a1 Department of Mathematics and Computer Science, University of Dundee, Dundee, DD1 4HN
a2 Mathematical Institute, University of St Andrews, North Haugh St Andrews, Fife, K Y 16 9SS
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 Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning 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 is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.
(Received May 28 1997)
* Supported by Dokuz Eylūl University, Izmir, Turkey.