turkmath.org

The limit q-Bernstein operator Bq plays a significant role in Approximation Theory as an exemplary model for the study of the convergence of positive linear operators based on the q-integers. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. It has been shown that Bq is a positive shape-preserving linear operator on C[0,1] with ∥Bq∥ = 1. Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this talk, both new and previously known results on the limit q-Bernstein operator are presented.