Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let Vϕ be a closed subspace of L2(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∈Zϕ(n)e−inγ on [−π,π]. In this paper we determine a positive number δϕ so that the set {n+δn}n∈Z is a set of stable sampling for the space Vϕ for any selection of the elements δn within the ranges ±δϕ. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈Vϕ 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.