Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Convexity and generalized Bernstein polynomials

Tim N. T. Goodmana1, Halil Oruça1* and George M. Phillipsa2

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 S0013091500020101_inline1, where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning S0013091500020101_inline2 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 S0013091500020101_inline3 is increasing, and if f is convex then S0013091500020101_inline4 is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n S0013091500020101_inline5 This supplements the well known classical result that S0013091500020101_inline6 when f is convex.

(Received May 28 1997)


* Supported by Dokuz Eylūl University, Izmir, Turkey.