Approximation properties of the $q$-Balázs-Szabados operators in the case $q\geq1$

Abstract

This paper deals with approximation properties of the newly defined $q$-generalization of the Balázs-Szabados operators in the case $q\geq1$. Quantitative estimates of the convergence and Voronovskaja type theorem are given. In particular, it is proved that the rate of approximation by the $q$-Balázs-Szabados ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical Balázs-Szabados ($q=1$) operators. The results are new even for the classical case $q=1$.