Towards a symmetrical theory of generalised functions:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
CWI
1991
|
| Schriftenreihe: | CWI tracts
79 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015230867&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | VI, 341 S. |
| ISBN: | 9061963966 |
Internformat
MARC
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| 100 | 1 | |a Lodder, Jan J. |d 1944- |e Verfasser |0 (DE-588)1145866018 |4 aut | |
| 245 | 1 | 0 | |a Towards a symmetrical theory of generalised functions |c J. J. Lodder |
| 264 | 1 | |a Amsterdam |b CWI |c 1991 | |
| 300 | |a VI, 341 S. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | TABLE OF CONTENTS
Preface t
Table of contents tit
1 Introduction 1
1.1 A historical perspective 2
1.2 Introduction to this work 4
1.3 Outline of the contents 5
2 Requirements and properties 7
2.1 Contents of the model 7
2.2 The scalar product 8
2.3 Operators 9
2.4 Products 12
2.5 Convolution products 15
2.6 Summary of the properties 16
2.7 Other properties 17
2.8 Outline of the construction 18
3 A trivial model 21
4 Preliminaries 23
4.1 A preliminary class of (generalised) functions 23
4.2 A preliminary product 27
4.3 Preliminary integration on the preliminary class 27
4.4 A preliminary scalar product 31
4.5 Preliminary operators 31
4.6 Preliminary convolutions 33
4.7 Summary of the preliminaries 33
5 Linear functionals on the preliminary class 35
5.1 The preliminary class as linear functionals 35
5.2 Analytic functionals 36
5.3 Analytic properties of the partie finie 39
5.4 Localized functionals 40
5.5 Translated arguments and exponentials 50
5.6 Linear combinations 52
6 Operators on the linear functionals 55
6.1 Multiplication 56
6.2 Differentiation 58
6.3 The Fourier operator 61
6.4 Operators on ordinary functions 67
7 Operators on the completed preliminary class 71
7.1 Completion of the preliminary class 71
7.2 The remaining scalar products 76
7.3 The operators X and 2? on the preliminary class 77
7.4 The Fourier transform on the preliminary class 80
continued
iv Table of contents
8 Products of linear functionals 83
8.1 Linear functionals on the linear functionals 83
8.2 The pointwise product on PC A 88
8.3 Associativity and commutativity 94
8.4 The convolution product on PCA 97
8.5 Operator and product properties 99
8.6 Summary of the product properties 101
9 Products on the completed preliminary class 103
9.1 Mappings 103
9.2 The pointwise product on PCA X PCA 111
9.3 The convolution product 118
9.4 Uniqueness of the products 120
10 Product properties 123
10.1 Associativity and commutativity of the products 123
10.2 Operator properties of the product 126
10.3 The scalar products on PCA 128
10.4 Summary of the product properties 132
11 The simple model 133
11.1 Local power functions 133
11.2 Scalar products 136
11.3 Operators on the generalised functions 136
11.4 The product of generalised functions 140
11.5 Convolution of generalised functions 144
11.6 The simple model 144
11.7 Generalised functions as ordinary functions 145
11.8 Generalised functions as distributions 147
11.9 Summary of the contents of the model 147
12 Properties and verification 149
12.1 Contents of the model 149
12.2 The scalar product 150
12.3 Integration on the preliminary class 150
12.4 Operator algebra 152
12.5 Operators and products 155
12.6 Selfadjoint properties 156
12.7 Parseval s equality 158
13 Values, limits, and the support 161
13.1 Values of generalised functions 161
13.2 Limiting values of generalised functions 166
13.3 The support of the generalised functions 168
14 Integration 171
14.1 Integration between arbitrary limits 171
14.2 Inverse operators 177
14.3 The fundamental theorem of the calculus 184
continued
Table of contents v
15 Translations of generalised functions 187
15.1 Translations 187
15.2 Wave number translations 190
15.3 Surface terms 191
15.4 Phase plane translations 193
16 Scale transformations and homogeneity 195
16.1 Definition of the scale transformations 195
16.2 Scaling of the scalar product and unitarity 199
16.3 Operator properties of the scale operator 200
16.4 Scaling of the product 202
16.5 Homogeneity of generalised functions 203
17 Indeterminacy calculus 207
17.1 Indeterminate generalised functions 207
17.2 Operators on indeterminate functions 211
