Generalized Bernstein operators defined by increasing nodes

TL;DR

This paper explores generalized Bernstein operators defined by increasing nodes, providing conditions for their existence based on the properties of functions they fix and the monotonicity of node sequences.

Contribution

It characterizes when such generalized Bernstein operators can be constructed, linking node monotonicity to the derivative of the ratio of the fixed functions.

Findings

Necessary condition for existence when nodes are non-decreasing: (f_1/f_0)' > 0.

Necessary condition for strictly increasing nodes: (f_1/f_0)' > 0 on [a,b].

Provides a framework for constructing Bernstein operators with prescribed fixing functions.

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 $[a,b]$.