Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

Nasiruzzaman, Md.; Aljohani, A. F.

doi:https://doi.org/10.1155/2020/8863664

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Sequence Spaces, Function Spaces and Approximation Theory

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Volume 2020 | Article ID 8863664 | https://doi.org/10.1155/2020/8863664

Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

Md. Nasiruzzaman¹and A. F. Aljohani¹

Academic Editor: Syed Abdul Mohiuddine

Received09 Sept 2020

Revised07 Oct 2020

Accepted11 Nov 2020

Published23 Nov 2020

Abstract

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.

1. Introduction

In the year 1950, a famous mathematician Szász [1] invented the positive linear operators for the continuous function on and that were extensively searched rather than Bernstein operators [2]. For and , Szász introduced the operators as follows: where is the space of continuous functions on . In recent years, Szász-Mirakjan operators were introduced by Sucu [3] by proposing an exponential function on Dunkl generalization by including a nonnegative number , such that where and a recursion formula for .

In 1969, Jakimovski and Leviatan introduced the sequence of Szász-Mirakjan-type positive linear operators by the use of Appell polynomials [4], such that where , , . Note that, if in (4), the Szász-Mirakjan operator (1) is obtained. Most recently, in [5], Nasiruzzaman and Aljohani have introduced the Szász-Mirakjan-Jakimovski-Leviatan-type operators involving the Dunkl generalization for the function by

Lemma 1. [5]. For the test function , if , the operators have , and the following identities:

There are several research articles mentioned regarding the Szász-Mirakjan-type operators, for instance, [6–13]. For some further related concepts and approximation, we refer to see [9, 10, 14–20].

2. Kantorovich Operators Involving Appell Polynomials and Their Moments

In this section, we construct the generalized operators of recent investigation [5] including the Kantorovich polynomial. For this purpose, we let as ; then, for all , , , , and , we define the operators as follows:

Lemma 2. For , let the test function be . Then, operators have the following identities:

Proof. To prove this Lemma, we take into account [5] Lemma 1. Thus, for all and , we can conclude that Thus, from (7) and (9), clearly we can write Therefore, by applying Lemma 1, we get the required results.

Lemma 3. For the central moments , we have the following identities:

3. Approximations in Weighted Space

In the present section, we follow the well-known results by Gadziev [21] and recall the results in weighted spaces with some additional conditions precisely, under the analogous of P.P. Korovkin’s theorem holds. In order to define the uniformly approximations, we take be the kind of functions which is continuous and strictly increasing with the assumptions and . For this reason, we let be a set of all such functions which are defined on and verifying the results where is a constant and depending only on function and equipped the norm with

Furthermore, we denote the set all continuous functions on by and its subsets be defined as .

It is well known for the sequence of linear positive operators (see [21]) maps into if and only if where is a positive constant. For , let us denote

Theorem 4. Let . Then, for every , operators (7) are uniformly convergent on each compact subset of such that where denotes the uniform convergence.

Proof. In view of Lemma 2, we use Korovkin’s theorem by [22]; then, it is enough to see that for each uniformly. Thus obviously, we get , , and , which completes the proof of Theorem 4.

Theorem 5 [21, 23]. Let the positive linear operators acting from to and for if it verifies that , then for every it satisfies

Theorem 6. For every , operators satisfy

Proof. It is enough to prove Theorem 6; we use the well-known Korovkin theorem and show Taking into account Lemma 2, then it is easy to see that For , we can write here If , then easily we get . Similarly, for , we conclude that Thus, we easily get , as .

Theorem 7. If . Then, operators follow that where the number .

Proof. By the virtue of and for any positive real , we easily obtain Thus, From Lemma 2, it follows that Now, for each , there exists for all such that Therefore, for all In view of (26) and (29), we get If we choose any so large, such that , then we get On the other hand, there exists such that Finally, take and on combining (31) and (32) with the above expression, we get This completes the proof of Theorem 7.

