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

Abstract

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 with and integrating by parts, we have where .
On the other hand, we have
Applying Holder’s inequality, Remark 2, and Lemma 1, we have
Also,
Combining the estimates (17)–(21), we get the desired results.
This completes the proof of Theorem.

3. Rate of Convergence for Stancu Type Generalization of

In 1968, Stancu introduces Bernstein-Stancu operators in [10], a sequence of the linear positive operators depending on two parameters and satisfying the condition . Recently many researchers applied this approach to many operators; for details see [11–17]. For , Stancu generalization of operators (1) is as follows: where , and are as defined in (1).

Lemma 5 (see [18]). If we define the central moments, for every as
then , , and
For we have the following recurrence relation:
From the recurrence relation, it can be easily verified that for all , we have .

Remark 6. Observe that preserve constant functions but not linear functions. If these operators reduce to the operators defined in (1). Notice that

Remark 7. From Lemma 3, taking to be sufficiently large and , we observe that where is positive constant.

Remark 8. Applying the Cauchy-Schwarz inequality and keeping the same condition as in Remark 7 for , , and , we derive from Lemma 5 that

Lemma 9. Let and be the kernel defined in (1). Then for being sufficiently large, we have (c). (d).

The proof is the same as Lemma 3, thus we omit the details.

Theorem 10. Let , , and . Then for being sufficiently large, we have where the auxiliary functions and were defined in Theorem 4.
The proof of the above theorem follows along the lines of Theorem 4; thus we omit the details.

Conflict of Interests

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

Acknowledgments

The authors are thankful to the anonymous referee for making valuable comments leading to the better presentation of the paper. Special thanks are due to Professor Dr. Abdelalim A. Elsadany, Editor of Journal of Difference Equations, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.