An introduction to frames and Riesz bases

Title page......Page 1
Series......Page 2
Date-line......Page 4
Series preface......Page 5
Dedication......Page 8
Contents......Page 9
Preface......Page 14
1 Frames in Finite-dimensional Inner Product Spaces......Page 19
1.1 Some basic facts about frames......Page 20
1.2 Frame bounds and frame algorithms......Page 28
1.3 Frames in $\\mathbb{C}_n$......Page 32
1.4 The discrete Fourier transform......Page 37
1.5 Pseudo-inverses and the singular value decomposition......Page 41
1.6 Finite-dimensional function spaces......Page 46
1.7 Exercises......Page 50
2.1 Sequences......Page 53
2.3 $L^2(\\mathbb{R})$ and $l^2(\\mathbb{N})$......Page 56
2.4 The Fourier transform......Page 58
2.5 Operators on $L^2(\\mathbb{R})$......Page 59
2.6 Exercises......Page 60
3 Bases......Page 63
3.1 Bases in Banach spaces......Page 64
3.2 Bessel sequences in Hilbert spaces......Page 68
3.3 Bases and biorthogonal systems in $\\mathcal{H}$......Page 72
3.4 Orthonormal bases......Page 74
3.5 The Gram matrix......Page 78
3.6 Riesz bases......Page 81
3.7 Fourier series and Gabor bases......Page 87
3.8 Wavelet bases......Page 90
3.9 Exercises......Page 94
4 Bases and their Limitations......Page 97
4.1 Gabor systems and the Balian-Low Theorem......Page 100
4.2 Bases and wavelets......Page 101
4.3 General shortcomings......Page 104
5 Frames in Hilbert Spaces......Page 105
5.1 Frames and their properties......Page 106
5.2 Frame sequences......Page 110
5.3 Frames and operators......Page 111
5.4 Frames and bases......Page 114
5.5 Characterization of frames......Page 119
5.6 The dual frames......Page 129
5.8 Continuous frames......Page 133
5.9 Frames and signal processing......Page 135
5.10 Exercises......Page 137
6.1 Conditions for a frame being a Riesz basis......Page 141
6.3 Frames containing a Riesz basis......Page 144
6.4 A frame which does not contain a basis......Page 146
6.5 A moment problem......Page 152
6.6 Exercises......Page 154
7 Frames of Translates......Page 155
7.1 Sequences in $\\mathbb{R}^d$......Page 156
7.2 Frames of translates......Page 158
7.3 Frames of integer-translates......Page 165
7.4 Irregular frames of translates......Page 171
7.5 The sampling problem......Page 174
7.6 Frames of exponentials......Page 175
7.7 Exercises......Page 181
8 Gabor Frames in $L^2(\\mathbb{R})$......Page 185
8.1 Continuous representations......Page 187
8.2 Gabor frames......Page 189
8.3 Necessary conditions......Page 192
8.4 Sufficient conditions......Page 194
8.5 The Wiener space $W$......Page 205
8.6 Special functions......Page 208
8.7 General shift-invariant systems......Page 210
8.8 Exercises......Page 216
9 Selected Topics on Gabor Frames......Page 219
9.1 Popular Gabor conditions......Page 220
9.2 Representations of the Gabor frame operator and duality......Page 222
9.3 The duals of a Gabor frame......Page 226
9.4 The Zak transform......Page 233
9.5 Tight Gabor frames......Page 237
9.6 The lattice parameters......Page 240
9.7 Irregular Gabor systems......Page 244
9.8 Applications of Gabor frames......Page 248
9.9 Wilson bases......Page 250
9.10 Exercises......Page 251
10.1 Translation and modulation on $l^2(\\mathbb{Z})$......Page 253
10.2 Discrete Gabor systems through sampling......Page 254
10.3 Gabor frames in $\\mathbb{C}^L$......Page 262
10.4 Shift-invariant systems......Page 263
10.5 Frames in $l^2(\\mathbb{Z})$ and filter banks......Page 264
10.6 Exercises......Page 266
11 General Wavelet Frames......Page 267
11.1 The continuous wavelet transform......Page 269
11.2 Sufficient and necessary conditions......Page 271
11.3 Irregular wavelet frames......Page 285
11.4 Oversampling of wavelet frames......Page 288
11.5 Exercises......Page 289
12 Dyadic Wavelet Frames......Page 291
12.1 Wavelet frames and their duals......Page 292
12.2 Tight wavelet frames......Page 295
12.3 Wavelet frame sets......Page 296
12.5 Exercises......Page 299
13 Frame Multiresolution Analysis......Page 301
13.1 Frame multiresolution analysis......Page 302
13.2 Sufficient conditions......Page 304
13.3 Relaxing the conditions......Page 308
13.4 Construction of frames......Page 310
13.5 Frames with two generators......Page 326
13.6 Some limitations......Page 328
13.7 Exercises......Page 329
14 Wavelet Frames via Extension Principles......Page 331
14.1 The general setup......Page 332
14.2 The unitary extension principle......Page 334
14.3 Applications to $B$-splines I......Page 341
14.4 The oblique extension principle......Page 346
14.5 Fewer generators......Page 349
14.6 Applications to $B$-splines II......Page 352
14.7 Approximation orders......Page 357
14.8 Construction of pairs of dual wavelet frames......Page 359
14.9 Applications to $B$-splines III......Page 362
14.10 Exercises......Page 363
15 Perturbation of Frames......Page 365
15.1 A Paley-Wiener Theorem for frames......Page 366
15.2 Compact perturbation......Page 372
15.3 Perturbation of frame sequences......Page 374
15.4 Perturbation of Gabor frames......Page 376
15.5 Perturbation of wavelet frames......Page 379
15.7 Exercises......Page 380
16.1 The first approach......Page 383
16.2 A general method......Page 387
16.3 Applications to Gabor frames......Page 394
16.4 Integer oversampled Gabor frames......Page 396
16.5 The finite section method......Page 397
16.6 Exercises......Page 400
17.1 Representations of locally compact groups......Page 401
17.2 Feichtinger-Groechenig theory......Page 406
17.3 Banach frames......Page 412
17.4 $p$-frames......Page 415
17.5 Gabor systems and wavelets in $L^p(\\mathbb{R})$ and related spaces......Page 418
17.6 Exercises......Page 419
A.1 Normed vector spaces and inner product spaces......Page 421
A.2 Linear algebra......Page 422
A.3 Integration......Page 423
A.4 Some special normed vector spaces......Page 424
A.5 Operators on Banach spaces......Page 425
A.6 Operators on Hilbert spaces......Page 426
A.7 The pseudo-inverse......Page 428
A.8 Some special functions......Page 430
A.9 B-splines......Page 431
A.10 Notes......Page 434
List of symbols......Page 437
References......Page 439
Index......Page 455