On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators

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 haveNow, the coefficients of , , and in the above expression tend to , , and , respectively, which follows by using induction hypothesis on and taking the limit as . Hence, in order to prove (34), it is sufficient to show that as , which follows along the lines of the proof of Theorem 6 and by using Remark 2 and Lemmas 1 and 4.

Remark 8. Particular case was discussed in Theorem in [4], which says that the coefficient of converges to but it converges to and we get this by putting in the above theorem.

Definition 9. The th order modulus of continuity for a function continuous on is defined by For , is usual modulus of continuity.

Theorem 10. Let for some and . Then, for sufficiently large, one has where and .

Proof. Let us assume that . For sufficiently small , we define the function corresponding to and as follows: where and is the second order forward difference operator with step length For , the functions are known as the Steklov mean of order 2, which satisfy the following properties [11]: (a) has continuous derivatives up to order 2 over .(b).(c).(d), where , , are certain constants which are different in each occurrence and are independent of and .
We can write by linearity properties of , Since , by property (c) of the function , we get Next, on an application of Theorem 7, it follows that Using the interpolation property due to Goldberg and Meir [12], for each , it follows that Therefore, by applying properties (c) and (d) of the function , we obtain Finally we will estimate , choosing , satisfying the conditions . Suppose denotes the characteristic function of the interval . Then By Lemma 5, we have Hence, Now, for and , we choose a satisfying
Therefore, by Lemma 4 and the Cauchy-Schwarz inequality, we have For sufficiently large , the second term tends to zero. Thus, Hence, by using Remark 2 and Lemma 1, we have where . Now choosing satisfying , we obtain Therefore, by property (c) of the function , we get Choosing , the theorem follows.

Remark 11. In the last decade the applications of -calculus in approximation theory are one of the main areas of research. In 2008, Gupta [13] introduced -Durrmeyer operators whose approximation properties were studied in [14]. More work in this direction can be seen in [15–17].
A Durrmeyer type -analogue of the is introduced as follows: where Notations used in (53) can be found in [18]. For the operators (53), one can study their approximation properties based on -integers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this research article.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article.