Convexity and generalized Bernstein polynomials

In a recent generalization of the Bernstein polynomials, the approximated function f 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 B(n)f by a one-parameter family of polynomials B(n)(o)f, where 0 < q less than or equal to 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning B(n)(q)f when f 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 B(n)(q)f is increasing, and if f is convex then B(n)(q)f is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n, B(n)(r)f less than or equal to B(n)(q)f for 0 < q less than or equal to r less than or equal to 1. This supplements the well known classical result that f less than or equal to B(n)f when f is convex.