An introduction to frames and Riesz Bases:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Cham]
Birkhäuser
[2016]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | xxv, 704 Seiten Diagramme 23.5 cm x 15.5 cm, 0 g |
| ISBN: | 3319256114 9783319256115 |
Internformat
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| 100 | 1 | |a Christensen, Ole |d 1966- |e Verfasser |0 (DE-588)12445836X |4 aut | |
| 245 | 1 | 0 | |a An introduction to frames and Riesz Bases |c Ole Christensen |
| 250 | |a Second edition | ||
| 264 | 1 | |a [Cham] |b Birkhäuser | |
| 264 | 1 | |c [2016] | |
| 264 | 4 | |c © 2016 | |
| 300 | |a xxv, 704 Seiten |b Diagramme |c 23.5 cm x 15.5 cm, 0 g | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Applied and numerical harmonic analysis | |
| 650 | 0 | 7 | |a Riesz-Basis |0 (DE-588)4688993-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Frame |g Mathematik |0 (DE-588)4528312-6 |2 gnd |9 rswk-swf |
| 653 | |a Graduate | ||
| 653 | |a Wavelets | ||
| 653 | |a Frame Theory | ||
| 653 | |a LCA Groups | ||
| 653 | |a Hilbert Spaces | ||
| 653 | |a Vector Spaces | ||
| 653 | |a Riesz Bases | ||
| 653 | |a Gabor Frames | ||
| 653 | |a Generalized Shift-invariant Systems | ||
| 689 | 0 | 0 | |a Frame |g Mathematik |0 (DE-588)4528312-6 |D s |
| 689 | 0 | 1 | |a Riesz-Basis |0 (DE-588)4688993-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 710 | 2 | |a Springer International Publishing |0 (DE-588)1064344704 |4 pbl | |
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Datensatz im Suchindex
| _version_ | 1806333138979258368 |
|---|---|
| adam_text |
Contents
Preface xi
1 Frames in Finite-Dimensional
Inner Product Spaces 1
1.1 Some Basic Facts About Frames . 3
1.2 Extensions to Tight Frames and Dual Frames. 10
1.3 Frame Bounds and Frame Algorithms. 12
1.4 Frames in Cn . 16
1.5 Frames and the Discrete Fourier Transform. 21
1.6 Pseudo-inverses and the Singular Value Decomposition . 28
1.7 Finite-Dimensional Function Spaces. 33
1.8 Fusion Frames. 37
1.9 Applications of Finite Frames. 38
1.10 Remarks on Recent Frame Constructions. 42
1.11 Exercises. 44 2 * * *
2 Infinite-Dimensional Vector Spaces
and Sequences 47
2.1 Banach Spaces and Sequences. 47
2.2 Operators on Banach Spaces. 50
2.3 Hilbert Spaces. 52
xix
XX
Contents
2.4 Operators on Hilbert Spaces. 53
2.5 The Pseudo-inverse Operator . 56
2.6 A Moment Problem. 58
2.7 The Spaces L*(R), L2(R), £P(N), and £2(N) 59
2.8 The Fourier Transform and Convolution. 51
2.9 Operators on T2(R). 63
2.10 Exercises. 65
3 Bases 67
3.1 Bases in Banach Spaces. 68
3.2 Bessel Sequences in Hilbert Spaces . 73
3.3 Bases and Biorthogonal Systems in H. 76
3.4 Orthonormal Bases. 79
3.5 The Gram Matrix. 82
3.6 Riesz Bases . 86
3.7 Riesz Sequences. 92
3.8 Fourier Series and Gabor Bases. 94
3.9 Wavelet Bases. 97
3.10 Sampling and Analog-Digital Conversion. 102
3.11 Exercises. 105
4 Bases and Their Limitations 109
4.1 Bases and the Expansion Property. 109
4.2 Gabor Systems and the Balian-Low Theorem. 113
4.3 Bases and Wavelets. 115
4.4 General Shortcomings. 118
4.5 Exercises. 118
5 Frames in Hilbert Spaces 119
5.1 Frames and Their Properties. 120
5.2 Frame Sequences . 127
5.3 Frames and Operators. 128
5.4 Frames and Bases. 130
5.5 Characterization of Frames. 136
5.6 Continuous Frames. 145
5.7 Exercises. 148 6
6 Tight Frames and Dual Frame Pairs 153
6.1 Tight Frames . 154
6.2 Extension of Bessel Sequences to Tight Frames . 155
6.3 The Dual Frames. 156
6.4 Extension Problems for Bessel Sequences. 161
6.5 Approximately Dual Frames. 162
6.6 Exercises. 163
Contents
xxi
7 Frames Versus Riesz Bases 165
7 i Conditions for a Frame Being a Riesz Basis . 166
7 2 Frames and Their Subsequences. 168
7 3 Riesz Frames and Near-Riesz Bases. 170
7.4 Frames Containing a Riesz Basis. 171
7.5 A Frame Which Does Not Contain a Basis. 173
7.6 A Moment Problem. 179
7.7 The Feichtinger Conjecture. 181
7.8 Exercises. 181
8 Selected Topics in Frame Theory 183
8.1 G-Frames. 184
8.2 Localization of Frames. 188
8.3 The R-Duals of a Frame. 190
8.4 Frame Theory via Unbounded Operators. 194
8.5 Frames and Signal Processing. 196
