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 l ²(ℤ) consisting of sequences vanishing outside $\\mathbb{S}$. For f = {f _l : 0 ⩽ l ⩽ L − 1} ⊂ l ²(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 = {χ _{E l} : 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.