Definition 8. For every and all , the modulus of continuity of the uniformly continuous function on defined as

Theorem 9 [24]. Let the sequence of positive linear operators and , then (1)for any and , it follows that(2)if any , then for all one has

Theorem 10. Let , then for all it follows the inequality where .

Proof. If we consider Lemma 2 and Theorem 9, then we can obtain where if we take then we are easily denumerable to get results.

Theorem 11. For any , if , then we have the inequality where .

Proof. If we consider Lemmas 2 and 3 and (2) of Theorem 9, then it is obvious to get that Put , then we easily get our desired results of Theorem 11.
From [25] for an arbitrary , the weighted modulus of continuity introduced such that Two main properties of this modulus of continuity are and where and weighted modulus of continuity of the function for .

Theorem 12. Let , then for all we have the inequality where , for and

Proof. We use expressions (41) and (42) and applying the Cauchy-Schwarz inequality to operators , we get We know the expression In view of Lemma 3, we can obtain where and are positive constant and Thus, from inequality (45), we get

If we choose and taking supremum , then we easily get the result.

4. Direct Approximation Results of

The present section gives some direct approximation results in space of -functional and in Lipschitz spaces. We take be the set of all continuous and bounded functions defined on .

Definition 13. For every and the -functional is defined such that For an absolute constant , one has Let denote the modulus of continuity of order two such that while the classical modulus of continuity is given by

Theorem 14. For an arbitrary , let an auxiliary operator be such that Then, for any operators (55), verify the inequality where is defined by Theorem 10.

Proof. For any , it is easy to verify that and We have For any , the Taylor series expression gives us Therefore, after applying the operators , on both sides we get We know the inequality Thus, we get This gives the complete proof.

Theorem 15. If , then for any operators by (7) satisfying where is defined by Theorem 10.

Proof. We prove Theorem 15 in view of Theorem 14. Therefore, for all and , we get If we take infimum for all , then in view of (50) it is easy to conclude that The proof is completed here.

Now, we give the local direct estimate for the operators defined by (7) via the well-known Lipschitz-type maximal function involving the parameters and number . Thus, from [26], we recall that where is a positive constant.

Theorem 16. For any satisfied by (70), operators hold the inequality where is obtained by Theorem 10.

Proof. Let for ; then, first we verify the results are true when . For any , it is easy to use the result and then we apply the Cauchy-Schwarz inequality. Thus, we can write From these conclusions, we get that the statement holds for . Now, we check if the statement is valid if . For this reason, we use monotonicity property to and apply the well-known Hölder inequality which completes the proof.

Here, we obtain the other local approximation results of in Lipschitz spaces. For all Lipschitz maximal function and , from [27] we recall that

Theorem 17. Let , then for all , where is obtained in and is defined by Theorem 10.

Proof. From the well-known Hölder inequality, we get Thus, we get the proof.

5. Voronovskaja-Type Approximation Theorems

In this section, we establish a quantitative Voronovskaja-type theorem for the operators .

Theorem 18. Let , then for each where .

Proof. From the expression of Taylor’s expansion of function in , we write where is the remainder term and with as . On applying the operators to (74), then use the Cauchy-Schwarz inequality. Thus, we get Since we have , therefore Thus, we have which completes the proof.

As a consequence of Theorem 18, we immediately get the corollary.

Corollary 19. For any , we have

6. Conclusion

Motivated by article [5], we have introduced a Kantorovich generalization of the Szász-Mirakjan operators by Dunkl analogue involving the Appell polynomials. These types of generalizations enable to give the generalized results rather than the earlier study demonstrations by [3, 5, 7]. Lastly, we have also discussed the Voronovskaja-type approximation theorems of these new operators.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors are very grateful and declare that they have no competing interest.

Authors’ Contributions

All authors read and agreed to the contents of this research article.

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Copyright

Copyright © 2020 Md. Nasiruzzaman and A. F. Aljohani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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