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 L2(ℝ). It is a generalization of Gabor bi-frame, and is new even if ℳ = L2(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 L2(S) of the form 𝒢(g,a,b) = {e2πimbxg(x − na) : m,n ∈ ℤ}. It is well-known that, if S≠ℝ, the projections of Gabor frames for L2(ℝ) onto L2(S) cannot cover all Gabor frames for L2(S). This paper presents a Zak transform-domain and a time-domain characterization of WGBFs for L2(S). These characterizations are new even if S = ℝ. Some examples are also provided to illustrate the generality of our theory.

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