Approximation by q -Bernstein Polynomials in the Case q → 1 +

Abstract

Let $B_{n,q}(f;x)$, $q\in (0,\infty )$ be the $q$-Bernstein polynomials of a function $f\in C[\mathrm{0,1}]$. It has been known that, in general, the sequence $\left(B_{n,q_n}(f)\right)$ with $q_n\to 1+$ is not an approximating sequence for $f\in C[\mathrm{0,1}]$, in contrast to the standard case $q_n\to 1-$. In this paper, we give the sufficient and necessary condition under which the sequence $\left(B_{n,q_n}(f)\right)$ approximates $f$ for any $f\in C[\mathrm{0,1}]$ in the case $q_n>1$. Based on this condition, we get that if for sufficiently large $n$, then $\left(B_{n,q_n}(f)\right)$ approximates $f$ for any $f\in C[\mathrm{0,1}]$. On the other hand, if $\left(B_{n,q_n}(f)\right)$ can approximate $f$ for any $f\in C[\mathrm{0,1}]$ in the case $q_n>1$, then the sequence $(q_n)$ satisfies ${\overline{\text{lim}}}_{n\to \mathrm{\infty }}n(q_n-1)\le \text{ln}2$.