On the [formula omitted]-Bernstein polynomials of piecewise linear functions in the case [formula omitted

Abstract

The aim of this paper is to present new results related to the approximation of continuous functions by their q-Bernstein polynomials in the case q>1. The first part of the paper is devoted to the behavior of the q-Bernstein polynomials of piecewise linear functions. This study naturally leads to the notion of truncated q-Bernstein polynomials introduced in the paper. The second part deals with the asymptotic estimates for the norms of the m-truncatedq-Bernstein polynomials, in the case where both n and q vary. The results of the paper are illustrated by numerical examples.