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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Totally positive functions in sampling theory and
time-frequency analysis - Karlheinz Groechenig (
University of Vienna)
DTSTART;TZID=Europe/London:20190621T095000
DTEND;TZID=Europe/London:20190621T104000
UID:TALK126310AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/126310
DESCRIPTION: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 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
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