科研之友
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摘要
Given L, N, M is an element of N and an NZ-periodic set S in Z, let l(2)(S) be the closed subspace of l(2)(Z) consisting of sequences vanishing outside S. For f = {f(l) : 0 <= l <= L - 1} subset of l(2)(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l(2)(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l(2)(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l(2)(S), and show that such condition is sufficient for the existence of g = {chi(El) : 0 <= l <= L - 1} with G(g, N, M) being a tight frame for l(2)(S). We characterize g with G(g, N, M) being respectively a frame for L(g, N, M) and l(2)(S). Moreover, for given frames G(g, N, M) in l(2)(S) and G(h, N, M) in L(h, N, M), we establish a criterion for the existence of an oblique Gabor dual of g in 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.
