The Convergence of <i>q</I>-bernstein Polynomials (0 &lt; <i>q</I> &lt; 1) and Limit <i>q</I>-bernstein Operators in Complex Domains

Abstract

Due to the fact that the convergence properties of q-Bernstein polynomials are not similar to those in the classical case q = 1, their study has become an area of intensive research with a wide scope of open problems and unexpected results. The present paper is focused on the convergence of q-Bernstein polynomials, 0 < q < 1, and related linear operators in complex domains. An analogue of the classical result on the simultaneous approximation is presented. The approximation of analytic functions With the help of the limit q-Bernstein operator is studied.