On Gabor frames generated by sign-changing windows and B-splines

For a class of compactly supported windows we characterize the frame property for a Gabor system {EmbTnag}m,n∈Z{EmbTnag}m,n∈Z, for translation parameters a   belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of “curves.” For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a,b)(a,b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Gröchenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain “hyperbolic strip.”