17.3 Indeterminate products 212
17.4 More indeterminacy 215
18 Indeterminacy and measurement 219
18.1 Measurement and unit systems 219
18.2 Indeterminate computations 223
18.3 Determinacy 226
19 Convergence of sequences 229
19.1 Sequences of generalised functions 229
19.2 Increasingly peaked sequences 232
19.3 Convergence on PC A 236
19.4 Dirac s limit property 237
19.5 Limits at infinity 240
19.6 Operators and limits 241
19.7 Completed limits 243
19.8 On topology 243
19.9 Conclusion 245
20 Summation and periodic functions 247
20.1 Preliminary summation 247
20.2 Scalar products of sequences 251
20.3 The comb of ^ functions 253
20.4 Periodic functions 256
20.5 Completion of the sequences 259
20.6 Conclusion 260
21 Hilbert transforms and causality 261
21.1 The HUbert transform 261
21.2 Functions of argument (a;±io) 265
21.3 Boundary values of analytic functions 269
21.4 Causality 272
continued
vi Table of contents
22 On regularization 275
22.1 Perturbations in quantum field theory 276
22.2 Standard regularizations 277
22.3 Cutoff regularization 278
22.4 Analytic regularizations 279
22.5 Arbitrariness and standardization 280
22.6 Convolutions and surface terms 281
22.7 Conclusion 283
23 Multiplication and the infinite 285
23.1 A practical impossibility argument 285
23.2 On the nature of the infinite 287
23.3 The impossibility result of Schwartz 288
23.4 On more general theories 288
23.5 Nonstandard analysis 289
23.6 Other distribution products. 291
23.7 Conclusion 291
24 Program, outlook, and conclusion 293
24.1 Larger models 293
24.2 On asymptotics 294
24.3 On foundations 296
24.4 Outlook 297
24.5 Propositions 299
A Laurent coefficients 301
B Binomial coefficients, Pochhammer symbols 302
C Laurent coefficients for Fourier transforms 305
D Generalised zeta functions 311
E Operator algebra 312
P Cantor s staircase function 315
Y Formula index 319
Y.I Definitions 319
Y.2 The operator X 324
Y.3 The operator T 326
Y.4 The operator T 327
Y.5 Mappings 328
Y.6 Translation operators 330
Y.7 Scale transformation operators 330
Y.8 Indeterminacy 331
Y.9 HUbert transforms 332
Y.10 Tables 332
Z Product tables 333
Literature 335
Index 337
Acknowledgements 342
|
| any_adam_object | 1 |
| author | Lodder, Jan J. 1944- |
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| author_facet | Lodder, Jan J. 1944- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV022016233 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T12:46:49Z |
| institution | BVB |
| isbn | 9061963966 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015230867 |
| oclc_num | 24870407 |
| open_access_boolean | |
| owner | DE-706 DE-188 |
| owner_facet | DE-706 DE-188 |
| physical | VI, 341 S. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | CWI |
| record_format | marc |
| series | CWI tracts |
| series2 | CWI tracts |
| spellingShingle | Lodder, Jan J. 1944- Towards a symmetrical theory of generalised functions CWI tracts Distribution (théorie des probabilités) ram Theory of distributions (Functional analysis) Symmetrie (DE-588)4058724-1 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| subject_GND | (DE-588)4058724-1 (DE-588)4070505-5 |
| title | Towards a symmetrical theory of generalised functions |
| title_auth | Towards a symmetrical theory of generalised functions |
| title_exact_search | Towards a symmetrical theory of generalised functions |
| title_full | Towards a symmetrical theory of generalised functions J. J. Lodder |
| title_fullStr | Towards a symmetrical theory of generalised functions J. J. Lodder |
| title_full_unstemmed | Towards a symmetrical theory of generalised functions J. J. Lodder |
| title_short | Towards a symmetrical theory of generalised functions |
| title_sort | towards a symmetrical theory of generalised functions |
| topic | Distribution (théorie des probabilités) ram Theory of distributions (Functional analysis) Symmetrie (DE-588)4058724-1 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| topic_facet | Distribution (théorie des probabilités) Theory of distributions (Functional analysis) Symmetrie Distribution Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015230867&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001902011 |
| work_keys_str_mv | AT lodderjanj towardsasymmetricaltheoryofgeneralisedfunctions |