On the Degree of Multivariate Bernstein Polynomial Operators

Abstract

Let σ be a d-dimensional simplex with vertices v 0, ..., v d and B n (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 B n (ƒ,·) ≥ B n+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 σ m 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 B mn (ƒ,·), ≥ B mn+1 (ƒ,·), n ∈ N, for any which is convex on every cell of σ m . The objective of this paper is to derive conditions under which this inequality becomes an identity.