Advances in Gabor Analysis

The 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.

1 Introduction.- 1.1 Recent Trends in Gabor Analysis.- 1.2 Outline of the Book.- 2 Uncertainty Principles for Time-Frequency Representations.- 2.1 Introduction.- 2.2 The Classical Uncertainty Principle.- 2.3 Time-Frequency Representations.- 2.4 Support Conditions.- 2.5 Essential Support Conditions.- 2.6 Hardy’s Uncertainty Principle.- 2.7 Beurling’s Theorem.- References.- 3 Zak Transforms with Few Zeros and the Tie.- 3.1 Introduction and Announcements of Results.- 3.2 Zak Transforms with Few Zeros.- 3.3 When is (?[0, c0),a, b) a Gabor Frame? “.- References.- 4 Bracket Products for Weyl-Heisenberg Frames.- 4.1 Introduction.- 4.2 Preliminaries.- 4.3 Pointwise Inner Products.- 4.4 a-Orthogonality.- 4.5 a-Factorable Operators.- 4.6 Weyl—Heisenberg Frames and the a-Inner Product.- References.- 5 A First Survey of Gabor Multipliers.- 5.1 Introduction.- 5.2 Notation and Conventions.- 5.3 Basic Theory of Gabor Multipliers.- 5.4 From Upper Symbol to Operator Ideal.- 5.5 Eigenvalue Behavior of Gabor Multipliers.- 5.6 Changing the Ingredients.- 5.7 From Gabor Multipliers to their Upper Symbol.- 5.8 Best Approximation by Gabor Multipliers.- 5.9 STFT-multipliers and Gabor Multipliers.- 5.10 Compactness in Function Spaces.- 5.11 Gabor Multipliers and Time-Varying Filters.- References.- 6 Aspects of Gabor Analysis and Operator Algebras.- 6.1 Introduction.- 6.2 Background.- 6.3 The Density (or Incompleteness) Property.- 6.4 Characterizing the Unique Gabor Dual Property.- 6.5 Gabor Frames for Subspaces.- References.- 7 Integral Operators, Pseudo differential Operators,and Gabor Frames.- 7.1 Introduction.- 7.2 Discussion and Statement of Results.- 7.3 The Modulation Spaces.- 7.4 Invariance Properties of the Modulation Space.- 7.5 Gabor Frames.- 7.6 An Easy Trace-Class Result.-7.7 Finite-Rank Approximations.- 7.8 Improving the Estimate.- 7.9 Conclusion and Observations.- References.- 8 Methods for Approximation of the Inverse (Gabor) Frame Operator.- 8.1 Introduction.- 8.2 The Double Projection Method.- 8.3 Projection Methods for Gabor Frames.- 8.4 On Sampling of Gabor Frames in L2(?).- References.- 9 Wilson Bases on the Interval.- 9.1 Introduction.- 9.2 Wilson Bases of L2(?).- 9.3 Wilson Bases for Periodic Functions.- 9.4 Wilson Bases on the Interval.- 9.5 Algorithms.- References.- 10 Localization Properties and Wavelet-Like Orthonormal Bases for the Lowest Landau Level.- 10.1 Introduction: Phase Space Localization.- 10.2 The Fractional Quantum Hall Effect.- 10.3 A Toy Model.- 10.4 Wavelet Bases for the LLL.- 10.5 Magnetic Translations and Multiresolution Analysis.- 10.6 Conclusion.- 10.7 Appendix: Two Mathematical Tools.- References.- 11 Optimal Stochastic Encoding and Approximation Schemes using Weyl—Heisenberg Sets.- 11.1 Introduction.- 11.2 Stochastic Processes and Statement of the Problems.- 11.3 Semi-optimal and Optimal Solutions.- 11.4 Non-Localization Results.- 11.5 Numerical Examples.- 11.6 Conclusions.- References.- 12 Orthogonal Frequency Division Multiplexing Based on Offset QAM.- 12.1 Introduction and Outline.- 12.2 Orthogonal Frequency Division Multiplexing Based on OQAM.- 12.3 Orthogonality Conditions for OFDM/OQAM Pulse Shaping Filters.- 12.4 Design of OFDM/OQAM Filters.- 12.5 Biorthogonal Frequency Division Multiplexing Based on Offset QAM.- 12.6 Conclusion.- 12.7 Appendix.- References.