. Special functions 403
Appendix D. Power type hyperfunctions with negative integral power 407
Appendix E. Upper type and lower type hyperfunctions 409
Appendix F. Hyperfunctions and generating functions 411
Appendix G. Convolutions 415
Appendix H. Hilbert transforms 421
Appendix I. Fourier transforms 427
Appendix J. Laplace transforms 433
xvi
Appendix K. Cosine transforms and sine transforms 435
Index 437
|
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| id | DE-604.BV009021147 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:28:43Z |
| institution | BVB |
| isbn | 0792315073 |
| language | English Japanese |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005965617 |
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| owner | DE-29T DE-703 DE-11 |
| owner_facet | DE-29T DE-703 DE-11 |
| physical | XIX, 438 S. graph. Darst. |
| publishDate | 1992 |
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| publishDateSort | 1992 |
| publisher | Kluwer |
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| series2 | Mathematics and its applications / Japanese series |
| spelling | Imai, Isao Verfasser aut Ōyō chōkansūron Applied hyperfunction theory by Isao Imai Dordrecht [u.a.] Kluwer 1992 XIX, 438 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications / Japanese series 8 Aus dem Japan. übers. Functies (wiskunde) gtt Functionaalanalyse gtt Hyperfonctions ram Hyperfunctions Hyperfunktion (DE-588)4161056-8 gnd rswk-swf Hyperfunktion (DE-588)4161056-8 s DE-604 Japanese series Mathematics and its applications 8 (DE-604)BV000613779 8 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005965617&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Imai, Isao Applied hyperfunction theory Functies (wiskunde) gtt Functionaalanalyse gtt Hyperfonctions ram Hyperfunctions Hyperfunktion (DE-588)4161056-8 gnd |
| subject_GND | (DE-588)4161056-8 |
| title | Applied hyperfunction theory |
| title_alt | Ōyō chōkansūron |
| title_auth | Applied hyperfunction theory |
| title_exact_search | Applied hyperfunction theory |
| title_full | Applied hyperfunction theory by Isao Imai |
| title_fullStr | Applied hyperfunction theory by Isao Imai |
| title_full_unstemmed | Applied hyperfunction theory by Isao Imai |
| title_short | Applied hyperfunction theory |
| title_sort | applied hyperfunction theory |
| topic | Functies (wiskunde) gtt Functionaalanalyse gtt Hyperfonctions ram Hyperfunctions Hyperfunktion (DE-588)4161056-8 gnd |
| topic_facet | Functies (wiskunde) Functionaalanalyse Hyperfonctions Hyperfunctions Hyperfunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005965617&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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