On the Degree of Multivariate Bernstein Polynomial Operators Original Research Article

Let σ be a d-dimensional simplex with vertices v0, ..., vd and Bn(ƒ,·) denote the nth degree Bernstein polynomial of a continuous function ƒ on σ. Dahmen and Micchelli (Stud. Sci. Hungar.23 (1988), 265-287) proved that Bn(ƒ,·) ≥ Bn+1(ƒ,·), n ∈ N, for any convex function ƒ on σ, and it is clear that a necessary and sufficient condition for the inequality to become an identity for all n ∈ N is that ƒ is an affine polynomial. Let σm be the mth simplicial subdivision of σ (which will be defined precisely later). By using a degree-raising formula, the result of Dahmen and Micchelli can be extended to Bmn(ƒ,·), ≥ Bmn+1(ƒ,·), n ∈ N, for any which is convex on every cell of σm. The objective of this paper is to derive conditions under which this inequality becomes an identity.