Preface |
Frames in Finite-dimensional Inner Product Spaces / 1: |
Some basic facts about frames / 1.1: |
Frame bounds and frame algorithms / 1.2: |
Frames in C[superscript n] / 1.3: |
The discrete Fourier transform / 1.4: |
Pseudo-inverses and the singular value decomposition / 1.5: |
Finite-dimensional function spaces / 1.6: |
Exercises / 1.7: |
Infinite-dimensional Vector Spaces and Sequences / 2: |
Sequences / 2.1: |
Banach spaces and Hilbert spaces / 2.2: |
L[superscript 2] (R) and l[superscript 2] (N) / 2.3: |
The Fourier transform / 2.4: |
Operators on L[superscript 2] (R) / 2.5: |
Bases / 2.6: |
Bases in Banach spaces / 3.1: |
Bessel sequences in Hilbert spaces / 3.2: |
Bases and biorthogonal systems in H / 3.3: |
Orthonormal bases / 3.4: |
The Gram matrix / 3.5: |
Riesz bases / 3.6: |
Fourier series and Gabor bases / 3.7: |
Wavelet bases / 3.8: |
Bases and their Limitations / 3.9: |
Gabor systems and the Balian-Low Theorem / 4.1: |
Bases and wavelets / 4.2: |
General shortcomings / 4.3: |
Frames in Hilbert Spaces / 5: |
Frames and their properties / 5.1: |
Frame sequences / 5.2: |
Frames and operators / 5.3: |
Frames and bases / 5.4: |
Characterization of frames / 5.5: |
The dual frames / 5.6: |
Tight frames / 5.7: |
Continuous frames / 5.8: |
Frames and signal processing / 5.9: |
Frames versus Riesz Bases / 5.10: |
Conditions for a frame being a Riesz basis / 6.1: |
Riesz frames and near-Riesz bases / 6.2: |
Frames containing a Riesz basis / 6.3: |
A frame which does not contain a basis / 6.4: |
A moment problem / 6.5: |
Exercise / 6.6: |
Frames of Translates / 7: |
Sequences in R[superscript d] / 7.1: |
Frames of translates / 7.2: |
Frames of integer-translates / 7.3: |
Irregular frames of translates / 7.4: |
The sampling problem / 7.5: |
Frames of exponentials / 7.6: |
Gabor Frames in L[superscript 2] (R) / 7.7: |
Continuous representations / 8.1: |
Gabor frames / 8.2: |
Necessary conditions / 8.3: |
Sufficient conditions / 8.4: |
The Wiener space W / 8.5: |
Special functions / 8.6: |
General shift-invariant systems / 8.7: |
Selected Topics on Gabor Frames / 8.8: |
Popular Gabor conditions / 9.1: |
Representations of the Gabor frame operator and duality / 9.2: |
The duals of a Gabor frame / 9.3: |
The Zak transform / 9.4: |
Tight Gabor frames / 9.5: |
The lattice parameters / 9.6: |
Irregular Gabor systems / 9.7: |
Applications of Gabor frames / 9.8: |
Wilson bases / 9.9: |
Gabor Frames in l[superscript 2] (Z) / 9.10: |
Translation and modulation on l[superscript 2] (Z) / 10.1: |
Discrete Gabor systems through sampling / 10.2: |
Gabor frames in C[superscript L] / 10.3: |
Shift-invariant systems / 10.4: |
Frames in l[superscript 2] (Z) and filter banks / 10.5: |
General Wavelet Frames / 10.6: |
The continuous wavelet transform / 11.1: |
Sufficient and necessary conditions / 11.2: |
Irregular wavelet frames / 11.3: |
Oversampling of wavelet frames / 11.4: |
Dyadic Wavelet Frames / 11.5: |
Wavelet frames and their duals / 12.1: |
Tight wavelet frames / 12.2: |
Wavelet frame sets / 12.3: |
Frames and multiresolution analysis / 12.4: |
Frame Multiresolution Analysis / 12.5: |
Frame multiresolution analysis / 13.1: |
Relaxing the conditions / 13.2: |
Construction of frames / 13.4: |
Frames with two generators / 13.5: |
Some limitations / 13.6: |
Wavelet Frames via Extension Principles / 13.7: |
The general setup / 14.1: |
The unitary extension principle / 14.2: |
Applications to B-splines I / 14.3: |
The oblique extension principle / 14.4: |
Fewer generators / 14.5: |
Applications to B-splines II / 14.6: |
Approximation orders / 14.7: |
Construction of pairs of dual wavelet frames / 14.8: |
Applications to B-splines III / 14.9: |
Perturbation of Frames / 14.10: |
A Paley-Wiener Theorem for frames / 15.1: |
Compact perturbation / 15.2: |
Perturbation of frame sequences / 15.3: |
Perturbation of Gabor frames / 15.4: |
Perturbation of wavelet frames / 15.5: |
Perturbation of the Haar wavelet / 15.6: |
Approximation of the Inverse Frame Operator / 15.7: |
The first approach / 16.1: |
A general method / 16.2: |
Applications to Gabor frames / 16.3: |
Integer oversampled Gabor frames / 16.4: |
The finite section method / 16.5: |
Expansions in Banach Spaces / 16.6: |
Representations of locally compact groups / 17.1: |
Feichtinger-Grochenig theory / 17.2: |
Banach frames / 17.3: |
p-frames / 17.4: |
Gabor systems and wavelets in L[superscript p] (R) and related spaces / 17.5: |
Appendix A / 17.6: |
Normed vector spaces and inner product spaces / A.1: |
Linear algebra / A.2: |
Integration / A.3: |
Some special normed vector spaces / A.4: |
Operators on Banach spaces / A.5: |
Operators on Hilbert spaces / A.6: |
The pseudo-inverse / A.7: |
Some special functions / A.8: |
B-splines / A.9: |
Notes / A.10: |
List of symbols |
References |
Index |
Preface |
Frames in Finite-dimensional Inner Product Spaces / 1: |
Some basic facts about frames / 1.1: |
Frame bounds and frame algorithms / 1.2: |
Frames in C[superscript n] / 1.3: |
The discrete Fourier transform / 1.4: |