The saturation of convergence for the complex q-durrmeyer polynomials

Abstract

The aim of this paper is to establish a saturation result for the complex q-Durrmeyer polynomials (Dn,qf)(z), where q∈(0,1), f∈C[0,1]. It is known that the sequence {(Dn,qf)(z)}n∈N converges uniformly on any compact set in C to the limit function (D∞,qf)(z), which, therefore, is entire. Previously, the rate of this convergence has been estimated as O(qn), n→∞. In the present article, this result is refined to derive Voronovskaya-type formula and to demonstrate that this rate is o(qn), n→∞ on a set possessing an accumulation point if and only if f takes on the same value at all qj, j∈N0.