Contents
Abstract ix
Introduction 1
Chapter 1. Basic Theory for Frames 5
1.1. A Dilation Viewpoint on Frames 5
1.2. The Canonical Dual Frame 14
1.3. Alternate Dual Frames 16
Chapter 2. Complementary Frames and Disjointness 21
2.1. Strong Disjointness, Disjointness and Weak Disjointness 21
2.2. Characterizations of Equivalence and Disjointness 24
2.3. Cuntz Algebra Generators 29
2.4. More on Alternate Duals 30
Chapter 3. Frame Vectors for Unitary Systems 39
3.1. The Local Commutant and Frame Vectors 39
3.2. Dilation Theorems for Frame Vectors 42
3.3. Equivalence Classes of Frame Vectors 47
Chapter 4. Gabor Type Unitary Systems 49
Chapter 5. Frame Wavelets, Super-wavelets and Frame Sets 55
5.1. Frame Sets 55
5.2. Super-wavelets 58
5.3. A Characterization of Super-wavelets 65
5.4. Some Frazier-Jawerth Frames 66
5.5. MRA Super-wavelets 70
5.6. Interpolation Theory 73
Chapter 6. Frame Representations for Groups 79
6.1. Basics 79
6.2. Frame Multiplicity 80
6.3. Parameterizations of Frame Vectors 84
6.4. Disjoint Group Representations 86
Chapter 7. Concluding Remarks 89
7.1. Spectral families of frames: 89
7.2. A Joint Project with Pete Casazza 90
7.3. A Matrix Completion Characterization of Frames 90
7.4. Some Acknowledgements 90
7.5. Density and Connectivity of Gabor Type Frames 91
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