Generalized wavelets and hypergroups:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
Gordon and Breach Science Publ.
1997
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 354 S. |
| ISBN: | 9056990802 |
Internformat
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| 245 | 1 | 0 | |a Generalized wavelets and hypergroups |c Khalifa Trimèche |
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| 300 | |a XII, 354 S. | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | IMAGE 1
GENERALIZED WAVELETS AND HYPERGROUPS
KHALIFA TRIMECHE
GORDON AND BREACH SCENCE PUBLISHERS AUSTRALIA * CANADA * CHINA * FRANCE
* GERMANY * INDIA * JAPAN * LUXEMBOURG MALAYSIA * THE NETHERLANDS *
RUSSIA * SINGAPORE * SWITZERLAND * THAILAND UNITED KINGDOM
I
IMAGE 2
CONTENTS
INTRODUCTION XI
1 PRODUCT FORMULAS AND GENERAHZED HYPERGROUPS 1
INTRODUCTION 1
1.1. NOTATIONS 2
LH. GENERAHZED TRANSLATION OPERATORS 3
L.III. GENERAHZED HYPERGROUPS AND HARMONIC ANALYSIS 6
L.NI.L. GENERAHZED HYPERGROUPS 6
1.IUE.2. HARMONIC ANALYSIS ON GENERAHZED HYPERGROUPS 8
1.IV. GENERAHZED HYPERGROUPS ASSOCIATED WITH JACOBI FUNCTIONS 11
L.IV.L. THE DIRECTCASE 11
1.IV.2. THE DUAL CASE 14
I.V. GENERAHZED HYPERGROUPS ASSOCIATED WITH THE UNIVERSAL COVERING
GROUPOF G = SU(N-L, 1) 17
L.V.L. PRELIMINARIES ON THE STRUCTURE OF GE, L = 0 , 1, 2, 17
L.V.2. THE SPHERICAL FUNCTIONS ON G^/K, I = 0 , 1, 2, 20
I.V.3. PRODUCT FORMULAS FOR THE SPHERICAL FUNCTIONS ON G F /K, = 0,1,2,
21
L.V.4. GENERAHZED HYPERGROUPS ASSOCIATED WITH G^, T - 0 , 1, 2, 24
L.VI. GENERAHZED HYPERGROUPS ASSOCIATED WITH DIFFERENTIAL OPERATORS 27
L.VII. GENERAHZED HYPERGROUPS ASSOCIATED WITH PARTIAL DIFFERENTIAL
OPERATORS .. 35
2 HYPERGROUPS 43
INTRODUCTION 43
V
IMAGE 3
VI CONTENTS
2.1. HISTORY OF HYPERGROUPS 44
2.II. DEFINITION AND PROPERTIES OF HYPERGROUPS 44
2.11.1. DEFINITION OF A HYPERGROUP 44
2.UE.2. HYPERGROUP ISOMORPHISM 45
2.UE.3. PRODUCT OF HYPERGROUPS 46
2.UE.4. HAAR MEASURE ON A HYPERGROUP 46
2.H.5. EXAMPLES OF HYPERGROUPS 47
2.HI. GENERALIZED TRANSLATION OPERATORS AND GENERALIZED CONVOLUTION
PRODUCT ON COMMUTATIVE HYPERGROUPS 50
2.IV. GENERALIZED FOURIER TRANSFORM ON COMMUTATIVE HYPERGROUPS 52
2.V. THE DUAL OF A HYPERGROUP 56
2.VI. DOUBLE COSET HYPERGROUPS 56
2.VII. HYPERGROUPS ARISING FROM THE HEISENBERG GROUP 58
2.VIII. HYPERGROUPS ASSOCIATED WITH PARTIAL DIFFERENTIAL OPERATORS 63
2.VHI.1. THE HYPERGROUP OF THE EXTERIOR OF THE UNIT DISC 63
2.VIUE.2. THE HYPERGROUP OF THE UNIT DISC 64
2.VIII.3. HYPERGROUPS PRODUCT ASSOCIATED WITH PARTIAL DIFFERENTIAL
OPERATORS 67
3 WAVELETS AND THE WINDOWED SPHERICAL FOURIER TRANSFORM ON GELFAND PAIRS
