# Mean value interpolation for points in general position

## Abstract:

The reduced interpolation conditions for mean value interpolation at a multiset of points in $\Rd$ are given explicitly, i.e., the interpolation conditions are expressed using (tangential) derivatives of the lowest possible orders averaged over the smallest possible (polytope) regions. For points in general position the result takes a particularly simple form. Here we give the Lagrange' form of the extension of the mean value interpolant to less smooth functions, and give an application of it to the finite element method. For points which are not in general position the statement and proof of the result are more technical. These are closely related to an interesting binomial identity which counts the number of allowable' submultisets of a finite multiset of points in $\Rd$. For example, if the multiset consists of $r$ points in general position each repeated $m$ times, then the identity gives $${mr-1+d\choose d} =\sum_{j=1}^d {r\choose j} \sum_{k_1=1}^m \cdots \sum_{k_j=1}^m {k_1+\cdots+k_j-1+d-j\choose d-j},$$ which reduces to the Chu--Vandermonde convolution identity when $m=1$. From our construction we are able to define multivariate analogues of Hermite--Fej\'er interpolation. We also outline the numerical calculation of the reduced functionals and the corresponding mean value interpolants.

Keywords: mean value interpolation, Kergin interpolation, Hakopian interpolation, multivariate interpolation, finite element method, binomial identity, Chu--Vandermonde identity, simplex spline, Kergin--Fej\'er interpolation

Math Review Classification: 05A19, 41A05, 65D07 (primary), 41A10, 41A63 (secondary)

Length: 25 pages

Comment: This is a substantially revised version, which includes large parts of a paper entitled `Chu-Vandermonde convolution identities and reduced Kergin interpolation' (which no longer exists)

Last updated: 27 April 2001

Status: Revision as requested by referees