An introduction to frames and Riesz bases

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An introduction to frames and Riesz bases / Ole Christensen

資料種別:: 図書
出版情報:: Boston : Birkhäuser, c2003
形態:: xx, 440 p ; 24 cm
シリーズ名:: Applied and numerical harmonic analysis / series editor, John J. Benedetto <BA30247287>
著者名:: Christensen, Ole, 1966- <DA13864603>
ISBN:: 9780817642952 [0817642951]
書誌ID:: BA60652728

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書誌詳細

主題:

Differential equations--Numerical solutions; Harmonic analysis; Riesz spaces

言語:

英語

目次情報:

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：

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