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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 $B_nf$ by a one-parameter family of polynomials $B^q_nf$, where $0\le q\le 1$. 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 $n$, $B_n^rf\le B^q_nf$ for $0<q\le r\le 1$. This supplements the well known classical result that $f\le B_nf$ when $f$ is convex.

41A10Approximation by polynomials
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