Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.017385

ARTICLE

Approximation by Szász Type Operators Involving Apostol-Genocchi Polynomials

Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, Gaziantep, TR-27310, Turkey

*Corresponding Author: Mine Menekşe Yılmaz. Email: menekse@gantep.edu.tr

Received: 07 May 2021; Accepted: 12 July 2021

Abstract: The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of order α. Applying the Korovkin theorem, we arrive at the convergence of the operator with the aid of moments and central moments. We determine the rate of convergence of the operator using several tools such as

Keywords: Apostol-Genocchi polynomials; rate of convergence; Korovkin theorem; modulus of continuity; Szász type operators; generating functions

The Weierstrass approximation theorem shows that the polynomials are uniformly dense in the space of continuous functions on a compact interval equipped with supremum norm [1]. Polynomials are useful tools that are easy to evaluate, differentiate and integrate, and the Weierstrass theorem also shows their importance in the approximation theory. Since Bernstein [2] proved the Weierstrass theorem using a polynomial class in 1911, some authors [3–6], defined linear positive operators for the same purpose. One of these operators is Szász operators that generalization of Bernstein polynomials to infinite interval [7]:

Many mathematicians have found various generalizations of the Szász positive linear operator and studied the approximate behaviour of these new operators. The idea of establishing an operator using the generating function first appeared in [8]. In [8], assuming that

where

In [20], Apostol–Genocchi numbers and polynomials of (real or complex) order α,

with

Generating functions for Apostol–Genocchi polynomials with their congruence properties involving these polynomials has been studied by many authors in recent years (see [21,22]).

Prakash et al. [23] established a sequence that includes Apostol-Genocchi polynomials of order α, and then Deo et al. [24] introduced the Durrmeyer form of Apostol-Genocchi polynomials with Baskakov type operators. In this study, motivated by [23,24], we define a generalization of Szász type operators involving Apostol–Genocchi polynomials of order α as follows

where

2 Convergence of the Operator

In this section, to begin with, we find the moments of the operator in Eq. (5) by using the generating function of Apostol–Genocchi polynomials. In addition, we prove the convergence of the operator

Lemma 2.1. The operator

Proof. By the aid of the generating function of the Apostol-Genocchi polynomials in Eq. (4), for Eqs. (6)–(8), we obtain

In view of Eqs. (9)–(11), we get the required result.

Remark 2.1. Using Lemma 2.1, we can give the central moments of the operator

Theorem 2.1. If

Proof. We have fact that

3 The Rate of Convergence of the Operator

The concept of modulus of continuity is the main instrument in approximation theory by positive linear operators. This concept works well in providing quantitive estimates. In this section, we use the usual modulus of continuity and the second modulus of continuity when measuring the rate of convergence. Since

Definition 3.1. Let f be uniformly continuous function on

Then for any δ > 0, and each

Definition 3.2. The second modulus of continuity of

Definition 3.3. ([26]) The Peetre’s

where δ > 0 and

It is well known that

(see [27]).

Theorem 3.1. If f

Proof. It follows from Lemma 2.1 and monotonicity property of operators

Using (14), we get the following from (19)

Applying the Cauchy–Schwarz inequality to the right side of (20), we get

By choosing

Definition 3.4. Let

where M > 0.

Next theorem satisfies an estimate for the error of the operator

Theorem 3.2. Let

where

Proof. Since

Using the Hölder inequality and from (24), we can write the following:

Therefore, we obtain (23) by the help of the (25).

Theorem 3.3. Let

where C is a constant and

Proof. Assume that

and

is the Taylor expansion of g. If we apply the operator Ln by (27) to both sides of the Eq. (28), and use the linearity property of the operator Ln, then we have the following:

Using (12) and (13) in (29), we get

where

Now we consider the term of

When infimum is taken over all

Using Eq. (18) in (32), we obtain (26). Therefore the proof is completed.

Voronovskaya proved a theorem giving asymptotic error terms for the Bernstein polynomials for functions that can be differentiable twice (see [28]). Based on this idea, we present the following theorem.

Theorem 3.4. If

for every x ≥ 0.

Proof. Let

where

Applying the operator

If we use the Cauchy–Schwarz inequality for the last term in (35), then we get

Observe that

uniformly with respect to x in every compact set of

Moreover, if we use (12), (13) and (38) in (35), we get (33). This completes the proof.

4 Weighted Approximation for the Operator

Let

Lemma 4.1. If

Proof. In view of Lemma 2.1, we can obtain desired result.

Now, we have fact that the operators

Theorem 4.1. For every

Proof. From [25], it is sufficient to verify the following cases:

Since

For r = 1, using Lemma 2.1, we get

as

then, for r = 2, the condition in Eq. (39) holds as

Corollary 4.1. For each

Proof.

Let’s examine the terms in (42). From the Theorem 2.1,

5 Numerical Example for

In this section we give some examples to obtain an upper bound for the error

Example 5.1. Let β = 1, m = η = 0, and n = 0,…,7. The approximation of

Example 5.2. Let β = 1, m = η = 1 and n = 0,…,7. The approximation of

Example 5.3. The approximation of

In the present paper, we have introduced a form of the operator using the generating function of Apostol-Genocchi polynomials of order α and obtained the approximation properties and rate of convergence of this operator. At the end of the paper, we have found an upper bound for the error

Acknowledgement: The author would like to thank the editor and referees for their many valuable comments and suggestions to improve the overall presentation of paper.

Funding Statement: The author received no specific funding for this study.

Conflicts of Interest: The author declares that he has no conflicts of interest to report regarding the present study.

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