Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let V ϕ be a closed subspace of L 2 ( R ) generated from the integer shifts of a continuous function ϕ with a certain decay at infinity and a non-vanishing property for the function Φ † ( γ ) = ∑ n ∈ Z ϕ ( n ) e − i n γ on [ − π , π ] . In this paper we determine a positive number δ ϕ so that the set { n + δ n } n ∈ Z is a set of stable sampling for the space V ϕ for any selection of the elements δ n within the ranges ± δ ϕ . We demonstrate the resulting sampling formula (called perturbation formula) for functions f ∈ V ϕ and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.