A continuity property of multivariate Lagrange interpolation

Let {St} be a sequence of interpolation schemes in Rn of degree d (i.e. for each St one has unique interpolation by a polynomial of total degree ≤ d) and total order ≤ l. Suppose that the points of St tend to 0 ∈ Rn as t→ ∞ and the Lagrange-Hermite interpolants, HSt , satisfy limt→∞HSt(x) = 0 for all monomials xα with |α| = d + 1. Theorem: limt→∞HSt (f) = T d(f) for all functions f of class Cl−1 in a neighborhood of 0. (Here T d(f) denotes the Taylor series of f at 0 to order d.) Specific examples are given to show the optimality of this result.