Problems in Time-Frequency Analysis

This project is devoted to several problems in time-frequency analysis, which is a modern branch of harmonic analysis. (i) The first topic is the fine structure of Gabor frames. These are frames generated by time-frequency shifts of a single ``window`` function. The problem is to determine all lattices that generate a Gabor frame for a fixed window function. Based on a recent discovery that explains the fine structure for totally positive window functions, there is reason to believe that a generic lattice with density exceeding one generates a frame and that the lack of the frame property is caused by specific obstructions. (ii) The second topic is the investigation of a generalized time-frequency analysis that includes not only time- frequency shifts, but also chirps and similar operators. The generalized time-frequency analyses arise from an irreducible, unitary, square-integrable representation of a nilpotent group. The objective is to study the frames and bases and the coorbit spaces related to such representations of nilpotent groups. The generalized time-frequency analysis is motivated by the discovery of orthonormal bases in the orbit of square-integrable representations of certain nilpotent groups. (iii) The third topic is the investigation of several specific problems about the time-frequency analysis of operators. These are the study of the time evolution of the Schroedinger equation with an initial condition in a modulation spaces; the invertibility of time-frequency localization operators; and a new concept of variable bandwidth and associated sampling theorems. The time-frequency analysis of operators is part of a long-term program of the local research group.

The project Problems in Time-frequency Analysis dealt with the representation of functions or signals by means of time-frequency information or samples.Concerning the recovery of signals or functions from time-frequency data, we have proved a new and optimal result for so-called totally positive generators. This result gives a partial solution to a 25-year old problem and was ranked a break-through by the referees.The time-frequency problem (Gabor analysis in mathematical language) is closely related to sampling in shift-invariant spaces. Even for this general class of functions, which goes far beyond the standard bandlimited signal, on average one needs one sample per time unit to recover a function in a shift-invariant space with Gaussian generator. This result validates the heuristic reasonings in the engineering literature.On a fundamental and more abstract level, we established the existence of critical density that is necessary for the complete recovery of a function. This is an information theoretic quantity which in simple cases amounts to the maximum frequency or bandwidth of a signal.In this context we studied a new concept of variable bandwidth and explored its properties. We established a sampling theorem for variable bandwidth (what in engineering would be called the analog-digital conversion) and found the correct expression for the critical sampling density.