For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system:
Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχ
S, a, b) of a frame 𝒢(g, a, b) in L
2(ℝ) onto L
2(S) is a frame for L
2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L
2(ℝ) for any g ∈ L
2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L
2(S). So the projections of Gabor frames in L
2(ℝ) onto L
2(S) cannot cover all Gabor frames in L
2(S). This paper considers Gabor systems in L
2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L
2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).
