Abstract

perturbation results .................. 42 Gabor flames 45 6.2 Sucient conditions ........................ 46 6.3 The dual of a Gabor frame .................... 53 Frames with a special structure 55 7.1 Wavelets .............................. 55 7.2 Frames of translates ....................... 56 7.3 Frames of exponentials ...................... 57 Multiresolution analysis 59 8.1 General theory .......................... 60 8.2 Riesz bases of scaling functions ................. 76 8.3 Decomposition and reconstruction ................ 86 8.4 Vanishing moments and regularity ................ 90 Orthonormal bases of compactly supported wavelets 96 9.1 Compactly supported scaling functions ............. 96 9.2 Sufficient conditions for orthonormality ............. 107 10 Biorthogonal wavelets 111 10.1 The Daubechies assymetry theorem ............... 112 10.2 Construction of the two dual MRA's .............. 116 10.3 L2()-expansions in biorthogonal frames and bases ...... 126 A An alternative proof of lemma 8.2.8 Preface The aim of the notes is to present parts of the modern theory for bases and frames in Hilbert spaces. Chapter 1-5 describes the theory on an abstract level, and explicite constructions appear in chapter 6 (Gabor frames) and chapter 7-10 (wavelets).

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