On the Sets of Convergence for Sequences of the q-Bernstein Polynomials with q>1

Abstract EN

The aim of this paper is to present new results related to the convergence of the sequence of the q-Bernstein polynomials {Bn,q(f;x)} in the case q>1, where f is a continuous function on [0,1].

It is shown that the polynomials converge to f uniformly on the time scale ?q={q-j}j=0∞∪{0}, and that this result is sharp in the sense that the sequence {Bn,q(f;x)}n=1∞ may be divergent for all x∈R∖?q.

Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established.

Throughout the paper, the results are illustrated by numerical examples.