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Gabor frames by sampling and periodization

Authors:	Peter L Søndergaard

Institution:	(1) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark

Abstract:	By sampling the window of a Gabor frame for $L^{2} {\left( \mathbb{R} \right)}$ belonging to Feichtinger’s algebra, $S_{0} {\left( \mathbb{R} \right)}$ , one obtains a Gabor frame for $l^{2} {\left( \mathbb{Z} \right)}$ . In this article we present a survey of results by R. Orr and A.J.E.M. Janssen and extend their ideas to cover interrelations among Gabor frames for the four spaces $L^{2} {\left( \mathbb{R} \right)}$ , $l^{2} {\left( \mathbb{Z} \right)}$ , $L^{2} {\left( {{\left {0,L} \right]}} \right)}$ and $\mathbb{C}^{L}$ . Some new results about general dual windows with respect to sampling and periodization are presented as well. This theory is used to show a new result of the Kaiblinger type to construct an approximation to the canonical dual window of a Gabor frame for $L^{2} {\left( \mathbb{R} \right)}$ .

Keywords:	Gabor systems frames sampling periodization Poisson summation
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