On the q-Bernstein polynomials of rational functions with real poles

The paper aims to investigate the convergence of the q  -Bernstein polynomials Bn,q(f;x)Bn,q(f;x) attached to rational functions in the case q>1q>1. The problem reduces to that for the partial fractions (x−α)−j(x−α)−j, j∈Nj∈N. The already available results deal with cases, where either the pole α   is simple or α≠q−mα≠q−m, m∈N0m∈N0. Consequently, the present work is focused on the polynomials Bn,q(f;x)Bn,q(f;x) for the functions of the form f(x)=(x−q−m)−jf(x)=(x−q−m)−j with j⩾2j⩾2. For such functions, it is proved that the interval of convergence of {Bn,q(f;x)}{Bn,q(f;x)} depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously.