On the Degree of Multivariate Bernstein Polynomial Operators yCharles

Let be a d-dimensional simplex with vertices v 0 ; ; v d and B n (f;) denote the n th degree Bernstein polynomial of a continuous function f on. Dahmen and Micchelli 4] proved that B n (f;) B n+1 (f;); n 2 N; for any convex function f on , and it is clear that a necessary and suucient condition for the inequality to become an identity for all n 2 N is that f is an aane polynomial. Let m be the m th simplicial subdivision of (which will be deened precisely later). By using a degree-raising formula, the result of Dahmen and Micchelli can be extended to B mn (f;) B mn+1 (f;); n 2 N; for any f which is convex on every cell of m. The objective of this paper is to derive conditions under which this inequality becomes an identity.