Invertibility of Laurent operators and shift invariant spaces with finitely many generators

In this paper, it is shown that for a fixed (Formula presented.) (Formula presented.) is a stable set of sampling for (Formula presented.) if and only if there exists (Formula presented.) such that (Formula presented.) a.e. on (Formula presented.), where (Formula presented.) is a symbol matrix of a block Laurent operator associated with the sample set of (Formula presented.). A similar result is proved when (Formula presented.) where (Formula presented.) and (Formula presented.) is a stable set of sampling for (Formula presented.) (single generator) as well as (Formula presented.) (multi generators). A perturbation of stable set of sampling is also discussed.