Affine Density in Wavelet Analysis (Lecture Notes in Mathematics Book 1914)

"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."--Publisher's description 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, Nonexistence Of Irregular Co-affine Frames, The Nyquist Phenomenon For Wavelet Systems, And Approximation Properties Of Irregular Wavelet Frames.--book Jacket. 1. Introduction -- 2. Wavelet And Gabor Frames -- 3. Weighted Affine Density -- 4. Qualitative Density Conditions -- 5. Quantitative Density Conditions -- 6. Homogenous Approximation Property -- 7. Weighted Beurling Density And Shift-invariant Gabor Systems. Gitta Kutyniok. Includes Bibliographical References (p. [127]-133) And Index. This Volume Provides A Thorough And Comprehensive Treatment Of Irregular Wavelet Frames. It Introduces And Employs A New Notion Of Affine Density As A Highly Effective Tool For Examining The Geometry Of Sequences Of Time-scale Indices. Coverage Includes Non-existence Of Irregular Co-affine Frames, The Nyquist Phenomenon For Wavelet Systems, And Approximation Properties Of Irregular Wavelet Frames. 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