The distance between two limit $q$-Bernstein operators

TL;DR

This paper estimates the operator norm distance between two limit q-Bernstein operators for different q and r, revealing it can range from 1 to 2 depending on their rational power relationship.

Contribution

It provides explicit bounds for the distance between limit q-Bernstein operators and characterizes when these bounds are attained.

Findings

Distance between operators can be exactly 1 or 2.

Distance depends on whether q and r are rational powers of each other.

Explicit formula for the distance when r=q^m.

Abstract

For $q\in(0,1),$ let $B_q$ denote the limit $q$-Bernstein operator. In this paper, the distance between $B_q$ and $B_r$ for distinct $q$ and $r$ in the operator norm on $C[0,1]$ is estimated, and it is proved that $1\leqslant \|B_q-B_r\|\leqslant 2,$ where both of the equalities can be attained. To elaborate more, the distance depends on whether or not $r$ and $q$ are rational powers of each other. For example, if $r^j\neq q^m$ for all $j,m\in \mathbb{N},$ then $\|B_q-B_r\|=2,$ and if $r=q^m, m\in \mathbb{N},$ then $\|B_q-B_r\|=2(m-1)/m.$