.. 71
INTRODUCTIAN 71
3.1. GELFAND PAIRS 72
3.1.1. DEFINITION AND EXAMPLES OF GELFAND PAIRS 72
3.1.2. SPHERICAL FUNCTIONS 74
3.II. WAVELETS AND THE WINDOWED SPHERICAL FOURIER TRANSFORM ON GELFAND
PAIRS (G,K) 79
3.II. 1. PLANCHEREI AND PARSEVAL FORMULAS FOR THE WINDOWED SPHERICAL
FOURIER TRANSFORM ON GELFAND PAIRS (G,K) 82
3.11.2. CHARACTERIZATION OF THE IMAGE OF THE WINDOWED SPHERICAL FOURIER
TRANSFORM ON GELFAND PAIRS (G,K) 83
3.11.3. INVERSION FORMULA FOR THE WINDOWED SPHERICAL FOURIER TRANSFORM
ON GELFAND PAIRS (G,K) 84
4 GENERALIZED WAVELETS AND GENERALIZED CONTINUOUS WAVELET TRANSFORMS
ON HYPERGROUPS 87
INTRODUCTIAN 87
4.1. GENERALIZED WAVELETS ON HYPERGROUPS SATISFYING FIRST TYPE
CONDITIONS . .. 89
4.UE. GENERALIZED CONTINUOUS WAVELET TRANSFORM ON HYPERGROUPS SATISFYING
FIRST TYPE CONDITIONS 93
IMAGE 4
CONTENTS VII
4.IUE. GENERALIZED WAVELETS AND GENERALIZED CONTINUOUS WAVELET TRANSFORM
ON HYPERGROUPS SATISFYING SECOND TYPE CONDITIONS 98
4.III.1. GENERALIZED WAVELETS 98
4.HI.2. GENERALIZED CONTINUOUS WAVELET TRANSFORM 99
4.IV. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS ON HYPERGROUPS
SATISFYING FIRST TYPE CONDITIONS 100
4.IV.1. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS 100
4.IV.2. GENERALIZED PARTIAL RECONSTRUCTIONS WHEN S = M 103
4.V. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS ON HYPERGROUPS
SATISFYING SECOND TYPE CONDITIONS . 105
4.V.I. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS 105
4.V.2. GENERALIZED PARTIAL RECONSTRUCTIONS WHEN S = M 106
5 GENERALIZED WAVELETS AND GENERALIZED CONTINUOUS WAVELET TRANSFORMS ON
SEMISIMPLE LIE GROUPS AND ON CARTAN MOTION GROUPS 109
INTRODUCTIAN 109
5.1. PRELIMINARIES ON REAL SEMISIMPLE LIE GROUPS 110
5.II. THE SPHERICAL FOURIER TRANSFORM AND THE ABEL TRANSFORM ON G 114
5.11.1. THE SPHERICAL FOURIER TRANSFORM OF FUNCTIONS 116
5.11.2. THE ABEL TRANSFORM ON G 117
5.III. GENERALIZED TRANSLATION OPERATORS ON G AND CONVOLUTION 118
5.HI.1. GENERALIZED TRANSLATION OPERATORS ON G 118
5.O.2. CONVOLUTION 120
5.IV. GENERALIZED WAVELETS ON G 121
5.V. GENERALIZED CONTINUOUS WAVELET TRANSFORM ON G 129
5.VI. PRELIMINARIES ON THE CARTAN MOTION GROUPS 136
5.VII. THE SPHERICAL FOURIER TRANSFORM AND THE ABEL TRANSFORM ON P 138
5.VII.1. THE SPHERICAL FOURIER TRANSFORM OF FUNCTIONS 138
5.VII.2. THE SPHERICAL FOURIER TRANSFORM OF DISTRIBUTIONS 139
5.VII.3. THE ABEL TRANSFORM ON P 140
5.VHI. CONVOLUTION 141
5.IX. GENERALIZED WAVELETS AND THE GENERALIZED CONTINUOUS WAVELET
TRANSFORM ON P 143
5.IX.1. GENERALIZED WAVELETS ON P 143
