Approximation properties of the $q$-Bal\'{a}zs-Szabados operators in the case $q\geq1$

TL;DR

This paper investigates the approximation capabilities of a new q-generalization of Balázs-Szabados operators for q≥1, providing quantitative convergence estimates and a Voronovskaja theorem, revealing improved approximation rates for q>1.

Contribution

It introduces and analyzes the approximation properties of the q-Balázs-Szabados operators for q≥1, including convergence rates and a Voronovskaja theorem, extending classical results.

Findings

Rate of approximation by q-Balázs-Szabados operators is of order q^{-n} for q>1.

The classical case q=1 has a different approximation order of 1/n.

Results are new even for the classical case q=1.

Abstract

This paper deals with approximation properties of the newly defined $q$-generalization of the Bal\'{a}zs-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 $q$-Bal\'{a}zs-Szabados ($q>1$) is of order $q^{-n}$ versus $1/n$ for the classical Bal\'{a}zs-Szabados ($q=1$) operators. The results are new even for the classical case $q=1$.