# Integral error formulae for the scale of mean value interpolations which includes Kergin and Hakopian interpolation

## Abstract:

In this paper, we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of Kergin and Hakopian. This formula involves only derivatives of order one higher than the degree of the interpolating polynomial space, and from it we can obtain sharp $L_\infty$-estimates. These $L_\infty$-estimates are precisely those that numerical analysts want, to guarantee that a scheme based on such an interpolation has the maximum possible order.

Keywords: scale of mean value interpolations, Kergin interpolation, Hakopian interpolation, Lagrange interpolation, Hermite interpolation, Hermite-Genocchi formula, multivariate divided difference, plane wave, lifting, Radon transform

Math Review Classification: 41A05, 41A63, 41A80 (primary), 41A10, 41A44, 44A12 (secondary)

Length: 18 pages

Comment: Written in TeX. This paper is the basis of Chapter 1 of Shayne Waldron's dissertation.

Last updated: 24 October 1997

Status: Appeared in Numer. Math. 77 (1997), no. 1, 105--122.