Ostrovska, SofiyaOstrovskii, IossifOstrovska, SofiyaMathematics2024-07-052024-07-05201020096-300310.1016/j.amc.2010.04.0202-s2.0-77955416547https://doi.org/10.1016/j.amc.2010.04.020https://hdl.handle.net/20.500.14411/1595Since in the case q > 1 the q-Bernstein polynomials B-n,B-q are not positive linear operators on C[0, 1], the investigation of their convergence properties for q > 1 turns out to be much harder than the one for 0 < q < 1. What is more, the fast increase of the norms parallel to B-n,B-q parallel to as n -> infinity, along with the sign oscillations of the q-Bernstein basic polynomials when q > 1, create a serious obstacle for the numerical experiments with the q-Bernstein polynomials. Despite the intensive research conducted in the area lately, the class of functions which are uniformly approximated by their q-Bernstein polynomials on [0, 1] is yet to be described. In this paper, we prove that if f : [0, 1] -> C is analytic at 0 and can be uniformly approximated by its q-Bernstein polynomials (q > 1) on [0, 1], then f admits an analytic continuation from [0, 1] into {z: vertical bar z vertical bar < 1}. (C) 2010 Elsevier Inc. All rights reserved.eninfo:eu-repo/semantics/closedAccessq-Integersq-Bernstein polynomialsUniform convergenceAnalytic functionAnalytic continuationArticleQ121716572WOS:000280580700007