A continuity property of multivariate Lagrange interpolation

Abstract

Let { S t } \{S_{t}\} be a sequence of interpolation schemes in R n {\mathbb {R}}^{n} of degree d d (i.e. for each S t S_{t} one has unique interpolation by a polynomial of total degree ≤ d ) \leq d) and total order ≤ l \leq l . Suppose that the points of S t S_{t} tend to 0 ∈ R n 0 \in {\mathbb {R}}^{n} as t → ∞ t \to \infty and the Lagrange-Hermite interpolants, H S t H_{S_{t}} , satisfy lim t → ∞ H S t ( x α ) = 0 \lim _{t\to \infty } H_{S_{t}} (x^{\alpha }) = 0 for all monomials x α x^{\alpha } with | α | = d + 1 |\alpha | = d+1 . Theorem: lim t → ∞ H S t ( f ) = T d ( f ) \lim _{t\to \infty } H_{S_{t}} (f) = T^{d} (f) for all functions f f of class C l − 1 C^{l-1} in a neighborhood of 0 0 . (Here T d ( f ) T^{d} (f) denotes the Taylor series of f f at 0 to order d d .) Specific examples are given to show the optimality of this result.