J. Math. Anal. Appl. 328 (2007) 101–105 www.elsevier.com/locate/jmaa Simultaneous approximation for Bézier variant of Szász–Mirakyan–Durrmeyer operators Vijay Gupta School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110075, India Received 3 January 2006 Available online 9 June 2006 Submitted by H.M. Srivastava Abstract We study the rate of convergence in simultaneous approximation for the Bézier variant of Szász– Mirakyan–Durrmeyer operators by using the decomposition technique of functions of bounded variation.  2006 Elsevier Inc. All rights reserved. Keywords: Rate of convergence; Bounded variation; Bézier variant; Simultaneous approximation 1. Introduction To approximate Lebesgue integrable functions on the interval [0, ∞), Szász–Mirakyan– Durrmeyer operators with the basis function p n,k (x) = e −nx (nx) k k! (see [1,3]) are defined by S n (f,x) = n ∞  k=0 p n,k (x) ∞  0 p n,k (t)f(t)dt, x ∈[0, ∞). (1) For α  1, the Bézier variant of operators (1) is defined by S n,α (f,x) = n ∞  k=0 Q (α) n,k (x) ∞  0 p n,k (t)f(t)dt, (2) E-mail address: vijay@nsit.ac.in. 0022-247X/$ – see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.05.021 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector