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# Title: Generalized Bernstein operators defined by increasing nodes

Abstract: We study certain generalizations of the classical Bernstein operators, defined via increasing sequences of nodes. Such operators are required to fix two functions, $f_0$ and $f_1$, such that $f_0 > 0$ and $f_1/ f_0$ is increasing on an interval $[a,b]$. A characterization regarding when this can be done is presented. From it we obtain, under rather general circumstances, the following necessary condition for existence: if nodes are non-{\guillemotleft}decreasing, then $(f_1/f_0)^\prime >0$ on $(a,b)$, while if nodes are strictly increasing, then $(f_1/f_0)^\prime >0$ on $[a,b]$.
 Comments: 11 pages, Example 2.8 has been corrected Subjects: Classical Analysis and ODEs (math.CA) MSC classes: 41A10 Cite as: arXiv:1803.05343 [math.CA] (or arXiv:1803.05343v2 [math.CA] for this version)

## Submission history

From: Jesus Munarriz Aldaz [view email]
[v1] Wed, 14 Mar 2018 15:10:33 GMT (11kb)
[v2] Thu, 15 Mar 2018 14:05:34 GMT (11kb)