# 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.

## Key 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 $[a,b]$.