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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 | |