2,551 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
Gabor analysis over finite Abelian groups
The topic of this paper are (multi-window) Gabor frames for signals over
finite Abelian groups, generated by an arbitrary lattice within the finite
time-frequency plane. Our generic approach covers simultaneously
multi-dimensional signals as well as non-separable lattices. The main results
reduce to well-known fundamental facts about Gabor expansions of finite signals
for the case of product lattices, as they have been given by Qiu, Wexler-Raz or
Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a
central role is given to spreading function of linear operators between
finite-dimensional Hilbert spaces. Another relevant tool is a symplectic
version of Poisson's summation formula over the finite time-frequency plane. It
provides the Fundamental Identity of Gabor Analysis.In addition we highlight
projective representations of the time-frequency plane and its subgroups and
explain the natural connection to twisted group algebras. In the
finite-dimensional setting these twisted group algebras are just matrix
algebras and their structure provides the algebraic framework for the study of
the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe
On the Schr\"odinger equation with potential in modulation spaces
This work deals with Schr\"odinger equations with quadratic and sub-quadratic
Hamiltonians perturbed by a potential. In particular we shall focus on bounded,
but not necessarily smooth perturbations. We shall give a representation of
such evolution as the composition of a metaplectic operator and a
pseudodifferential operator having symbol in certain classes of modulation
spaces. About propagation of singularities, we use a new notion of wave front
set, which allows the expression of optimal results of propagation in our
context. To support this claim, many comparisons with the existing literature
are performed in this work.Comment: 25 page
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