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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 $$0\leq q\leq 1$$. 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)

##### MSC:
 41A10 Approximation by polynomials
##### Keywords:
Bernstein polynomials
Full Text:
##### References:
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