Multivariate polynomial interpolation and sampling in Paley-Wiener spaces

摘要

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 {(τ_n,f(τ_n))}n for certain classes of unequally spaced data nodes {τn}n and corresponding ?_2 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 L_2 and uniform convergence on Rd to f. A simpler computational version of this recovery formula is also given at the cost of replacing L_2 and uniform convergence on R{double-struck}~d with L_2 and uniform convergence on increasingly large subsets of R{double-struck}~d. As a special case, the polynomial interpolants of given ?_2 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.