Preface ........................................................ xi
Introduction .................................................... 1
1 Normed vector spaces ......................................... 3
1.1 Definitions ............................................. 4
1.2 Inner products and norms ............................... 10
1.3 Finite-dimensional ℓp spaces ............................ 14
1.4 Digging deeper: completion of inner product spaces ..... 20
1.5 Hilbert spaces, L2 and ℓ2 ............................... 25
1.6 Orthogonal projections, Gram-Schmidt
orthogonalization ...................................... 39
1.7 Linear operators and matrices, LS approximations ....... 46
1.8 Additional exercises ................................... 64
2 Analytic tools .............................................. 73
2.1 Improper integrals ..................................... 73
2.2 The gamma functions and beta functions ................. 78
2.3 The sine function and its improper relatives ........... 79
2.4 Infinite products ...................................... 83
2.5 Additional exercises ................................... 86
3 Fourier series .............................................. 92
3.1 Definitions, real Fourier series and complex Fourier
series ................................................. 92
3.2 Examples ............................................... 96
3.3 Intervals of varying length, odd and even functions .... 97
3.4 Convergence results .................................... 99
3.5 More on pointwise convergence, Gibbs phenomena ........ 107
3.6 Further properties of Fourier series .................. 113
3.7 Digging deeper: arithmetic summability and Fejйr's
theorem ............................................... 116
3.8 Additional exercises .................................. 123
4 The Fourier transform ...................................... 127
4.1 Fourier transforms as integrals ....................... 127
4.2 The transform as a limit of Fourier series ............ 129
4.3 L2 convergence of the Fourier transform ............... 135
4.4 The Riemann-Lebesgue lemma and pointwise convergence .. 140
4.5 Relations between Fourier series and integrals:
sampling .............................................. 146
4.6 Fourier series and Fourier integrals: periodization ... 152
4.7 The Fourier integral and the uncertainty principle .... 154
4.8 Digging deeper ........................................ 157
4.9 Additional exercises .................................. 161
5 Compressive sampling ....................................... 164
5.1 Introduction .......................................... 164
5.2 Algebraic theory of compressive sampling .............. 168
5.3 Analytic theory of compressive sampling ............... 172
5.4 Probabilistic theory of compressive sampling .......... 183
5.5 Discussion and practical implementation ............... 201
5.6 Additional exercises .................................. 206
6 Discrete transforms ........................................ 208
6.1 Z transforms .......................................... 208
6.2 Inverse Z transforms .................................. 211
6.3 Difference equations .................................. 213
6.4 Discrete Fourier transform and relations to Fourier
series ................................................ 214
6.5 Fast Fourier transform (FFT) .......................... 222
6.6 Approximation to the Fourier transform ................ 223
6.7 Additional exercises .................................. 224
7 Linear filters ............................................. 230
7.1 Discrete linear filters ............................... 230
7.2 Continuous filters .................................... 233
7.3 Discrete filters in the frequency domain .............. 235
7.4 Other operations on discrete signals .................. 238
7.5 Additional exercises .................................. 240
8 Windowed Fourier and continuous wavelet transforms.
Frames ..................................................... 242
8.2 Bases and frames, windowed frames ..................... 251
8.3 Affine frames ......................................... 268
8.4 Additional exercises .................................. 270
9 Multiresolution analysis ................................... 272
9.1 Haar wavelets ......................................... 272
9.2 The multiresolution structure ......................... 284
9.3 Filter banks and reconstruction of signals ............ 296
9.4 The unitary two-channel filter bank system ............ 304
9.5 A perfect reconstruction filter bank with N = 1 ....... 306
9.6 Perfect reconstruction for two-channel filter banks ... 307
9.7 Halfband filters and spectral factorization ........... 309
9.8 Maxflat filters ....................................... 312
9.9 Low pass iteration and the cascade algorithm .......... 317
9.10 Scaling functions by recursion: dyadic points ......... 320
9.11 The cascade algorithm in the frequency domain ......... 329
9.12 Some technical results ................................ 332
9.13 Additional exercises .................................. 335
10 Discrete wavelet theory .................................... 341
10.1 L2 convergence ........................................ 345
10.2 Accuracy of approximation ............................. 354
10.3 Smoothness of scaling functions and wavelets .......... 359
10.4 Additional exercises .................................. 365
11 Biorthogonal filters and wavelets .......................... 367
11.1 Resume of basic facts on biorthogonal filters ......... 367
11.2 Biorthogonal wavelets: multiresolution structure ...... 370
11.3 Splines ............................................... 382
11.4 Generalizations of filter banks and wavelets .......... 390
11.5 Finite length signals ................................. 395
11.6 Circulant matrices .................................... 397
11.7 Additional exercises .................................. 400
12 Parsimonious representation of data ........................ 401
12.1 The nature of digital images .......................... 402
12.2 Pattern recognition and clustering .................... 418
12.3 Image representation of data .......................... 426
12.4 Image compression ..................................... 429
12.5 Additional exercises .................................. 433
References ................................................. 437
Index ......................................................... 443
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