On sampling in shift invariant spaces

A necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived. Then this result is applied, respectively, to the regular sampling and the perturbation of regular sampling in shift invariant spaces. A simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained. Furthermore, an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces. The derived estimate is easy to calculate, and shown to be optimal in some shift invariant spaces. The algorithm to calculate the reconstruction frame is also presented.