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| Preface | |
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| Frames in Finite-dimensional Inner Product Spaces | |
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| Some basic facts about frames | |
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| Frame bounds and frame algorithms | |
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| Frames in C[superscript n] | |
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| The discrete Fourier transform | |
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| Pseudo-inverses and the singular value decomposition | |
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| Finite-dimensional function spaces | |
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| Exercises | |
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| Infinite-dimensional Vector Spaces and Sequences | |
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| Sequences | |
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| Banach spaces and Hilbert spaces | |
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| L[superscript 2] (R) and l[superscript 2] (N) | |
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| The Fourier transform | |
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| Operators on L[superscript 2] (R) | |
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| Exercises | |
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| Bases | |
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| Bases in Banach spaces | |
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| Bessel sequences in Hilbert spaces | |
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| Bases and biorthogonal systems in H | |
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| Orthonormal bases | |
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| The Gram matrix | |
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| Riesz bases | |
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| Fourier series and Gabor bases | |
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| Wavelet bases | |
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| Exercises | |
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| Bases and their Limitations | |
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| Gabor systems and the Balian-Low Theorem | |
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| Bases and wavelets | |
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| General shortcomings | |
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| Frames in Hilbert Spaces | |
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| Frames and their properties | |
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| Frame sequences | |
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| Frames and operators | |
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| Frames and bases | |
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| Characterization of frames | |
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| The dual frames | |
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| Tight frames | |
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| Continuous frames | |
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| Frames and signal processing | |
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| Exercises | |
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| Frames versus Riesz Bases | |
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| Conditions for a frame being a Riesz basis | |
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| Riesz frames and near-Riesz bases | |
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| Frames containing a Riesz basis | |
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| A frame which does not contain a basis | |
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| A moment problem | |
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| Exercise | |
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| Frames of Translates | |
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| Sequences in R[superscript d] | |
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| Frames of translates | |
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| Frames of integer-translates | |
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| Irregular frames of translates | |
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| The sampling problem | |
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| Frames of exponentials | |
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| Exercises | |
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| Gabor Frames in L[superscript 2] (R) | |
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| Continuous representations | |
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| Gabor frames | |
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| Necessary conditions | |
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| Sufficient conditions | |
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| The Wiener space W | |
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| Special functions | |
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| General shift-invariant systems | |
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| Exercises | |
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| Selected Topics on Gabor Frames | |
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| Popular Gabor conditions | |
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| Representations of the Gabor frame operator and duality | |
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| The duals of a Gabor frame | |
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| The Zak transform | |
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| Tight Gabor frames | |
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| The lattice parameters | |
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| Irregular Gabor systems | |
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| Applications of Gabor frames | |
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| Wilson bases | |
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| Exercises | |
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| Gabor Frames in l[superscript 2] (Z) | |
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| Translation and modulation on l[superscript 2] (Z) | |
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| Discrete Gabor systems through sampling | |
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| Gabor frames in C[superscript L] | |
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| Shift-invariant systems | |
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| Frames in l[superscript 2] (Z) and filter banks | |
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| Exercises | |
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| General Wavelet Frames | |
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| The continuous wavelet transform | |
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| Sufficient and necessary conditions | |
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| Irregular wavelet frames | |
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| Oversampling of wavelet frames | |
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| Exercises | |
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| Dyadic Wavelet Frames | |
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| Wavelet frames and their duals | |
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| Tight wavelet frames | |
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| Wavelet frame sets | |
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| Frames and multiresolution analysis | |
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| Exercises | |
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| Frame Multiresolution Analysis | |
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| Frame multiresolution analysis | |
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| Sufficient conditions | |
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| Relaxing the conditions | |
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| Construction of frames | |
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| Frames with two generators | |
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| Some limitations | |
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| Exercises | |
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| Wavelet Frames via Extension Principles | |
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| The general setup | |
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| The unitary extension principle | |
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| Applications to B-splines I | |
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| The oblique extension principle | |
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| Fewer generators | |
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| Applications to B-splines II | |
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| Approximation orders | |
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| Construction of pairs of dual wavelet frames | |
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| Applications to B-splines III | |
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| Exercises | |
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| Perturbation of Frames | |
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| A Paley-Wiener Theorem for frames | |
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| Compact perturbation | |
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| Perturbation of frame sequences | |
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| Perturbation of Gabor frames | |
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| Perturbation of wavelet frames | |
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| Perturbation of the Haar wavelet | |
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| Exercises | |
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| Approximation of the Inverse Frame Operator | |
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| The first approach | |
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| A general method | |
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| Applications to Gabor frames | |
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| Integer oversampled Gabor frames | |
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| The finite section method | |
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| Exercises | |
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| Expansions in Banach Spaces | |
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| Representations of locally compact groups | |
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| Feichtinger-Grochenig theory | |
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| Banach frames | |
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| p-frames | |
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| Gabor systems and wavelets in L[superscript p] (R) and related spaces | |
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| Exercises | |
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| Appendix A | |
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| Normed vector spaces and inner product spaces | |
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| Linear algebra | |
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| Integration | |
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| Some special normed vector spaces | |
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| Operators on Banach spaces | |
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| Operators on Hilbert spaces | |
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| The pseudo-inverse | |
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| Some special functions | |
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| B-splines | |
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| Notes | |
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| List of symbols | |
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| References | |
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| Index | |