8.6 Exercises. 198
9 Frames of Translates 199
9.1 Sequences in WLd. 200
9.2 Frames of Translates. 203
9.3 Frames of Integer-Translates. 210
9.4 The Canonical Dual Frame. 213
9.5 Frames of Translates and Oblique Duals. 218
9.6 Irregular Frames of Translates. 226
9.7 Sampling Theory and Applications. 229
9.8 Frames of Exponentials . 232
9.9 Exercises. 238
10 Shift-Invariant Systems in L2(R) 241
10.1 Frame Properties of Shift-Invariant Systems. 241
10.2 Representations of the Flame Operator. 253
10.3 Exercises. 256
11 Gabor Flames in L2(R) 257
11.1 Continuous Representations. 259
11.2 Gabor Frames {jE/m6Tna#}m?n€Z for L2(R). 262
11.3 Necessary Conditions. 266
llA Sufficient Conditions. 268
11.5 The Wiener Space W. 275
11.6 The Frame Set and Special Functions. 279
11.7 Gabor Frames Generated by B-Splines. 283
11.8 Exercises . . 285
xxii Contents
12 Gabor Frames and Duality 287
12.1 Popular Gabor Conditions. 288
12.2 Representations of the Gabor Frame Operator
and Duality. 289
12.3 The Duals of a Gabor Frame. 295
12.4 The Canonical Dual Window. 302
12.5 Explicit Construction of Dual Frame Pairs. 306
12.6 Windows with Short Support and High Regularity . 310
12.7 Extension of Bessel Sequences to Dual Pairs. 318
12.8 Approximately Dual Gabor Frames. 319
12.9 Tight Gabor frames. 320
12.10 Exercises. 324
13 Selected Topics on Gabor Frames 327
13.1 The Duality Principle. 328
13.2 The Zak Transform. 330
13.3 The Lattice Parameters. 336
13.4 Irregular Gabor Systems. 340
13.5 Localized Gabor Frames. 345
13.6 Wilson Bases. 347
13.7 Time—Frequency Localization of Gabor Expansions . . . 348
13.8 Applications of Gabor Frames. 354
13.9 Exercises. 356
14 Gabor Frames in £2(Z), L2(0, L), CL 359
14.1 Translation and Modulation on £2(Z). 360
14.2 Dual Gabor Frames in £2(Z). 361
14.3 Dual Gabor Frames in f2(Z) Through Sampling . 362
14.4 Discrete Gabor Frames Through Sampling. 365
14.5 Gabor Frames for L2(0yL) via Periodization. 372
14.6 Gabor Frames in CL . 375
14.7 Shift-Invariant Systems. 380
14.8 Frames in £2(Z) and Filter Banks. 381
14.9 Gabor frames in £2(Xd). 383
14.10 Exercises. 383
15 General Wavelet Frames in L2(R) 385
15.1 The Continuous Wavelet Transform. 387
15.2 Sufficient and Necessary Conditions. 389
15.3 Dual Pairs of Wavelet Frames. 401
15.4 Exercises . 405
16 Dyadic Wavelet Frames for £2(M) 407
16.1 Wavelet Frames and Their Duals . 408
16.2 Tight Wavelet Frames. . 411
Contents xxiii
16.3 Wavelet Frame Sets. 412
16.4 Frames and Multiresolution Analysis. 415
16.5 Exercises. 415
17 Frame Multiresolution Analysis 417
17.1 Frame Multiresolution Analysis. 418
17.2 Sufficient Conditions. 419
17.3 Relaxing the Conditions. 423
17.4 Construction of Frames. 425
17.5 Frames with Two Generators. 442
17.6 Some Limitations. 444
17.7 Exercises. 444
18 Wavelet Frames via Extension Principles 445
18.1 The General Setup . 446
18.2 The Unitary Extension Principle. 448
18.3 Applications to 5-splines I. 454
18.4 The Oblique Extension Principle . 459
18.5 Fewer Generators. 463
18.6 Applications to 5-splines II. 466
18.7 Approximation Orders. 471
18.8 Construction of Pairs of Dual Wavelet Frames. 473
18.9 Applications to 5-splines III. 476
18.10 The MRA Literature and Applications . 477
18.11 Exercises. 478
19 Selected Topics on Wavelet Frames 479
19.1 Irregular Wavelet Frames. 480
19.2 Oversampling of Wavelet Frames . 482
19.3 An Open Extension Problem. 483
19.4 The Signal Processing Perspective. 485
19.5 Exercises. 492
20 Generalized Shift-Invariant Systems in L2(Md) 493
20.1 Analysis in M.d and Notation. 494
20.2 The Case of One Generator . 497
20.3 Frames with Multiple Generators. 501
20.4 Dual Pairs of F’ames with Multiple Generators. 503
20.5 Gabor Systems in 52(Md). 507
20.6 Wavelet Systems in L2(Rd). 511
20.7 Exercises. 516
21 Frames on Locally Compact Abelian Groups 519
21.1 LCA Groups. 521
21.2 Fourier Analysis on LCA Groups . 526
XXIV
Contents
21.3 Gabor Systems on LCA Groups. 530
21.4 Basic Frame Calculations in L2Xg)*. 532
21.5 Explicit Gabor Frame Constructions in L2{G) . 537
21.6 GSI Systems on LCA Groups. 544
21.7 Generalized Translation-Invariant Systems. 548
