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 \(B_nf\) by a one-parameter family of polynomials \(B^q_nf\), where \(0\leq q\leq 1\). This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and give new results concerning \(B_n^qf\) 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 \(B_n^qf\) is increasing, and if \(f\) is convex then \(B^q_nf\) is convex, generalizing well known results when \(q=1\). It is also shown that if \(f\) is convex, then for any positive integer \(n\), \(B_n^rf\leq B^q_nf\) for \(0<q\leq r\leq 1\). This supplements the well known classical result that \(f\leq B_nf\) when \(f\) is convex.
Reviewer: E.Deeba (Houston)


41A10 Approximation by polynomials
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