Quasi-Interpolation in Shift Invariant Spaces

Let s ≥ 1 be an integer, φ s → be a compactly supported function, and S φ denote the linear span of φ · − k k ∈ s . We consider the problem of approximating a continuous function f s → on compact subsets of s from the classes S φ h· , h > 0, based on samples of the function at scattered sites in s . We demonstrate how classical polynomial inequalities lead to the construction of local, quasi-interpolatory operators for this purpose