Gabor frames by sampling and periodization

By sampling the window of a Gabor frame for [Equation] belonging to Feichtinger's algebra, [Equation], one obtains a Gabor frame for [Equation]. 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 [Equation], [Equation], [Equation] and [Equation]. 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 [Equation].