5.IX.2. GENERALIZED CONTINUOUS WAVELET TRANSFORM ON P 145
6 HARMONIE ANALYSIS, GENERALIZED WAVELETS AND THE GENERALIZED
CONTINUOUS WAVELET TRANSFORM ON CHEBLI-TRIMECHE HYPERGROUPS 147
INTRODUCTIAN 147
6.1. EIGENFUNCTIONS OF THE OPERATOR A 148
*
IMAGE 5
VIII CONTENTS
6.1.1. THE OPERATOR A 148
6.1.2. GENERALIZED MEHLER FORMULA 149
6.1.3. GENERALIZED RIEMANN-LIOUVILLE INTEGRAL TRANSFORM 153
6.1.4. SPECTRAL MEASURE 155
6.II. HARMONIE ANALYSIS ASSOCIATED WITH THE OPERATOR A 163
6.II.1. GENERALIZED FOURIER TRANSFORM 163
6.UE.2. GENERALIZED WEYL INTEGRAL TRANSFORM 171
6.UE.3. GENERALIZED TRANSLATION OPERATORS ASSOCIATED WITH THE OPERATOR A
175 6.II.4. GENERALIZED FOURIER TRANSFORM OF DISTRIBUTIONS 184
6.III. GENERALIZED WAVELETS AND THE GENERALIZED CONTINUOUS WAVELET
TRANSFORM ON R+ 190
6.111.1. GENERALIZED WAVELETS ON K+ 190
6.111.2. GENERALIZED CONTINUOUS WAVELET TRANSFORM ON K+ 196
7 HARMONIE ANALYSIS, GENERALIZED WAVELETS AND THE GENERALIZED CONTINUOUS
WAVELET TRANSFORM ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR 207
INTRODUCTION 207
7.1. SPHERICAL MEAN OPERATOR AND ITS DUAL 208
7.1.1. SPHERICAL MEAN OPERATOR 208
7.1.2. THE DUAL OF THE SPHERICAL MEAN OPERATOR 210
7.1.3. INTEGRAL TRANSFORMS WITH KERNEL DEFINED BY A BESSEL FUNETION AND
APPLICATIONS 212
7.H. GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH THE SPHERICAL MEAN
OPERATOR 218
7.11.1. GENERALIZED FOURIER TRANSFORM OF FUNETIONS 218
7.11.2. GENERALIZED FOURIER TRANSFORM OF DISTRIBUTIONS 226
7.III. TRANSMUTATION OPERATORS 230
7.IV. GENERALIZED CONVOLUTION PRODUET 232
7.V. GENERALIZED WAVELETS ON [0, +OO[XR 236
7.VI. GENERALIZED CONTINUOUS WAVELET TRANSFORM 238
8 HARMONIE ANALYSIS, GENERALIZED WAVELETS AND THE GENERALIZED CONTINUOUS
WAVELET TRANSFORM ASSOCIATED WITH LAGUERRE FUNCTIONS 243
INTRODUCTION 243
8.1. HARMONIE ANALYSIS ASSOCIATED WITH LAGUERRE FUNETIONS 244
8.1.1. LAGUERRE FUNETIONS 244
8.1.2. GENERALIZED TRANSLATION OPERATORS ASSOCIATED WITH LAGUERRE
FUNETIONS 247
8.1.3. GENERALIZED CONVOLUTION PRODUET ASSOCIATED WITH LAGUERRE
FUNETIONS 250
IMAGE 6
CONTENTS IX
8.1.4. GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH LAGUERRE FUNCTIONS
252
8.IL GENERALIZED WAVELETS AND THE GENERALIZED CONTINUOUS WAVELET
TRANSFORM ASSOCIATED WITH LAGUERRE FUNCTIONS 263
8.11.1. GENERALIZED WAVELETS ON X 263
8.11.2. GENERALIZED CONTINUOUS WAVELET TRANSFORM ON X 265
8.III. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS AND PARTIAL
RECONSTRUCTIONS ASSOCIATED WITH LAGUERRE FUNCTIONS 268
8.III.1. GENERALIZED CONTINUOUS MULTISCALE ANALYSIS ON X 268