21.8 Co-compact Gabor Systems. 554
21.9 Exercises. 556
22 Perturbation of Frames 557
22.1 A Paley-Wiener Theorem for Frames. 558
22.2 Compact Perturbation . 565
22.3 Perturbation of Frame Sequences. 567
22.4 Perturbation of Gabor frames. 570
22.5 Perturbation of Wavelet Frames. 573
22.6 Perturbation of the Haar Wavelet. 574
22.7 Exercises. 574
23 Approximation of the Inverse Frame Operator 577
23.1 The First Approach. 578
23.2 The Casazza-Christensen Method. 582
23.3 Convergence Estimates for Localized Frames. 589
23.4 Applications to Gabor Frames. 591
23.5 Integer Oversampled Gabor Frames. 594
23.6 The Finite Section Method. 595
23.7 The Finite Section Method for Gabor Frames. 598
23.8 Exercises. 599
24 Expansions in Banach Spaces 601
24.1 Representations of Locally Compact Groups. 602
24.2 Feichtinger-Grochenig Theory. 606
24.3 Banach Frames. 611
24.4 p- frames. 616
24.5 Gabor Systems and Wavelets in Lp(R) and Related
Spaces. 619
24.6 Exercises. 620
A Appendix 623
A.l Linear Algebra. 623
A. 2 Integration. 624
A.3 Locally Compact Groups. 625
A.4 Some Infinite-Dimensional Vector Spaces. 626
A.5 Modulation Spaces. 627
A.6 Feichtinger’s algebra So. 629
A.7 Some Special Functions. . 632
Contents
xxv
A.8 B-Splines.
A. 9 Exponential B-Splines .
A.10 Splines on Locally Compact Abelian Groups .
A. 11 Notes.
633
639
641
643
List of Symbols
645
References
647
Index
691 |
| any_adam_object | 1 |
| author | Christensen, Ole 1966- |
| author_GND | (DE-588)12445836X |
| author_facet | Christensen, Ole 1966- |
| author_role | aut |
| author_sort | Christensen, Ole 1966- |
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| building | Verbundindex |
| bvnumber | BV043324351 |
| classification_rvk | SK 470 SK 600 SK 450 |
| ctrlnum | (OCoLC)951596979 (DE-599)DNB1076584055 |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-08-03T02:52:01Z |
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| institution_GND | (DE-588)1064344704 |
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| spelling | Christensen, Ole 1966- Verfasser (DE-588)12445836X aut An introduction to frames and Riesz Bases Ole Christensen Second edition [Cham] Birkhäuser [2016] © 2016 xxv, 704 Seiten Diagramme 23.5 cm x 15.5 cm, 0 g txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis Riesz-Basis (DE-588)4688993-0 gnd rswk-swf Frame Mathematik (DE-588)4528312-6 gnd rswk-swf Graduate Wavelets Frame Theory LCA Groups Hilbert Spaces Vector Spaces Riesz Bases Gabor Frames Generalized Shift-invariant Systems Frame Mathematik (DE-588)4528312-6 s Riesz-Basis (DE-588)4688993-0 s DE-604 Springer International Publishing (DE-588)1064344704 pbl Erscheint auch als Online-Ausgabe 978-3-319-25613-9 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=b8f90f0e15ab4fde8ca92f05f427a087&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028744695&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Christensen, Ole 1966- An introduction to frames and Riesz Bases Riesz-Basis (DE-588)4688993-0 gnd Frame Mathematik (DE-588)4528312-6 gnd |
| subject_GND | (DE-588)4688993-0 (DE-588)4528312-6 |
| title | An introduction to frames and Riesz Bases |
| title_auth | An introduction to frames and Riesz Bases |
| title_exact_search | An introduction to frames and Riesz Bases |
| title_full | An introduction to frames and Riesz Bases Ole Christensen |
| title_fullStr | An introduction to frames and Riesz Bases Ole Christensen |
| title_full_unstemmed | An introduction to frames and Riesz Bases Ole Christensen |
| title_short | An introduction to frames and Riesz Bases |
| title_sort | an introduction to frames and riesz bases |
| topic | Riesz-Basis (DE-588)4688993-0 gnd Frame Mathematik (DE-588)4528312-6 gnd |
| topic_facet | Riesz-Basis Frame Mathematik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=b8f90f0e15ab4fde8ca92f05f427a087&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028744695&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT christensenole anintroductiontoframesandrieszbases AT springerinternationalpublishing anintroductiontoframesandrieszbases |