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 \(L^{2} {\left( \mathbb{R} \right)}\).
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Communicated by Charles A. Micchelli
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Søndergaard, P.L. Gabor frames by sampling and periodization. Adv Comput Math 27, 355–373 (2007). https://doi.org/10.1007/s10444-005-9003-y
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DOI: https://doi.org/10.1007/s10444-005-9003-y