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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 185948, 19 pages
http://dx.doi.org/10.1155/2012/185948
Research Article

On the Sets of Convergence for Sequences of the ๐‘ž -Bernstein Polynomials with ๐‘ž > 1

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 ๐‘ž -Bernstein polynomials { ๐ต ๐‘› , ๐‘ž ( ๐‘“ ; ๐‘ฅ ) } in the case ๐‘ž > 1 , where ๐‘“ is a continuous function on [ 0 , 1 ] . It is shown that the polynomials converge to ๐‘“ uniformly on the time scale ๐• ๐‘ž = { ๐‘ž โˆ’ ๐‘— } โˆž ๐‘— = 0 โˆช { 0 } , and that this result is sharp in the sense that the sequence { ๐ต ๐‘› , ๐‘ž ( ๐‘“ ; ๐‘ฅ ) } โˆž ๐‘› = 1 may be divergent for all ๐‘ฅ โˆˆ ๐‘… โงต ๐• ๐‘ž . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.