Frames and point distributions in time-frequency analysis

This project sits at the crossroads of pure and applied mathematics, in the field of Applied Harmonic Analysis, whose relation with neighboring sciences resembles that of cell membrane, in a permanent osmotic relation with other corners of Mathematics and Science at large, including Signal Analysis, Acoustics, Data Science, Quantum Physics, Information Theory and Communications Engineering. A recurrent theme of the proposed research will be the rigorous mathematical study of heuristics, numerical experiments and methods using point set distributions that have proved to be useful on the applied side of time- frequency and time-scale analysis. This study will lead, in a first stage, to theoretical mathematical papers, turning into theorems the experimentally observed phenomena. In a second stage, this enhanced understanding of existing methodology, is expected to lead to the development of new methods and to the improvement of existing ones. These point distributions can be deterministic, as in classical frames and sampling, and random, as in random sampling and compressive sensing. We will first study methods using point distributions on the euclidean plane R. One important hypothesis that we plan to put into firm mathematical ground, is that the well-spreadness of a finite number of points distributed by repulsive processes (more precisely, determinantal point processes) inside a bounded region, is likely to improve the conditioning profile of sampling and recovery methods of random sampling. Another object of study will be Gaussian Analytic Functions (GAF`s), whose zeros match the zeros of white noise spectrograms and scalograms with analiticity-induced windows (Gaussian for STFT, Klauder/Cauchy for wavelets). But the project goals go beyond GAF`s. Indeed, for the analysis of high-resolution time-frequency methods involving averages of spectrograms with Hermite windows, like Bayran-Baraniuk Wigner-Ville spectral estimator, Xiao-Flandrin multitaper reassignment and Daubechies- Wang-Wu, ConceFT, since the zeros of spectrograms with Hermite windows match the zeros of a GAF raised to higher Landau Levels, where the functions are analytic no more. This leads naturally to Gaussian Polyanalytic Functions, which poses significant challenges in their analysis, precisely due to their non-analiticity.