On the Approximation of Analytic Functions by the Q-Bernstein Polynomials in the Case Q > 1

Abstract

Since for q > 1, the q-Bernstein polynomials Bn,q are not positive linear operators on C[0, 1], the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. In this paper, new results on the approximation of continuous functions by the q-Bernstein polynomials in the case q > 1 are presented. It is shown that if f Ε C[0, 1] and admits an analytic continuation f(z) into {z : |z| < a}, then Bn,q (f; z) → f(z) as n → λ, uniformly on any compact set in {z : |z| < a}. Copyright © 2010, Kent State University.