CAS Seminar

Tobias Kloos

(TU Dortmund)

"Zak transform of totally positive functions and lower framebounds for Gabor frames"

Date: Wed, April 24, 2013
Time: 14:15
Place: Research I Seminar Room

Abstract: In time-frequency analysis the Gabor transformation is a very popular method to analyse the behaviour of a given signal in time and frequency. An important problem in Gabor analysis is to find window functions, such that the translates and modulates induce a frame. Moreover, it is desirable to know all lattices, where a given function constitutes a frame.

Until recently, the framesets of only a few functions, like the one- and two-sided exponential, the Gaussian and the hyperbolic secant, were known. In 2012, Gröchenig and Stöckler proved that every totally positive function of finite order induces a Gabor frame on all possible lattices for such windows.

Estimates of the lower framebounds depending on the lattices are important in order to describe the numerical stability. Finding framebounds independent of the order may provide a method to prove that also some totally positive functions of infinite order are inducing frames.

For this reason, we show that the Zak transform of totally positive functions of finite order can be expressed in terms of the Zak transform of an exponential B-spline. Due to this identity and certain well-known properties of exponential B-splines, we show that the Zak transform of TP functions of finite order has exactly one zero in [0,1)x[0,1). Moreover, we present some analytic lower bounds for frames induced by even two-sided exponentials. This is joint work with Joachim Stöckler from TU Dortmund.