Generalized Baskakov-Durrmeyer type operators

In the present paper, we study some approximation properties of the Durrmeyer type modification of generalized Baskakov operators introduced by Erencin (Appl Math Comput 218(3):4384-4390, 2011). First, we establish a Lorentz-type lemma for the derivatives of the kernel of the generalized Baskakov operators and then obtain a recurrence relation for the moments of their Durrmeyer type modification. Next, we discuss some direct results in simultaneous approximation by these operators e.g. pointwise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity. Finally, we estimate the error in the approximation of functions having derivatives of bounded variation. © 2014 Springer-Verlag Italia.