Gabor frames by sampling and periodization

By sampling the window of Gabor frame for L-2 (R) belonging to Feichtingers algebra S-0 (R), one obtains a Gabor frame for l(2) (Z). 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 (R), l(2) (Z), L-2 ([O,L]) and C-L. Some new results about the 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 (R).