On the q-Bernstein polynomials of the logarithmic function in the case q > 1

Abstract

Abstract The q-Bernstein basis used to construct the q-Bernstein polynomials is an extension of the Bernstein basis related to the q-binomial probability distribution. This distribution plays a profound role in the q-boson operator calculus. In the case q > 1, q-Bernstein basic polynomials on [0, 1] combine the fast increase in magnitude with sign oscillations. This seriously complicates the study of q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present new results related to the q-Bernstein polynomials Bn , q of discontinuous functions in the case q > 1. The behavior of polynomials Bn , q (f; x) for functions f possessing a logarithmic singularity at 0 has been examined.