Frames and bases: an introductory course
"During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classi...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston ; Basel ; Berlin
Birkhäuser
2008
|
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classical sufficient conditions for obtaining a frame. Consequently, there is a need for an updated book on frames, which moves the focus from the classical approach to a more constructive one. Based on a streamlined presentation of the author's previous work, An Introduction to Frames and Riesz Bases, this new textbook fills a gap in the literature, developing frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field." -- Book cover. |
| Beschreibung: | xviii, 313 Seiten Diagramme |
| ISBN: | 9780817646776 9780817646783 |
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| 520 | 3 | |a "During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classical sufficient conditions for obtaining a frame. Consequently, there is a need for an updated book on frames, which moves the focus from the classical approach to a more constructive one. Based on a streamlined presentation of the author's previous work, An Introduction to Frames and Riesz Bases, this new textbook fills a gap in the literature, developing frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field." -- Book cover. | |
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Datensatz im Suchindex
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| adam_text | Contents
ANHA Series Preface vii
Preface xv
1 Frames in Finite-dimensional
Inner Product Spaces 1
1.1 Basic frames theory..................... 2
1.2 Frames in C ........................ 12
1.3 The discrete Fourier transform............... 17
1.4 Pseudo-inverses and the singular value decomposition . . 20
1.5 Applications in signal transmission............ 25
1.6 Exercises........................... 30
2 Infinite-dimensional Vector Spaces
and Sequences 33
2.1 Normed vector spaces and sequences........... 33
2.2 Operators on Banach spaces................ 36
2.3 Hubert spaces........................ 37
2.4 Operators on Hubert spaces................ 38
2.5 The pseudo-inverse operator................ 40
2.6 A moment problem..................... 42
2.7 The spaces LP(K), L2(R), and £2(N) ........... 43
2.8 The Fourier transform and convolution.......... 46
2.9 Operators on L2(R)..................... 47
2.10 Exercises........................... 49
xii Contents
3 Bases 51
3.1 Bessel sequences in Hilbert spaces............. 52
3.2 General bases and orthonormal bases........... 55
3.3 Riesz bases.......................... 59
3.4 The Gram matrix...................... 64
3.5 Fourier series and trigonometric polynomials....... 69
3.6 Wavelet bases........................ 72
3.7 Bases in Banach spaces................... 78
3.8 Sampling and analog-digital conversion.......... 83
3.9 Exercises........................... 86
4 Bases and their Limitations 89
4.1 Bases in L2(0,1) and in general Hilbert spaces...... 89
4.2 Gabor bases and the Balian-Low Theorem........ 92
4.3 Bases and wavelets..................... 93
5 Frames in Hilbert Spaces 97
5.1 Frames and their properties................ 98
5.2 Frames and Riesz bases................... 105
5.3 Frames and operators.................... 108
5.4 Characterization of frames................. 112
5.5 Various independency conditions.............. 116
5.6 Perturbation of frames................... 121
5.7 The dual frames....................... 126
5.8 Continuous frames ..................... 129
5.9 Frames and signal processing................ 130
5.10 Exercises........................... 133
6 B-splines 139
6.1 The B-splines........................ 140
6.2 Symmetric B-splines .................... 146
6.3 Exercises........................... 148
7 Frames of Translates 151
7.1 Frames of translates..................... 152
7.2 The canonical dual frame.................. 162
7.3 Compactly supported generators.............. 165
7.4 Frames of translates and oblique duals.......... 166
7.5 An application to sampling theory............. 175
7.6 Exercises........................... 176
8 Shift-Invariant Systems 179
8.1 Frame-properties of shift-invariant systems........ 179
8.2 Representations of the frame operator........... 191
8.3 Exercises........................... 194
Contents xiii
9 Gabor Frames in L2(R) 195
9.1 Basic Gabor frame theory................. 196
9.2 Tight Gabor frames..................... 210
9.3 The duals of a Gabor frame................ 212
9.4 Explicit construction of dual frame pairs......... 216
9.5 Popular Gabor conditions ................. 220
9.6 Representations of the Gabor frame operator and duality 224
9.7 The Zak transform..................... 227
9.8 Time-frequency localization of Gabor expansions .... 231
9.9 Continuous representations................. 237
9.10 Exercises........................... 240
10 Gabor Frames in £2(Z) 243
10.1 Translation and modulation on Í2( L)........... 243
10.2 Gabor systems in £2(Z) through sampling ........ 244
10.3 Shift-invariant systems................... 251
10.4 Exercises........................... 252
11 Wavelet Frames in L2(R) 253
11.1 Dyadic wavelet frames ................... 254
11.2 The unitary extension principle.............. 260
11.3 The oblique extension principle.............. 276
11.4 Approximation orders.................... 285
11.5 Construction of pairs of dual wavelet frames....... 286
11.6 The signal processing perspective............. 290
11.7 A survey on general wavelet frames............ 296
11.8 The continuous wavelet transform............. 300
11.9 Exercises........................... 303
List of Symbols 305
References 307
Index 311
|
| adam_txt |
Contents
ANHA Series Preface vii
Preface xv
1 Frames in Finite-dimensional
Inner Product Spaces 1
1.1 Basic frames theory. 2
1.2 Frames in C" . 12
1.3 The discrete Fourier transform. 17
1.4 Pseudo-inverses and the singular value decomposition . . 20
