The successor book to Gabor Analysis and Algorithms:

book cover
Advances in Gabor Analysis
H.G. Feichtinger, University of Vienna (Ed.)
T. Strohmer, University of California at Davis (Ed.)
Birkhäuser 2002 * Hardcover * 368 pages * 22 Illustrations
Series: Applied and Numerical Harmonic Analysis

This unified, self-contained volume provides insight into the richness of Gabor analysis and its potential for future development in applied mathematics and engineering. Mathematicians and engineers treat a range of topics covering theory as well as applications in eleven survey chapters. Taken as a whole, the work demonstrates interactions and connections among different areas in which Gabor analysis plays a critical role, including harmonic analysis, operator theory, quantum physics, numerical analysis, electrical engineering, and signal/image processing.

Key features of the work:

Graduate students, professionals, and researchers in pure and applied mathematics, mathematical physics, electrical and communications engineering will find Advances in Gabor Analysis a comprehensive resource.


Foreword / Henry Landau

1. Introduction / H.G. Feichtinger & T. Strohmer
1.1 Recent trends in Gabor Analysis
1.2 Outline of the book

2. Uncertainty Principles for Time-Frequency Representations / K.Gröchenig
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

3. Zak Transforms with Few Zeros and the Tie / A.J.E.M. Janssen
3.1 Introduction and Announcements of Results
3.2 Zak Transforms with Few Zeros
3.3 When is $(\chi _{[0,c_0)},a,b)$ a Gabor Frame?

4. Bracket Products for Weyl-Heisenberg Frames / P. Casazza & M.C. Lammers
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

5. A First Survey of Gabor Multipliers / H.G. Feichtinger & K. Nowak
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

6. Aspects of Gabor Analysis and Operator Algebra / J.P. Gabardo & D. Han
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

7. Integral Operators, Pseudodifferential Operators and Gabor Frames / C. Heil
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

8. Approximation of the Inverse (Gabor) Frame Operator / O. Christensen, T. Strohmer
8.1 Introduction
8.2 The Double Projection Method
8.3 Projection Methods for Gabor Frames
8.4 On Sampling of Gabor Frames in ${{\boldsymbol L}^2({\@mathbb R})}$

9. Folding Operators, Wilson Bases, and Zak Transforms / K. Bittner
9.1 Introduction
9.2 Wilson Bases of $L^2(\@mathbb {R})$
9.3 Wilson Bases for Periodic Functions
9.4 Wilson Bases on the Interval
9.5 Algorithms

10. Localization Properties and Wavelet-Like Orthonormal Bases for the Lowest Landau Level / J.P. Antoine & F. Bagarello
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

11. Optimal Stochastic Approximations and Encoding Schemes Using Weyl-Heisenberg Sets / R. Balan & I. Daubechies
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

12. Orthogonal Frequency Division Multiplexing Based on Offset-QAM / H. Bölcskei
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