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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:TALK126307AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/126307
DESCRIPTION:Totally positive functions play an important role
in approximation theory and statistics. In this ta
lk I will present recent new applications of total
ly positive functions (TPFs) in sampling theory an
d time-frequency analysis. \; (i) We study
the sampling problem for shift-invariant spaces ge
nerated by a TPF. These spaces arise the span of t
he integer shifts of a TPF and are often used as a
substitute for bandlimited functions. \;&nbs
p\; We give a complete characterization of sampli
ng sets for a shift-invariant space with a TPF ge
nerator of Gaussian type in the style of Beurling.
\; (ii) A related problem is the question
of Gabor frames\, i.e.\, the spanning properties o
f time-frequency shifts of a given function. It is
conjectured that the lattice shifts of a TPF gene
rate a frame\, if and only if the density of the l
attice \; exceeds 1. At this time this conject
ure has been proved \; for two important subcl
asses of TPFs. For \; rational lattices it is
true for arbitrary TPFs. \; So far\, TPFs seem
to be the only window functions for which the fin
e structure of the associated Gabor \; frames
is tractable. \; (iii) Yet another question
in time-frequency analysis is the existence of ze
ros of the Wigner distribution (or the radar ambig
uity function). So far all examples of zero-free a
mbiguity functions are related to TPFs\, e.g.\, th
e ambiguity function of the Gaussian is zero free.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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