Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Abstract

In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate q-Bernstein polynomials for a function analytic in the polydisc for arbitrary fixed q > 1. We give quantitative Voronovskaja type estimates for the bivariate q-Bernstein polynomials for q > 1. In the univariate case the similar results were obtained by S.Ostrovska: q-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232-255, and S.G.Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.