A continuity property of multivariate Lagrange interpolation

Abstract:

Let $\{S_{t}\}$ be a sequence of interpolation schemes in ${\mathbb {R}}^{n}$ of degree $d$ (i.e. for each $S_{t}$ one has unique interpolation by a polynomial of total degree $\leq d)$ and total order $\leq l$. Suppose that the points of $S_{t}$ tend to $0 \in {\mathbb {R}}^{n}$ as $t \to \infty$ and the Lagrange-Hermite interpolants, $H_{S_{t}}$, satisfy $\lim _{t\to \infty } H_{S_{t}} (x^{\alpha }) = 0$ for all monomials $x^{\alpha }$ with $|\alpha | = d+1$. Theorem: $\lim _{t\to \infty } H_{S_{t}} (f) = T^{d} (f)$ for all functions $f$ of class $C^{l-1}$ 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.