1.5 Applications in signal transmission. 25
1.6 Exercises. 30
2 Infinite-dimensional Vector Spaces
and Sequences 33
2.1 Normed vector spaces and sequences. 33
2.2 Operators on Banach spaces. 36
2.3 Hubert spaces. 37
2.4 Operators on Hubert spaces. 38
2.5 The pseudo-inverse operator. 40
2.6 A moment problem. 42
2.7 The spaces LP(K), L2(R), and £2(N) . 43
2.8 The Fourier transform and convolution. 46
2.9 Operators on L2(R). 47
2.10 Exercises. 49
xii Contents
3 Bases 51
3.1 Bessel sequences in Hilbert spaces. 52
3.2 General bases and orthonormal bases. 55
3.3 Riesz bases. 59
3.4 The Gram matrix. 64
3.5 Fourier series and trigonometric polynomials. 69
3.6 Wavelet bases. 72
3.7 Bases in Banach spaces. 78
3.8 Sampling and analog-digital conversion. 83
3.9 Exercises. 86
4 Bases and their Limitations 89
4.1 Bases in L2(0,1) and in general Hilbert spaces. 89
4.2 Gabor bases and the Balian-Low Theorem. 92
4.3 Bases and wavelets. 93
5 Frames in Hilbert Spaces 97
5.1 Frames and their properties. 98
5.2 Frames and Riesz bases. 105
5.3 Frames and operators. 108
5.4 Characterization of frames. 112
5.5 Various independency conditions. 116
5.6 Perturbation of frames. 121
5.7 The dual frames. 126
5.8 Continuous frames . 129
5.9 Frames and signal processing. 130
5.10 Exercises. 133
6 B-splines 139
6.1 The B-splines. 140
6.2 Symmetric B-splines . 146
6.3 Exercises. 148
7 Frames of Translates 151
7.1 Frames of translates. 152
7.2 The canonical dual frame. 162
7.3 Compactly supported generators. 165
7.4 Frames of translates and oblique duals. 166
7.5 An application to sampling theory. 175
7.6 Exercises. 176
8 Shift-Invariant Systems 179
8.1 Frame-properties of shift-invariant systems. 179
8.2 Representations of the frame operator. 191
8.3 Exercises. 194
Contents xiii
9 Gabor Frames in L2(R) 195
9.1 Basic Gabor frame theory. 196
9.2 Tight Gabor frames. 210
9.3 The duals of a Gabor frame. 212
9.4 Explicit construction of dual frame pairs. 216
9.5 Popular Gabor conditions . 220
9.6 Representations of the Gabor frame operator and duality 224
9.7 The Zak transform. 227
9.8 Time-frequency localization of Gabor expansions . 231
9.9 Continuous representations. 237
9.10 Exercises. 240
10 Gabor Frames in £2(Z) 243
10.1 Translation and modulation on Í2("L). 243
10.2 Gabor systems in £2(Z) through sampling . 244
10.3 Shift-invariant systems. 251
10.4 Exercises. 252
11 Wavelet Frames in L2(R) 253
11.1 Dyadic wavelet frames . 254
11.2 The unitary extension principle. 260
11.3 The oblique extension principle. 276
11.4 Approximation orders. 285
11.5 Construction of pairs of dual wavelet frames. 286
11.6 The signal processing perspective. 290
11.7 A survey on general wavelet frames. 296
11.8 The continuous wavelet transform. 300
11.9 Exercises. 303
List of Symbols 305
References 307
Index 311 |
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| index_date | 2024-07-02T20:34:06Z |
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| spelling | Christensen, Ole 1966- (DE-588)12445836X aut Frames and bases an introductory course Ole Christensen Boston ; Basel ; Berlin Birkhäuser 2008 xviii, 313 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis "During the last several years, frames have become increasingly popular; they have appeared in a large number of applications, and several concrete constructions of frames of various types have been presented. Most of these constructions were based on quite direct methods rather than the classical sufficient conditions for obtaining a frame. Consequently, there is a need for an updated book on frames, which moves the focus from the classical approach to a more constructive one. Based on a streamlined presentation of the author's previous work, An Introduction to Frames and Riesz Bases, this new textbook fills a gap in the literature, developing frame theory as part of a dialogue between mathematicians and engineers. Newly added sections on applications will help mathematically oriented readers to see where frames are used in practice and engineers to discover the mathematical background for applications in their field." -- Book cover. Mathematik Bases (Linear topological spaces) Frames (Vector analysis) Signal processing Mathematics Basis Mathematik (DE-588)4228195-7 gnd rswk-swf Frame Mathematik (DE-588)4528312-6 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Frame Mathematik (DE-588)4528312-6 s Harmonische Analyse (DE-588)4023453-8 s DE-604 Basis Mathematik (DE-588)4228195-7 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016451500&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Christensen, Ole 1966- Frames and bases an introductory course Mathematik Bases (Linear topological spaces) Frames (Vector analysis) Signal processing Mathematics Basis Mathematik (DE-588)4228195-7 gnd Frame Mathematik (DE-588)4528312-6 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4228195-7 (DE-588)4528312-6 (DE-588)4023453-8 |
| title | Frames and bases an introductory course |
| title_auth | Frames and bases an introductory course |
| title_exact_search | Frames and bases an introductory course |
| title_exact_search_txtP | Frames and bases an introductory course |
| title_full | Frames and bases an introductory course Ole Christensen |
| title_fullStr | Frames and bases an introductory course Ole Christensen |
| title_full_unstemmed | Frames and bases an introductory course Ole Christensen |
| title_short | Frames and bases |
| title_sort | frames and bases an introductory course |
| title_sub | an introductory course |
| topic | Mathematik Bases (Linear topological spaces) Frames (Vector analysis) Signal processing Mathematics Basis Mathematik (DE-588)4228195-7 gnd Frame Mathematik (DE-588)4528312-6 gnd Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Mathematik Bases (Linear topological spaces) Frames (Vector analysis) Signal processing Mathematics Basis Mathematik Frame Mathematik Harmonische Analyse |
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