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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 $\left[0,1\right]$, 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 $0\le q\le 1$. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and give new results concerning ${B}_{n}^{q}f$ 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 $n$, ${B}_{n}^{r}f\le {B}_{n}^{q}f$ for $0. This supplements the well known classical result that $f\le {B}_{n}f$ when $f$ is convex.
##### MSC:
 41A10 Approximation by polynomials
##### Keywords:
Bernstein polynomials