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Convexity and generalized Bernstein polynomials. (English) Zbl 0930.41010
Authors’ abstract. 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 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 q f, where 0q1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and give 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 fB n q f for 0<qr1. This supplements the well known classical result that fB n f when f is convex.
MSC:
41A10Approximation by polynomials