Tobias Kloos

Tobias Kloos, Joachim Stöckler, Karlheinz Gröchenig

The usefulness of Gabor frames depends on the easy computability of a suitable dual window. This question is addressed under several aspects: several versions of Schulz's iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability. For Gabor frames with totally positive windows or with exponential B-splines a direct algorithm yields a family of exact dual windows with compact support. It is shown that these dual windows converge exponentially fast to the canonical dual window.

Tobias Kloos

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.