On the <i>q</I>-bernstein Polynomials of Piecewise Linear Functions in the Case <i>q</I> &gt; 1

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-truncated q-Bernstein polynomials, in the case where both n and q vary. The results of the paper are illustrated by numerical examples. (C) 2012 Elsevier Ltd. All rights reserved.