Approximation by \(q\)-Durrmeyer operators (Q1032561)

The \(q\)-Durrmeyer operators were recently introduced by \textit{V. Gupta} [Appl. Math. Comput. 197, No.~1, 172--178 (2008; Zbl 1142.41008)]. In the present paper, the authors establish new and interesting approximation properties of the mentioned operators. The first main result,contained in Theorem 1, expresses the degree of local approximation of a function \(f\), continuous on the compact interval \([0,1]\) using the \(q\)-Durrmeyer operators in terms of first and respectively second order moduli of smoothness. the second main result, contained in Theorem 3, expresses the degree of global approximation in terms of second order modulus of smoothness and respectively Ditzian-Totik moduli of first order. Finally, a simultaneous approximation theorem for the case when the approximated function is a polynomial is established.