Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

Abstract

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate {(τn,f(τn))}n for certain classes of unequally spaced data nodes {τn}n and corresponding ℓ2 sampled data {f(τn)}n. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function f which demonstrates L2 and uniform convergence on Rd to f. A simpler computational version of this recovery formula is also given at the cost of replacing L2 and uniform convergence on Rd with L2 and uniform convergence on increasingly large subsets of Rd. As a special case, the polynomial interpolants of given ℓ2 data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.