## An Introduction to Frames and Riesz BasesThe 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 |

437 | |

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### Common terms and phrases

Anal apply arbitrary assume B-splines Banach frame Banach spaces bases basis for H Bessel sequence biorthogonal bounded operator bounds A,B canonical dual Chapter Christensen ckfk coefficients compact support consider construct converges Corollary Daubechies defined definition denote dual frame dual wavelet elements equation example Exercise exist constants expansion fc=i fc=l fcez Feichtinger finite-dimensional fk}kLi follows Fourier series Fourier transform frame for H frame for L2(R frame multiresolution analysis frame operator frame sequence function g Gabor frames given Grochenig Hilbert space implies inner product interval inverse Janssen k}kez Lebesgue Lebesgue point Lemma Let g lower frame bound Math matrix multiresolution analysis norm Note obtain orthogonal projection orthonormal basis p-frame parameters perturbation pre-frame operator properties Proposition prove representation result Riesz basis Riesz sequence Schauder basis Section shift-invariant systems shows splines subspace theory tight frame translation trigonometric polynomial unitary vector space wavelet frames wavelet set

### Popular passages

Page 432 - Convergence and summability of Gabor expansions at the Nyquist density,