Multivariate Bernstein operators and redundant systems

The Bernstein operator BnBn for a simplex in RdRd 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 RdRd. We call the associated polynomials a Bernstein frame, as they span the polynomials of degree ≤n≤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.