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Totally positive functions in sampling theory and time-frequency analysis

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ASCW03 - Approximation, sampling, and compression in high dimensional problems

Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and time-frequency analysis.   (i) We study the sampling problem for shift-invariant spaces generated by a TPF . These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions.   We give a complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussian type in the style of Beurling.   (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice  exceeds 1. At this time this conjecture has been proved  for two important subclasses of TPFs. For  rational lattices it is true for arbitrary TPFs.  So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor  frames is tractable.   (iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free.

This talk is part of the Isaac Newton Institute Seminar Series series.

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