Convexity and Variation Diminishing Property of Bernstein Polynomials over Triangles

Abstract

Let Bⁿ(f;P) denote the Bernstein polynomials over triangle T and ${\hat f_n}$ denote the Bézier net associated with Bⁿ(f;P). A certain type of variations of ${\hat f_n}$ is introduced by GOODMAN quite recently. In the present paper the corresponding variation of Bⁿ(f;P) is defined by integration of the absolute value of the Laplacian of BP(f;P) over T. It is shown that the variation of ${\hat f_n}$ is always greater or equal to the variation of Bⁿ(f;P). The equality holds if and only if ${\hat f_n}$ is either convex (or concave) over T. The convexity of ${\hat f_n}$ implies the convexity of Bⁿ(f;P). As an application we receive a simple proof of a theorem due to Chang and Davis.