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 [Formula: see text] of [Formula: see text]. It is a generalization of Gabor bi-frame, and is new even if [Formula: see text]. A WGBF for [Formula: see text] contains all information of [Formula: see text] to some extent. Let [Formula: see text], [Formula: see text], and [Formula: see text] be an [Formula: see text]-periodic subset of [Formula: see text] with positive measure. This paper is devoted to characterizing WGBFs for [Formula: see text] of the form [Formula: see text] It is well-known that, if [Formula: see text], the projections of Gabor frames for [Formula: see text] onto [Formula: see text] cannot cover all Gabor frames for [Formula: see text]. This paper presents a Zak transform-domain and a time-domain characterization of WGBFs for [Formula: see text]. These characterizations are new even if [Formula: see text]. Some examples are also provided to illustrate the generality of our theory.