Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets

Abstract

Given L, N, M ∈ ℕ and an Nℕ-periodic set \(\mathbb{S}\) in ℤ, let \(l^2 \left( \mathbb{S} \right)\) be the closed subspace of l2(ℤ) consisting of sequences vanishing outside \(\\mathbb{S}\). For f = {fl: 0 ⩽ l ⩽ L − 1} ⊂ l2(ℤ), we denote by \(\mathcal{G}\)(f, N, M) the Gabor system generated by f, and by \(\mathcal{L}\)(f, N, M) the closed linear subspace generated by \(\mathcal{G}\)(f, N, M). This paper addresses density results, frame conditions for a Gabor system \(\mathcal{G}\)(g, N, M) in \(l^2 \left( \mathbb{S} \right)\), and Gabor duals of the form \(\mathcal{G}\)(a, N, M) in some \(\mathcal{L}\)(h, N, M) for a frame \(\mathcal{G}\)(g, N, M) in \(l^2 \left( \mathbb{S} \right)\) (so-called oblique duals). We obtain a density theorem for a Gabor system \(\mathcal{G}\)(g, N, M) in \(l^2 \left( \mathbb{S} \right)\), and show that such condition is sufficient for the existence of g = {χEl: 0 ⩽ l ⩽ L − 1} with \(\mathcal{G}\)(g, N, M) being a tight frame for \(l^2 \left( \mathbb{S} \right)\). We characterize g with \(\mathcal{G}\)(g, N, M) being respectively a frame for \(\mathcal{L}\)(g, N, M) and \(l^2 \left( \mathbb{S} \right)\). Moreover, for given frames \(\mathcal{G}\)(g, N, M) in \(l^2 \left( \mathbb{S} \right)\) and \(\mathcal{G}\)(h, N, M) in \(\mathcal{L}\)(h, N, M), we establish a criterion for the existence of an oblique Gabor dual of g in \(\mathcal{L}\)(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.