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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:TALK126307@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Totally positive functions play an important role in approxima
 tion theory and statistics. In this talk I will present recent new applica
 tions of totally positive functions (TPFs) in sampling theory and time-fre
 quency analysis.  &nbsp\;  (i) We study the sampling problem for shift-inv
 ariant spaces generated by a TPF. These spaces arise the span of the integ
 er shifts of a TPF and are often used as a  substitute for bandlimited fun
 ctions.&nbsp\;&nbsp\; We give a complete  characterization of sampling set
 s  for a shift-invariant space with a TPF generator of Gaussian type in th
 e style of Beurling.  &nbsp\;  (ii) A related problem is the question of G
 abor frames\, i.e.\, the spanning properties of time-frequency shifts of a
  given function. It is conjectured that the lattice shifts of a TPF genera
 te a frame\, if and only if the density of the lattice&nbsp\; exceeds 1. A
 t this time this conjecture has been proved&nbsp\; for two important subcl
 asses of TPFs. For&nbsp\; rational lattices it is true for arbitrary TPFs.
 &nbsp\; So far\, TPFs seem to be the only window functions for which the f
 ine structure of the associated Gabor&nbsp\; frames is tractable.  &nbsp\;
   (iii) Yet another question in time-frequency analysis is the existence o
 f zeros of the Wigner distribution (or the radar ambiguity function). So f
 ar all examples of zero-free ambiguity functions are related to TPFs\, e.g
 .\, the ambiguity function of the Gaussian is zero free.
LOCATION:Seminar Room 1\, Newton Institute
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