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

## by Shayne Waldron

## 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.

## Availability:

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