8.IH.2. GENERALIZED PARTIAL RECONSTRUCTIONS ON X 269
9 GENERALIZED RADON TRANSFORMS ON GENERALIZED HYPERGROUPS 271
INTRODUCTION 271
9.1. GENERAL METHODS TO CONSTRUCT GENERALIZED RADON TRANSFORMS AND THEIR
DUALS ON GENERALIZED HYPERGROUPS 272
9.1.1. FIRST METHOD 272
9.1.2. SECOND METHOD 275
9.11. GENERAL METHODS TO OBTAIN INVERSION FORMULAS FOR THE GENERALIZED
RADON TRANSFORMS AND THEIR DUALS 277
9.II. 1. GENERALIZED RADON TRANSFORM AND ITS DUAL CONSTRUCTED BY THE
FIRST METHOD 277
9.UE.2. GENERALIZED RADON TRANSFORM CONSTRUCTED BY THE SECOND METHOD 279
9.IIL GENERALIZED RADON TRANSFORM AND ITS DUAL ON GENERALIZED
HYPERGROUPS ASSOCIATED WITH JACOBI FUNCTIONS 280
9.IUE.1. GENERALIZED RADON TRANSFORM AND ITS DUAL 280
9.IUE.2. INVERSION FORMULAS FOR THE TRANSFORMS XA,SS AND XA,^ 282
9.IV. GENERALIZED RADON TRANSFORM AND ITS DUAL ON GENERALIZED
HYPERGROUPS ASSOCIATED WITH PARTIAL DIFFERENTIAL OPERATORS 284
9.IV.1. CASE OF GENERALIZED HYPERGROUPS 284
9.IV.2. CASE OF HYPERGROUPS 287
9.V. GENERALIZED RADON TRANSFORM AND ITS DUAL ON DOUBLE COSET
HYPERGROUPS . 294 9.V.I. GENERALIZED RADON TRANSFORM AND ITS DUAL 294
9.V.2. INVERSION FORMULAS FOR THE OPERATORS A AND A* 295
9.VI. GENERALIZED RADON TRANSFORM AND ITS DUAL ON CHEBLI-TRIMECHE
HYPERGROUPS 296
9.VI.1. GENERALIZED RADON TRANSFORM AND ITS DUAL 296
9.VI.2. INVERSION FORMULAS FOR THE OPERATORS X AND X 296
9.VUE. GENERALIZED RADON TRANSFORM ON THE HYPERGROUP ASSOCIATED WITH THE
SPHERICAL MEAN OPERATOR 301
9.VII.1. GENERALIZED RADON TRANSFORM AND ITS DUAL 301
FC
IMAGE 7
*
X CONTENTS
9.VUE.2. INVERSION FORMULAS FOR THE OPERATORS R AND R 301
9.VIII. GENERALIZED RADON TRANSFORM ON THE HYPERGROUP ASSOCIATED WITH
LAGUERRE FUNCTIONS 313
9.VIUE.1. GENERALIZED RADON TRANSFORM 313
9.VIUE.2. INVERSION FORMULA FOR THE OPERATORS DIA 314
10 INVERSION OF GENERALIZED RADON TRANSFORM USING GENERALIZED WAVELETS .
.. 319
INTRODUCTIAN 319
10.1. GENERAL METHODS TO CONSTRUCT THE INVERSE OF THE GENERALIZED RADON
TRANSFORMS AND THEIR DUALS USING GENERALIZED WAVELETS 320
10.1.1. GENERALIZED RADON TRANSFORM AND ITS DUAL CONSTRUCTED BY THE
FIRST METHOD GIVEN IN THE SUBSECTION 9.1.1 320
10.1.2. GENERALIZED RADON TRANSFORM CONSTRUCTED BY THE SECOND METHOD
GIVEN IN THE SUBSECTION 9.1.2 322
10.11. INVERSION OF THE GENERALIZED RADON TRANSFORM AND ITS DUAL ON
DOUBLE COSET HYPERGROUPS USING GENERALIZED WAVELETS 322
10.111. INVERSION OF THE GENERALIZED RADON TRANSFORM AND ITS DUAL ON
CHEBLI-TRIMECHE HYPERGROUPS USING GENERALIZED WAVELETS 325
10.111.1. INVERSION BY THE METHOD GIVEN IN THE SUBSECTION 10.1.1 325
