A continuity property of multivariate Lagrange interpolation

Let {S-t} be a sequence of interpolation schemes in R-n of degree d (i.e. for each St one has unique interpolation by a polynomial of total degree less than or equal to d) and total order less than or equal to l. Suppose that the points of S-t tend to 0 epsilon R-n as t --> infinity and the Lagrange-Hermite interpolants, H-S?t satisfy lim(t-->infinity) HSt (x(alpha)) = 0 for all monomials x(alpha) with \alpha\ = d + 1. Theorem: lim(t-->infinity) H-S?t(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.