The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

Abstract

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.