# Multivariate Bernstein operators and redundant systems

## Tan Do and Shayne Waldron

## Abstract:

The Bernstein operator $B_n$ for a simplex in $\R^d$ is
naturally defined via the Bernstein basis obtained
from the barycentric coordinates given by its vertices.
Here we consider a generalisation of this basis and the Bernstein
operator, which is obtained from generalised barycentric coordinates
that are given for
more general configurations of points in $\R^d$.
We call the associated polynomials a Bernstein frame, as they
span the polynomials of degree $\le n$, but may not be a basis.
By using this redundant system we are able to give
geometrically motivated proofs of some basic properties of the
corresponding generalised
Bernstein operator, such
as the fact it is degree reducing and converges for all polynomials.
We also consider the conditions for polynomials in this Bernstein form
to join smoothly.

**Keywords:**
multivariate Bernstein operator,
barycentric coordinates,
frames,
redundant system
Stirling numbers,
blossoming,
de Casteljau algorithm

**Math Review Classification:**
Primary 41A10, 41A36 65D17;
Secondary 15A18, 42C15

**Length:** 20 pages

**Last Updated:** 19 January 2015

## Availability: