Zeit-Frequenz Analyse, Zufälligkeit und Abtastung

In time-frequency analysis a function is studied simultaneously in the spatial and frequency domains. The limit scale where joint time-frequency analysis is possible is given by the Fourier uncertainty principle. This project addresses a number of problems, motivated by specific applications, where the uncertainty limit is approached.
The goal of this project is two fold. One the one hand, we will apply time-frequency analysis to new contexts, building on real-variable techniques and signal processing intuition to tackle problems traditionally in the realm of complex variable methods and special functions. On the other hand, we will contribute to a number of long-standing problems in time-frequency analysis by systematically combining techniques from Banach algebras, approximation theory, Beurling's balayage, computational harmonic analysis, and harmonic analysis in phase-space.
Some specific topics of the project are:
(a) Sharp sampling theorems in the spirit of Beurling's theory for bandlimited functions, and characterization of Gabor frames up to the limit set by the uncertainty principle;
(b) Time-frequency techniques to study random point processes with repulsive behavior (such as the polyanalytic Ginibre ensembles and zeros of Gaussian Analytic Functions);
(c) The fundamental limits of sampling when performance is measured by
length rather than by number of evaluations (mobile sensing);
(d) Analysis of the joint profile of the solutions to the time-frequency localization problem and applications to the performance of multi-taper spectral
estimators; (e) The impact of sampling geometry on condition numbers near the critical (Nyquist) density, quantitative analysis of the spectrum of frame operators, and finite-dimensional sampling/reconstruction schemes for analog problems.