ON THE DEGREE OF MULTIVARIATE BERNSTEIN POLYNOMIAL OPERATORS

[摘要] Let sigma be a d-dimensional simplex with vertices v0, ..., v(d) and B(n)(f,.) denote the nth degree Bernstein polynomial of a continuous function f on sigma. Dahmen and Micchelli (Stud Sci. Hungar. 23 (1988), 265-287) proved that B(n)(f,.) greater-than-or-equal-to B(n+1)(f,.), n is-an-element-of N, for any convex function f on sigma, and it is clear that a necessary and sufficient condition for the inequality to become an identity for all n is-an-element-of N is that f is an affine polynomial. Let sigma(m) be the mth simplicial subdivision of a (which will be defined precisely later). By using a degree-raising formula, the result of Dahmen and Micchelli can be extended to B(mn)f,.) greater-than-or-equal-to B(mn+1)(f,.), n is-an-element-of N, for any f which is convex on every cell of sigma(m). The objective of this paper is to derive conditions under which this inequality becomes an identity. (C) 1994 Academic Press, Inc.

[发布日期] 1994-07-01