An Introduction to Frames and Riesz BasesThe Applied and Numerical Harmonic Analysis ( ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract har monic analysis to basic applications. The title of the series reflects the im portance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbi otic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flour ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as sig nal processing, partial differential equations (PDEs), and image processing is reflected in our state of the art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. |
Contents
| 8 | |
1 | 14 |
5 | 23 |
7 | 32 |
Banach spaces and Hilbert spaces | 38 |
Bases and their Limitations | 78 |
Frames in Hilbert Spaces | 87 |
Frames versus Riesz Bases | 123 |
Dyadic Wavelet Frames | 272 |
Exercises | 281 |
Frame Multiresolution Analysis | 293 |
1 | 312 |
Perturbation of Frames | 347 |
List of symbols | 357 |
Approximation of the Inverse Frame Operator | 365 |
Expansions in Banach Spaces | 383 |
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Common terms and phrases
arbitrary Banach spaces bases basis for H Bessel sequence biorthogonal bounded operator canonical dual chapter Ck fk coefficients compact support consider construct converges Corollary defined definition denote dual frame e}kez elements EmbTna9}m,nez equation equivalent example Exercise exist constants follows Fourier transform frame for H frame for L²(R frame multiresolution analysis frame operator frame sequence function g Gabor frames Gabor system given Hilbert space implies interval invertible Janssen l²(N L²(R l²(Z Lebesgue Lebesgue point Lemma Let f Let g lower frame bound lower frame condition m,nez matrix multiresolution analysis norm Note obtain orthonormal basis p-frame parameters pre-frame operator properties Proposition prove representation result Riesz basis Riesz sequence S-¹g satisfies condition Schauder basis Section sequence f sequence with bound shift-invariant systems shows subspace T}kez tight frame trigonometric polynomial vector space wavelet frames Σ Σ ΣΣ