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# Minicourse «Gabor analysis for rational functions»

St. Petersburg State University and Euler International Mathematical Institute are happy to announce a new minicourse, headed by Yurii S. Belov.  Lectures scheduled to Wednesdays starting 14 April, 7pm. To attend the lectures, please join zoom channel 511-327-649. If you do not know the password, please write an email to Yurii Belov. Video records of the course are available here.

### Course description

This series of lectures is aimed to present a recent progress in Gabor analysis for rational functions and is based on the works by A. Kulikov, Yu. Lyubarskii and the author. Given a function we call a collection of time-frequency shifts of ,

a Gabor system.

Gabor systems constitute one of the most important and natural objects in modern signal analysis, and they have numerous applications in quantum mechanics
and mathematical physics. Such a system provides a natural representation of a given signal in terms of its samples in the sense of classical Fourier analysis.

To produce a stable reconstruction of an arbitrary function by its {\it Gabor samples }

we require for the corresponding Gabor system to be a {\it a frame}. One of the main questions of Gabor analysis is to describe all rectangular lattices which generate a Gabor frame in for
a given function .

The complete answer is known only in a select few cases ( being Gaussian, one-sided exponential etc.) We start by giving an exposition of classical results by Daubechies, Grossman, Seip, Janssen, Grochenig and others. Then we develop a thorough investigation (featuring complete proofs) of frames generated by rational functions. Finally we prove the results about irregular sampling for Cauchy kernel (simplest rational function from ).

Program:

• Gabor systems: what they are, how do they appear and why we need them. %Overview.
• Main criterion for rational functions. Finite-diagonal matrices.
• Hunting for positivity. Herglotz functions.
• Irrational densities. Daubechies conjecture.
• Near the criticial hyperbola. Large densities.
• Irregular sampling. Toeplitz approach.
• Sum of two Cauchy kernels. Estimates of frame bounds.

This course is accessible to second year students. The prerequisites are: basic complex analysis and linear algebra.

Everyone is welcome!

Lecture 1. Notes

Lecture 2. Notes

Lecture 3. Notes

Lecture 4. Notes

Lecture 5. Notes

Lecture 6. Notes

Lecture 7. Notes