Abstract
Time-frequency analysis deals with localized Fourier transforms. Gabor analysis is a branch of time-frequency analysis. It shares with the wavelet transform methods the ability to describe the smoothness of a given function in a location-dependent way.
The main tool is the sliding window Fourier transform or short-time Fourier transform (STFT). It describes the correlation of a signal with the time-frequency shifts of a fixed function. Thus, it characterizes a function by its transform over the time-frequency plane (TF-plane).
The heart of the Gabor analysis is a study of so-called Gabor systems - systems of time-frequency shifts (with respect to a rectangular lattice) of a given function g, which we call a window.
Since Gabor system can never be an orthogonal basis for smooth and rapidly decaying windows, we need to develop a concept of Gabor frames - overcomplete systems with a control over the l^2 norm of scalar products.
The latest part of the story started in 1988 when I. Daubechies and D. Grossman conjectured that the Gaussian Gabor system forms a frame for all rectangular lattices with density bigger then 1. This was proved (independently) by Yu. Lyubarskii and K. Seip in 1992. Afterwards this result was extended to some other window functions.
We study both the classical part of the theory and recent advances. This includes techniques from complex and harmonic analysis and theory of dynamical systems.
Audience
Graduate, Undergraduate
Prerequisite
Complex analysis, Measure theory
Syllabus
1. Classical Fourier analysis. The Poisson summation formula. Plancherel's theorem.
2. Uncertainty Principles. Heisenberg's inequality. Donoho's and Stark's theorem.
3. Time-frequency analysis. Short Time Fourier Transform. Inversion formulas.
4. Liebs' Uncertainty Principle.
5. Hilbert Spaces of entire functions. Reproducing kernels. Fock space.
6. Bargmann transform.
7. Frames. Dual frames. Riesz bases.
8. Gabor frames. Dual Gabor frames.
9. The Wiener space. Boundedness of the frame operator.
10. Correlation functions. Walnut's representation.
11. Gabor frames for large densities.
12. Janssen's representation.
13. The Wexler-Raz biorthogonality relations.
14. The Ron-Shen Duality principle.
15. Structure of dual windows.
16. The Zak transform. Gabor frames for critical density.
17. Gabor frames for rational densities.
18. The Balian-Low theorem. Argument principle.
19. Gabor frames for Gaussians.
20. Gabor frames for Cauchy kernels.
21. Gabor frames for hyperbolic secant.
22. Gabor frames for rational functions.
23. Gabor frames for Haar function.
24. Irregular Gabor frames.
25. Tight frames with density 2.
25. Wilson bases.
26. Asymptotic of lower bounds for Gabor frames.
27. Gabor frames and spaces of analytic functions.
28. Totally positive functions of finite type.
29. Gabor frames and totally positive functions.
30. Pseudodifferential operators.
Reference
1. K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, MA, 2001
2. C. Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl., 13(2):113-166, (2007).
3. A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in L^2(R^d), Duke Math. J., 89(2):237:282, (1997).
4. K. Seip, Density theorems for sampling and interpolation in the BargmannFock space. I, J. Reine Angew. Math. 429 (1992) 91-106.
5. K. Grochenig, J. Stockler, Gabor frames and totally positive functions, Duke Mathematical Journal, 162 (6), 1003-1031, (2011).
6. K. Grochenig, J.L. Romero, J. Stockler, Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions, Inventiones mathematicae, 211 (3), 1119-1148, (2016).
7. Yu. Belov, A. Kulikov, Yu. Lyubarskii, Gabor frames for rational functions, Inventiones mathematicae, 231, No. 2, 431-466 (2023).
8. X. Dai, M. Zhu, Frame set for gabor systems with Haar window, https://arxiv.org/pdf/2205.06479.pdf
Lecturer Intro.
Yurii Belov is a professor at St. Petersburg State University and vice-chair of educational program "Mathematics" headed by Stanislav Smirnov. He got his PhD degree in 2007 (Norwegian University of Science and Technology) and Dr.Sci. degree in 2016 (St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia). He was a postdoc at Norwegian University of Science and Technology. Yurii Belov was awarded by the St. Petersburg Mathematical Society the prize for young mathematicians and won the "Young Russian Mathematics" contest (twice). In 2016 he got the L. Euler award from the Government of St. Petersburg.
