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

The paper aims to investigate the convergence of the q-Bernstein polynomials B n , q ( f ; x ) attached to rational functions in the case q > 1 . The problem reduces to that for the partial fractions ( x − α ) − j , j ∈ N . The already available results deal with cases, where either the pole α is simple or α ≠ q − m , m ∈ N 0 . Consequently, the present work is focused on the polynomials B n , q ( f ; x ) for the functions of the form f ( x ) = ( x − q − m ) − j with j ⩾ 2 . For such functions, it is proved that the interval of convergence of { B n , 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.