This paper deals with approximating properties of the newly defined -generalization of the genuine Bernstein-Durrmeyer polynomials in the case , which are no longer positive linear operators on . Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in , , the rate of approximation by the genuine -Bernstein-Durrmeyer polynomials is of order versus for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for . This paper represents an answer to the open problem initiated by Gal in (2013, page 115).
Nazim I. Mahmudov. "Approximation by Genuine -Bernstein-Durrmeyer Polynomials in Compact Disks in the Case ." Abstr. Appl. Anal. 2014 (SI10) 1 - 11, 2014. https://doi.org/10.1155/2014/959586