An Introduction to Frames and Riesz Bases

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Springer Science & Business Media, Dec 13, 2002 - Mathematics - 440 pages
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The Applied and Numerical Harmonic Analysis ( ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract har monic analysis to basic applications. The title of the series reflects the im portance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbi otic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flour ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as sig nal processing, partial differential equations (PDEs), and image processing is reflected in our state of the art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.
 

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Contents

Frames in Finitedimensional Inner Product Spaces
1
11 Some basic facts about frames
2
12 Frame bounds and frame algorithms
10
13 Frames in Cn
14
14 The discrete Fourier transform
19
15 Pseudoinverses and the singular value decomposition
23
16 Finitedimensional function spaces
28
17 Exercises
32
Gabor Frames in
235
102 Discrete Gabor systems through sampling
236
103 Gabor frames in CL
244
104 Shiftinvariant systems
245
105 Frames in 2Z and filter banks
246
106 Exercises
248
General Wavelet Frames
249
111 The continuous wavelet transform
251

Infinitedimensional Vector Spaces and Sequences
35
22 Banach spaces and Hilbert spaces
38
24 The Fourier transform
40
25 Operators on L²R
41
26 Exercises
42
Bases
45
31 Bases in Banach spaces
46
32 Bessel sequences in Hilbert spaces
50
33 Bases and biorthogonal systems in H
54
34 Orthonormal bases
56
35 The Gram matrix
60
36 Riesz bases
63
37 Fourier series and Gabor bases
69
38 Wavelet bases
72
39 Exercises
76
Bases and their Limitations
79
41 Gabor systems and the BalianLow Theorem
82
42 Bases and wavelets
83
43 General shortcomings
86
Frames in Hilbert Spaces
87
51 Frames and their properties
88
52 Frame sequences
92
53 Frames and operators
93
54 Frames and bases
96
55 Characterization of frames
101
56 The dual frames
111
57 Tight frames
115
59 Frames and signal processing
117
510 Exercises
119
Frames versus Riesz Bases
123
62 Riesz frames and near Riesz bases
126
64 A frame which does not contain a basis
128
65 A moment problem
134
66 Exercises
136
Frames of Translates
137
71 Sequences in Rd
138
72 Frames of translates
140
73 Frames of integertranslates
147
74 Irregular frames of translates
153
75 The sampling problem
156
76 Frames of exponentials
157
77 Exercises
163
Gabor Frames in L²2
167
81 Continuous representations
169
82 Gabor frames
171
83 Necessary conditions
174
84 Sufficient conditions
176
85 The Wiener space W
187
86 Special functions
190
87 General shiftinvariant systems
192
88 Exercises
198
Selected Topics on Gabor Frames
201
91 Popular Gabor conditions
202
92 Representations of the Gabor frame operator and duality
204
93 The duals of a Gabor frame
208
94 The Zak transform
215
95 Tight Gabor frames
219
96 The lattice parameters
222
97 Irregular Gabor systems
226
98 Applications of Gabor frames
230
99 Wilson bases
232
910 Exercises
233
112 Sufficient and necessary conditions
253
113 Irregular wavelet frames
267
114 Oversampling of wavelet frames
270
115 Exercises
271
Dyadic Wavelet Frames
273
121 Wavelet frames and their duals
274
122 Tight wavelet frames
277
123 Wavelet frame sets
278
124 Frames and multiresolution analysis
281
Frame Multiresolution Analysis
283
131 Frame multiresolution analysis
284
132 Sufficient conditions
286
133 Relaxing the conditions
290
134 Construction of frames
292
135 Frames with two generators
308
136 Some limitations
310
137 Exercises
311
Wavelet Frames via Extension Principles
313
141 The general setup
314
142 The unitary extension principle
316
143 Applications to 5splines I
323
144 The oblique extension principle
328
145 Fewer generators
331
146 Applications to Bsplines II
334
147 Approximation orders
339
148 Construction of pairs of dual wavelet frames
341
149 Applications to Bsplines III
344
1410 Exercises
345
Perturbation of Frames
347
151 A PaleyWiener Theorem for frames
348
152 Compact perturbation
354
153 Perturbation of frame sequences
356
154 Perturbation of Gabor frames
358
155 Perturbation of wavelet frames
361
156 Perturbation of the Haar wavelet
362
Approximation of the Inverse Frame Operator
365
162 A general method
369
163 Applications to Gabor frames
376
164 Integer oversampled Gabor frames
378
165 The finite section method
379
166 Exercises
382
Expansions in Banach Spaces
383
172 FeichtingerGrochenig theory
388
173 Banach frames
394
174 pframes
397
175 Gabor systems and wavelets in LPR and related spaces
400
176 Exercises
401
Appendix
403
A2 Linear algebra
404
A3 Integration
405
A4 Some special normed vector spaces
406
A 5 Operators on Banach spaces
407
A6 Operators on Hilbert spaces
408
A7 The pseudoinverse
410
A8 Some special functions
412
A9 Bsplines
413
A10 Notes
416
List of symbols
419
References
421
Index
437
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