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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 185948, 19 pages
doi:10.1155/2012/185948
On the Sets of Convergence for Sequences of the -Bernstein Polynomials with
Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
Received 27 March 2012; Accepted 19 June 2012
Academic Editor: Ngai-Ching Wong
Copyright © 2012 Sofiya Ostrovska and Ahmet Yaşar Özban. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The aim of this paper is to present new results related to the convergence of the sequence of the -Bernstein polynomials in the case , where is a continuous function on . It is shown that the polynomials converge to uniformly on the time scale , and that this result is sharp in the sense that the sequence may be divergent for all . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.