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Lecture Notes in Mathematics

By: Gitta Kutyniok

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Paperback | 12 July 2007

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156 Pages

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23.5 x 15.88 x 0.64

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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.

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Geometry Computing & I.T.Computer Science Mathematical Theory of Computation

Introduction	p. 1
Irregular Wavelet and Gabor Frames	p. 1
Density for Gabor Systems	p. 3
Geometry of Time-Scale Indices	p. 5
Overview of Main Results	p. 7
Wavelet and Gabor Frames	p. 11
Frame Theory	p. 11
Wavelet Analysis	p. 13
Time-Frequency Analysis	p. 16
Amalgam Spaces	p. 19
Weighted Affine Density	p. 21
Definitions	p. 21
Basic Properties	p. 24
The Notion of Affine Density by Sun and Zhou	p. 27
Comparison of Both Notions of Affine Density	p. 32
Qualitative Density Conditions	p. 35
Existence of an Upper Frame Bound	p. 35
Existence of a Lower Frame Bound	p. 37
Examples of Wavelet Systems	p. 42
Density of Sequences in R+	p. 45
Affine Density and the Local Integrability Condition	p. 48
Amalgam Spaces on R {0}	p. 48
A Density Version of the Local Integrability Condition	p. 50
A Characterization of Wavelet Parseval Frames	p. 56
Quantitative Density Conditions	p. 59
Outline and Comparison with Previous Work	p. 59
Density of Product Sequences	p. 61
A Fundamental Relationship	p. 63
The Nyquist Phenomenon	p. 75
Sufficient Density Conditions for Wavelet Frames	p. 76
Existence of Special Weight Functions	p. 81
Co-Affine Systems	p. 82
Homogeneous Approximation Property	p. 87
Amalgam Spaces and the Continuous Wavelet Transform	p. 87
The Basic Class B[subscript 0]	p. 89
The Homogeneous Approximation Property for Wavelet Frames	p. 93
The Comparison Theorem for Wavelet Frames	p. 97
Density Results for Wavelet Schauder Bases	p. 100
Weighted Beurling Density and Shift-Invariant Gabor Systems	p. 105
Motivation and Outline of Chapter	p. 105
Weighted Beurling Density	p. 107
Littlewood-Paley Type Inequalities	p. 110
Equivalent Definition of Weighted Beurling Density	p. 115
Beurling Density, Frame Bounds, and Gabor Generators	p. 119
Shift-Invariant Gabor Systems	p. 121
References	p. 127
Index	p. 135
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Mathematics

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