On the analyticity of functions approximated by their q-Bernstein polynomials when q > 1

Abstract

Since in the case q > 1 the q-Bernstein polynomials B n, 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 ∥ B n, q ∥ as n → ∞, 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: ∣ z∣ < 1}.