414 research outputs found
Co-compact Gabor systems on locally compact abelian groups
In this work we extend classical structure and duality results in Gabor
analysis on the euclidean space to the setting of second countable locally
compact abelian (LCA) groups. We formulate the concept of rationally
oversampling of Gabor systems in an LCA group and prove corresponding
characterization results via the Zak transform. From these results we derive
non-existence results for critically sampled continuous Gabor frames. We obtain
general characterizations in time and in frequency domain of when two Gabor
generators yield dual frames. Moreover, we prove the Walnut and Janssen
representation of the Gabor frame operator and consider the Wexler-Raz
biorthogonality relations for dual generators. Finally, we prove the duality
principle for Gabor frames. Unlike most duality results on Gabor systems, we do
not rely on the fact that the translation and modulation groups are discrete
and co-compact subgroups. Our results only rely on the assumption that either
one of the translation and modulation group (in some cases both) are co-compact
subgroups of the time and frequency domain. This presentation offers a unified
approach to the study of continuous and the discrete Gabor frames.Comment: Paper (v2) shortened. To appear in J. Fourier Anal. App
Balian-Low Theorems in Several Variables
Recently, Nitzan and Olsen showed that Balian-Low theorems (BLTs) hold for
discrete Gabor systems defined on . Here we extend these results
to a multivariable setting. Additionally, we show a variety of applications of
the Quantitative BLT, proving in particular nonsymmetric BLTs in both the
discrete and continuous setting for functions with more than one argument.
Finally, in direct analogy of the continuous setting, we show the Quantitative
Finite BLT implies the Finite BLT.Comment: To appear in Approximation Theory 16 conference proceedings volum
Total positive Funktionen und exponentielle B-Splines in der Zeit-Frequenz-Analyse
Die vorliegende Arbeit behandelt die Anwendung von Schoenbergs total positiven Funktionen, sowie exponentieller B-Splines in der Zeit-Frequenz-Analyse. Wir werden aufzeigen, dass sich diese Funktionen sehr gut als Fenster der Gabor-Transformation eignen und darĂĽber hinaus anwendungsorientierte Algorithmen zur Implementierung angeben.
Nach einer kurzen Einführung in die Thematik betrachten wir zunächst die Zak-Transformierten der genannten Funktionen und charakterisieren für eine Teilklasse der total positiven Funktionen ihre Nullstellenmengen. Dies liefert bereits Gabor-Frames mit ganzzahligem oversampling und gibt Hinweise über die Existenz im Fall von rationalem oversampling. Anschließend beschäftigen wir uns mit Gabor-Systemen auf beliebigen separablen Gittern und legen einige Situationen dar, in welchen die Systeme der betrachteten Funktionen einen Frame liefern. In diesen Fällen beschreiben wir Algorithmen zur Konstruktion unendlich vieler verschiedener Duale mit kompakten Trägern, welche gegen den kanonischen Dual konvergieren. Weiter geben wir einen kurzen Einblick in die sich ergebenden Möglichkeiten zur Bildung von Gabor-Frames über nicht-separablen Gittern. Abschließend erläutern wir, wie die gewonnenen Erkenntnisse genutzt werden können, um diskrete Gabor-Frames und deren Duale zu konstruieren.This thesis deals with the applicability of Schoenberg's totally positive functions and exponential B-splines in time-frequency analysis. We show that these functions provide excellent windows for the Gabor transform and give some application-oriented algorithms.
After a brief introduction to the topic, we consider the Zak transform of totally positive functions and exponential B-splines. We characterize the zero set of this transform for a special subclass of totally positive functions, which directly leads to the existence of Gabor frames with integer oversampling, and gives some information about the case of rational oversampling. Afterwards, we deal with Gabor systems on arbitrary separable lattices and present some concrete situations, where the considered functions yield a frame. In these cases, we describe algorithms for constructing infinitely many different duals with compact support, which converge to the canonical dual. We also provide a brief insight how to handle Gabor systems on non-separable lattices. Finally, we explain to construct discrete Gabor frames and their duals in the aforementioned situations
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