# Mean value interpolation for points in general position

## by Shayne Waldron

## 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

## Availability:

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