Affine Density in Wavelet Analysis

Gitta Kutyniok (auth.)354072916X, 9783540729167, 9783540729495

In wavelet analysis, irregular wavelet frames have recently come to the forefront of current research due to questions concerning the robustness and stability of wavelet algorithms. A major difficulty in the study of these systems is the highly sensitive interplay between geometric properties of a sequence of time-scale indices and frame properties of the associated wavelet systems.

This volume provides the first thorough and comprehensive treatment of irregular wavelet frames by introducing and employing a new notion of affine density as a highly effective tool for examining the geometry of sequences of time-scale indices. Many of the results are new and published for the first time. Topics include: qualitative and quantitative density conditions for existence of irregular wavelet frames, non-existence of irregular co-affine frames, the Nyquist phenomenon for wavelet systems, and approximation properties of irregular wavelet frames.

Table of contents :
Front Matter….Pages I-XII
Introduction….Pages 1-10
Wavelet and Gabor Frames….Pages 11-20
Weighted Affine Density….Pages 21-33
Qualitative Density Conditions….Pages 35-57
Quantitative Density Conditions….Pages 59-86
Homogeneous Approximation Property….Pages 87-104
Weighted Beurling Density and Shift-Invariant Gabor Systems….Pages 105-125
Back Matter….Pages 127-142