Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators. 1. Introduction The integral modification of Baskakov operators having weight function of some beta basis function are defined as the following: for , , where being the Dirac delta function. The operators defined by (1) were introduced by Gupta [1]; these operators are different from the usual Baskakov-Durrmeyer operators. Actually these operators satisfy condition , where and are constants. In [1], the author estimated some direct results in simultaneous approximation for these operators (1). In particular case , the operators (1) reduce to the operators studied in [2, 3]. In recent years a lot of work has been done on such operators. We refer to some of the important papers on the recent development on similar type of operators [4–9]. The rate of convergence for certain Durrmeyer type operators and the generalizations is one of the important areas of research in recent years. In present article, we extend the studies and here we estimate the rate of convergence for functions having derivative of bounded variation. We denote ; then, in particular, we have By we denote the class of absolutely continuous functions defined on the interval such that,(i) , .(ii)having a derivative on the interval coinciding a.e. with a function which is of bounded variation on every finite subinterval of . It can be observed that all function possess for each a representation 2. Rate of Convergence for Lemma 1 (see [1]). Let the function , , be defined as Then it is easily verified that, for each , , and , and also the following recurrence relation holds: From the recurrence relation, it can be easily be verified that for all , we have . Remark 2. From Lemma 1, using Cauchy-Schwarz inequality, it follows that Lemma 3. Let and be the kernel defined in (1). Then for being sufficiently large, one has ？(a) . ？(b) . Proof. First we prove (a); by using Lemma 1, we have The proof of (b) is similar; we omit the details. Theorem 4. Let , , and . Then for being sufficiently large, we have where the auxiliary function is given by denotes the total variation of on . Proof. By the application of mean value theorem, we have Also, using the identity where we can see that Also, Substitute value of from (12) in (11) and using (14) and (15), we get Using Lemma 1 and Remark 2, we obtain On applying Lemma 3