Abstract
We discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter . We compute the moments of these modified operators. Also, we establish pointwise convergence, Voronovskaja type asymptotic formula, and an error estimation in terms of second order modification of continuity of the function for the operators .
1. Introduction
For , , , and , we consider a certain integral type generalized Baskakov operators as where being the Dirac delta function.
The operators defined by (1) are the generalization of the integral modification of well-known Baskakov operators having weight function of some beta basis function. As a special case, that is, , the operators (1) reduce to the operators very recently studied in [1, 2]. Inverse results of same type of operators were established in [3]. Also, if , the operators (1) reduce to the operators recently studied in [4] and if and , the operators (1) reduce to the operators studied in [5]. The -analog of the operators (1) is discussed in [6]. We refer to some of the important papers on the recent development on similar type of the operators [7–9]. The present a paper that deals with the study of simultaneous approximation for the operators .
2. Moments and Recurrence Relations
Lemma 1. If one defines the central moments, for every , as then , , and for ; one has the following recurrence relation:From the recurrence relation, it can be easily verified that, for all , one has , where denotes the integral part of
Proof. Taking derivative of the above, Using , we get We can write as To estimate using , we have Next, to estimate using the equality, , we have Again putting , we get Now, integrating by parts, we get Proceeding in the similar manner, we obtain the estimate as Combining (6)–(12), we getHence, This completes the proof of Lemma 1.
Remark 2 (see [10]). For , if the th order moment is defined as then , , and
Consequently, for all , we have
Remark 3. It is easily verified from Lemma 1 that for each
Lemma 4 (see [10]). The polynomials exist independent of and such that
Lemma 5. If is times differentiable on , such that , as , then for and one has
Proof. First Now, using the identities for , we have Integrating by parts, we getThus the result is true for . We prove the result by induction method. Suppose that the result is true for ; then Thus, using the identities (20), we haveIntegrating by parts, we obtain This completes the proof of Lemma 5.
3. Direct Theorems
This section deals with the direct results; we establish here pointwise approximation, asymptotic formula, and error estimation in simultaneous approximation.
We denote forsome, and the norm on the class is defined as It can be easily verified that the operators are well defined for .
Theorem 6. Let and let exist at a point . Then one has
Proof. By Taylor’s expansion of , we have where as . Hence, First, to estimate , using binomial expansion of and Remark 3, we have Next, applying Lemma 4, we obtain The second term in the above expression tends to zero as . Since as for given , there exists a such that whenever . If , where is any integer, then we can find a constant , such that for . Therefore,Applying the Cauchy-Schwarz inequality for integration and summation, respectively, we obtainUsing Remark 2 and Lemma 1, we get
Again using the Cauchy-Schwarz inequality and Lemma 1, we get Collecting the estimation of , we get the required result.
Theorem 7. Let . If exists at a point , then
Proof. Using Taylor’s expansion of , we have where as and for .
Applying Lemma 1, we haveFirst, we have