approximation theory and statistics. In this talk I will present recent new

applications of totally positive functions (TPFs) in sampling the ory and

time-frequency analysis.

(i) We study the sampling problem for shift-invar iant

spaces generated by a TPF. These spaces ar ise the span of the integer shifts of

a TPF and are often used as a

substitute for bandlim ited functions. We give a complete

characte rization of sampling sets

for a shift-invar iant space with a TPF generator of

Gaussian typ e in the style of Beurling.

(ii) A related problem is the question of Gabor frames\,< br>i.e.\, the spanning properties of time-frequenc y 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 ex ceeds 1.

At this time this conjecture has been proved

for two important subclasses of TPFs. Fo r rational lattices it is true for arbitrary

TP Fs. So far\, TPFs seem to be the only

window fu nctions for which the fine structure of the associ ated Gabor frames is tractable.

(ii i) Yet another question in time-frequency analysis is

the existence of zeros of the Wigner distri bution (or the radar ambiguity

function). So fa r all examples of zero-free ambiguity functions ar e related to

TPFs\, e.g.\, the ambiguity functi on of the Gaussian is zero free. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR