International Scholarly Research Network ISRN Mathematical Analysis Volume 2011, Article ID 184374, 11 pages doi:10.5402/2011/184374 Research Article On L p -Approximation by Iterative Combination of Bernstein-Durrmeyer Polynomials T. A. K. Sinha, 1 P. N. Agrawal, 2 and Asha Ram Gairola 2 1 Department of Mathematics, SMD College Poonpoon, Patna, Bihar, India 2 Department of Mathematics, IIT Roorkee, Roorkee 247667, India Correspondence should be addressed to P. N. Agrawal, pna iitr@yahoo.co.in Received 11 November 2010; Accepted 9 December 2010 Academic Editors: O. Gu` es and R. Stenberg Copyright q 2011 T. A. K. Sinha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We improve the degree of approximation by Bernstein-Durrmeyer polynomials taking their iterates and obtain error estimate in higher-order approximation. 1. Introduction The Bernstein-Durrmeyer polynomials M n ( f ; t ) n 1 n ν0 p n,ν t 1 0 p n,ν uf udu, 1.1 where p n,ν t n v t ν 1 - t n-ν , t ∈ 0, 1, were introduced by Durrmeyer 1 and extensively studied by Derriennic 2 and several other researchers. It turns out that the order of approximation by these operators is, at best, On -1 however smooth the function may be. In order to improve this rate of approximation, we consider an iterative combination T n,k f ; t of the operators M n f ; t. This technique of improving the rate of convergence was given by Micchelli 3 who first used it to improve the order of approximation by Bernstein polynomials B n f ; t. Recently, this technique has been applied to obtain some direct and inverse theorems in ordinary and simultaneous approximation by several sequences of linear positive operators in uniform norm c.f., e.g., 4–6. The object of this paper is to study some direct theorems in L p -approximation by the operators T n,k f ; t.