The q-Bernstein polynomials of the Cauchy kernel with a pole on [0,1] in the case q>1

摘要

The problem to describe the Bernstein polynomials of unbounded functions goes back to Lorentz. The aim of this paper is to investigate the convergence properties of the q-Bernstein polynomials Bn,q(f;x) of the Cauchy kernel 1x-α with a pole α∈[0,1] for q>1. The previously obtained results allow one to describe these properties when a pole is different from q-m for some m∈0,1,2,…. In this context, the focus of the paper is on the behavior of polynomials Bn,q(f;x) for the functions of the form fm(x)=1/(x-q-m),x≠q-m and fm(q-m)=a,a∈R. Here, the problem is examined both theoretically and numerically in detail.