Convexity and generalized bernstein polynomials

Abstract

In a recent generalization of the Bernstein polynomials, the approximated function / is evaluated at points spaced at intervals which are in geometric progression on [0,1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials Bnqf, where 0 < q < 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning Bnqf when/ is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then Bnqf is increasing, and if f is convex then Bnqf is convex, generalizing well known results when q -1. It is also shown that if f is convex then, for any positive integer n, Bnq f < Bnq,f for 0 < q < r < 1. This supplements the well known classical result that f < Bnf when f is convex.