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SUMMARY:Totally positive functions in sampling theory and   time-frequency
  analysis - Karlheinz Groechenig (University of Vienna)
DTSTART:20190621T085000Z
DTEND:20190621T094000Z
UID:TALK126310@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Totally positive functions play an important role in<br>approx
 imation theory and statistics. In this talk I will present recent new<br>a
 pplications of totally positive functions (TPFs) in sampling theory and<br
 >time-frequency analysis.<br><br><br><br>(i) We study the sampling problem
  for shift-invariant<br>spaces generated by a TPF. These spaces arise the 
 span of the integer shifts of<br>a TPF and are often used as a<br><br>subs
 titute for bandlimited functions. We give a complete<br><br>characterizati
 on of sampling sets<br><br>for a shift-invariant space with a TPF generato
 r of<br>Gaussian type in the style of Beurling.<br><br><br><br>(ii) A rela
 ted problem is the question of Gabor frames\,<br>i.e.\, the spanning prope
 rties of time-frequency shifts of a given function. It<br>is conjectured t
 hat the lattice shifts of a TPF generate a frame\, if and only<br>if the d
 ensity of the lattice exceeds 1.<br>At this time this conjecture has been 
 proved<br>for two important subclasses of TPFs. For rational lattices it i
 s true for arbitrary<br>TPFs. So far\, TPFs seem to be the only<br>window 
 functions for which the fine structure of the associated Gabor frames is t
 ractable.<br><br><br><br>(iii) Yet another question in time-frequency anal
 ysis is<br>the existence of zeros of the Wigner distribution (or the radar
  ambiguity<br>function). So far all examples of zero-free ambiguity functi
 ons are related to<br>TPFs\, e.g.\, the ambiguity function of the Gaussian
  is zero free.
LOCATION:Seminar Room 1\, Newton Institute
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