11 research outputs found
Characterization and computation of canonical tight windows for Gabor frames
Let be a Gabor frame for for given window .
We show that the window that generates the canonically
associated tight Gabor frame minimizes among all windows
generating a normalized tight Gabor frame. We present and prove versions of
this result in the time domain, the frequency domain, the time-frequency
domain, and the Zak transform domain, where in each domain the canonical
is expressed using functional calculus for Gabor frame operators. Furthermore,
we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames.
Finally, a Newton-type method for a fast numerical calculation of \ho is
presented. We analyze the convergence behavior of this method and demonstrate
the efficiency of the proposed algorithm by some numerical examples
Affine and quasi-affine frames for rational dilations
In this paper we extend the investigation of quasi-affine systems, which were originally introduced by Ron and Shen [J. Funct. Anal. 148 (1997), 408–447] for integer, expansive dilations, to the class of rational, expansive dilations. We show that an affine system is a frame if, and only if, the corresponding family of quasi-affine systems are frames with uniform frame bounds. We also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi-affine frame such that its affine counterpart is not a frame
A Wiener Lemma for the discrete Heisenberg group
Analysis and Stochastic