Frames in Hilber spaces extend the concept of an orthonormal basis. They are more flexible for construction, allowing non-orthogonaliry and redundancy (overcompleteness), and still guarantee perfect and stable reconstruction via a dual frame. For signal processing, frames of specific structure called Gabor frames play an essential role. It is well known that the canonical dual of a Gabor frame is also with Gabor structure, but it may fail some other nice properties desired for applications, e.g. compact support, smoothness, good time-frequency localization, etc. Thus other dual frames are also of interest.
Although not all the dual frames of an overcomplete Gabor frame need to have the Gabor structure, yet there are infinitely many dual Gabor frames and there are well known characterizations of all these duals. For computational purpoces, the duals that have compact support are of significant importance.
In this talk we will report on recent results on the topic. We present a characterization of all the dual Gabor frames with compact support (when such ones exist). We also consider an iterative procedure for approximation of the canonical dual via compactly supported dual frames. In certain cases, each iteration step of this procedure gives a dual window from certain modulation spaces or from the Schwartz space.