### CAS Seminar

# Tobias Kloos

## "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.