Abstract

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.