Weak Gabor bi-frames on periodic subsets of the real line

In this paper, we introduce the concept of weak Gabor bi-frame (WGBF) in a general closed subspace M of L-2(R). It is a generalization of Gabor bi-frame, and is new even if M = L-2(R). A WGBF for M contains all information of M to some extent. Let a, b > 0, and S be an aZ-periodic subset of R with positive measure. This paper is devoted to characterizing WGBFs for L-2(S) of the form G(g, a, b) = {e(2 pi mbx)g(x - na) : m, n is an element of Z}. It is well-known that, if S not equal R, the projections of Gabor frames for L-2(R) onto L-2(S) cannot cover all Gabor frames for L-2(S). This paper presents a Zak transform-domain and a time-domain characterization of WGBFs for L-2(S). These characterizations are new even if S = R. Some examples are also provided to illustrate the generality of our theory.