Simultaneous Approximation for Generalized Srivastava-Gupta Operators

Abstract

We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.

1. Introduction

To approximate integrable functions on the interval , Srivastava and Gupta [1] introduced a general sequence of linear positive operators as follows: for a function , where is the class of locally integrable functions defined on and satisfying the growth condition

The general sequence of operators has many interesting properties in approximation theory, which is an interesting area of research in the present era, and several researchers have studied these operators; we can mention some important studies on these operators (see [1–3]). In [4], author introduced King and Stancu type generalization of Srivastava-Gupta operators and presented some direct results. Also, Verma and Agrawal [5] introduced a new generalization of Srivastava-Gupta operators and studied the rate of convergence for the functions having the derivatives of bounded variation (BV). The rate of convergence for the functions having the derivatives of (BV) is an active area of research and many researchers studied this direction. We refer the readers to [6–10] and references therein.

Stancu [11, 12] introduced generalizations of Bernstein polynomials with one and two parameters (resp.), satisfying the condition , as for any . Stancu type generalization of approximation operators present better approach depending on . Therefore, this kind of generalizations and their approximation properties have been studied intensively. We refer the readers to [13–17] and references therein. Mishra et al. [18, 19], V. N. Mishra, and L. N. Mishra [20] have established very interesting results on approximation properties of various functional classes using different types of positive linear summability operators.

The purpose of this paper is to introduce a new Stancu type generalization of the operators defined in [5] as By the definition of operators, it is clear that is positive and linear. For , reduces to operators defined in [5]. In this study we obtain the rate of convergence for the functions having the derivatives of bounded variation. Also, in the end of the paper, we study the simultaneous approximation.

2. Auxiliary Results

In order to prove our main results, we need the following lemmas.

Lemma 1. Let the th order moment be defined as where , and then, for , we have the following recurrence relation: Furthermore, is polynomial of degree in and

Proof. By definition of , taking the derivative of , we get Hence, using the identity we have We can write as To estimate using , we have Next to estimate using the equality we have Putting , we get Now integrating by parts, we get Proceeding in a similar manner, we obtain the estimate as Combining the equations, we have which is the desired result.
Moments for can be easily obtained by using the above recurrence relation.

Remark 2. For sufficiently large , , and , it can be seen from Lemma 1 that where for the convenient notation.

Remark 3. By using Cauchy-Schwarz inequality, it follows from Remark 2 that, for sufficiently large , , and ,

Lemma 4. Let and ; then, for sufficiently large , we have

Proof. We give the proof for only first inequality, and the other is similar. Using Remark 2 with , for sufficiently large and and , we have

Lemma 5. Suppose is times differentiable on such that , for some integer as . Then, for any , and , we have

Proof. Using the identity One can observe that, even in case , the above identity is true with the condition . Thus, applying (26), we have which means that the identity is satisfied for . Let us suppose that the result holds for ; that is, Also, from (26) we can write and, integrating by parts the last integral, we have Hence we have in which the result is true for , and hence by mathematical induction the proof of the lemma is completed.

3. Main Results

Throughout the paper by we denote the class of absolutely continuous functions on (where is a some positive integer) satisfying the conditions:(i) and ,(ii)the function has the first derivative on the interval which coincide almost everywhere with a function which is of bounded variation on every finite subinterval of . It can be observed that for all functions we can have the representation

Theorem 6. Let , , and . Then, for and sufficiently large , we have where is a constant which may be different on each occurrence.

Proof. Using the mean value theorem, we have Also, using the identity where we have Thus, using the above identities, we can write Also, it can be verified that Combining (38)–(40), we get Applying Remark 2 and Lemma 1 in above equation, we have In order to complete the proof of the theorem, it suffices to estimate the terms and . Applying Remark 2 with , we get For estimating the integral we proceed as follows: since implies that so by Schwarz inequality and Lemma 1, By using Hölder’s inequality and Remark 2 (), we get the estimate as follows: Collecting the estimates from (43)–(46), we obtain On the other hand, to estimate by applying Lemma 4 with and integration by parts, we have where .
Combining (41), (47), and (48), we get the desired result.

Corollary 7. Let , , and . Then, for and sufficiently large, one has where denotes the total variation of on and the auxiliary function is defined by

4. Conclusion

The results of our lemmas and theorems are more general rather than the results of any other previously proved lemmas and theorems, which will enrich the literature of applications of quantum calculus in operator theory and convergence estimates in the theory of approximations by positive linear operators. The researchers and professionals working or intend to work in areas of mathematical analysis and its applications will find this research paper to be quite useful. Consequently, the results so established may be found useful in several interesting situations appearing in the literature on mathematical analysis, pure and applied mathematics, and mathematical physics. Some interesting applications of the positive approximation linear operators can be seen in [21–24].

Conflict of Interests

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

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 paper. Special thanks are due to Professor Józef Banaś, Editor of the Journal of Function Spaces, for his efforts to send the reports of the paper timely. The authors are also grateful to all the editorial board members and reviewers of esteemed journal, that is, Journal of Function Spaces. The second author Lakshmi Narayan Mishra acknowledges the Ministry of Human Resource Development (MHRD), New Delhi, India, for supporting this research paper at the Department of Mathematics, National Institute of Technology (NIT), Silchar, Assam. The third author Vishnu Narayan Mishra acknowledges that this paper project was supported by Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat (Gujarat), India. All the authors carried out the proof of theorems. Each author contributed equally in the development of the paper. Vishnu Narayan Mishra conceived of the study and participated in its design and coordination. All the authors read and approved the final version of paper.