On some approximation properties of $q$-parametric Bernstein polynomials

Abstract

In 1997 Phillips introduced generalized Bernstein polynomials ${{B}_{n}}(f,q;x)$ based on the $q$-integers and $q$-binomial coefficients for any $q>0$. For $q=1$, generalized Bernstein polynomials ${{B}_{n}}(f,q;x)$ coincide with the classical ones ${{B}_{n}}(f;x)$. For $q\ne 1$, one gets a new class of polynomials having many interesting properties. In this paper we obtain the asymptotic formulae on the approximation of function $f(t)={{t}^{i}}\,\,\,\,(i\in Z)$ by the generalized Bernstein polynomials ${{B}_{n}}(f,q;x)$ in the case $q$ is fixed as $n\to +\infty $.