10.111.2. INVERSION BY THE CALDERON FORMULA ASSOCIATED WITH THE OPERATOR
A 328
10.IV. INVERSION OF THE GENERALIZED RADON TRANSFORM AND ITS DUAL ON THE
HYPERGROUP ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR 332
10.IV.1. INVERSION FORMULA FOR THE CLASSICAL WAVELET TRANSFORM ON [0,
+OO[XR 333
10.IV.2. INVERSION OF THE GENERALIZED RADON TRANSFORM AND ITS DUAL . . .
333
10.V. INVERSION OF THE GENERALIZED RADON TRANSFORM ON THE HYPERGROUP
ASSOCIATED WITH LAGUERRE FUNCTIONS 336
BIBLIOGRAPHY 339
INDEX 351
|
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| id | DE-604.BV012165583 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:22:51Z |
| institution | BVB |
| isbn | 9056990802 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008241163 |
| oclc_num | 38915109 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-703 DE-824 DE-83 |
| owner_facet | DE-91G DE-BY-TUM DE-703 DE-824 DE-83 |
| physical | XII, 354 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Gordon and Breach Science Publ. |
| record_format | marc |
| spelling | Trimèche, Khalifa Verfasser aut Generalized wavelets and hypergroups Khalifa Trimèche Amsterdam Gordon and Breach Science Publ. 1997 XII, 354 S. txt rdacontent n rdamedia nc rdacarrier Hypergroups Wavelets (Mathematics) Hypergruppe (DE-588)4561851-3 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf Wavelet (DE-588)4215427-3 s DE-604 Hypergruppe (DE-588)4561851-3 s GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008241163&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Trimèche, Khalifa Generalized wavelets and hypergroups Hypergroups Wavelets (Mathematics) Hypergruppe (DE-588)4561851-3 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4561851-3 (DE-588)4215427-3 |
| title | Generalized wavelets and hypergroups |
| title_auth | Generalized wavelets and hypergroups |
| title_exact_search | Generalized wavelets and hypergroups |
| title_full | Generalized wavelets and hypergroups Khalifa Trimèche |
| title_fullStr | Generalized wavelets and hypergroups Khalifa Trimèche |
| title_full_unstemmed | Generalized wavelets and hypergroups Khalifa Trimèche |
| title_short | Generalized wavelets and hypergroups |
| title_sort | generalized wavelets and hypergroups |
| topic | Hypergroups Wavelets (Mathematics) Hypergruppe (DE-588)4561851-3 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Hypergroups Wavelets (Mathematics) Hypergruppe Wavelet |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008241163&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT trimechekhalifa generalizedwaveletsandhypergroups |