Abstract and Applied Analysis

Volume 2012 (2012), Article ID 680340, 15 pages

doi:10.1155/2012/680340

Research Article

On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

Sezgin Sucu,¹ Gürhan İçöz,² and Serhan Varma¹

¹Department of Mathematics, Faculty of Science, Ankara University, Tandogan, 06100 Ankara, Turkey

²Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey

Received 25 April 2012; Revised 19 July 2012; Accepted 2 August 2012

Academic Editor: Giovanni Galdi

Copyright © 2012 Sezgin Sucu et al. 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.

Abstract.

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.

1. Introduction

The approximation theory, which is concerned with the approximation of functions by simpler calculated functions, is a branch of mathematical analysis. In 1885, Weierstrass identified the set of continuous functions on a closed and bounded interval through uniform approximation by polynomials. Later, Bernstein gave the first impressive example for these polynomials.

In 1953, Korovkin [1] published his celebrated theorem on the approximation of sequences of linear positive operators. This theorem contains a simple and easily applicable criterion to check if a sequence of linear positive operators converges uniformly to the function. One of the well-known examples of linear positive operators is Szasz operators [2]

where , , and whenever the above sum converges. Many researchers have dealt with the generalization of Szasz operators in a natural way.

Later,  Jakimovski and Leviatan [3] presented a generalization of Szasz operators with Appell polynomials. Let be an analytic function in the disc and assume that . The Appell polynomials have generating functions of the form

Under the assumption for , Jakimovski and Leviatan introduced the linear positive operators via

and gave the approximation properties of these operators.

Remark 1.1. For , in view of the generating functions (1.2), we easily find and from (1.3) we meet again the Szasz operators given by (1.1).

Then, Ismail [4] obtained another generalization of the Szasz operators (1.1) and also Jakimovski and Leviatan operators (1.3) by means of Sheffer polynomials. Let and be analytic functions in the disc where and are real. The Sheffer polynomials have generating functions of the type

Using the following assumptions:

Ismail investigated the approximation properties of linear positive operators given by

Remark 1.2. For , one can observe that the generating functions (1.4) reduce to (1.2) and from this fact, the operators (1.6) return to the operators (1.3).

Remark 1.3. For and , it is easy to get the Szasz operators from the operators (1.6).

Recently, Varma et al. [5] constructed linear positive operators including Brenke-type polynomials. Brenke-type polynomials [6] have generating functions of the form

where and are analytic functions

and have the following explicit expression:

Using the following assumptions

Varma et al. introduced the following linear positive operators involving the Brenke-type polynomials

where and .

Remark 1.4. Let . In this case, the operators (1.11) (resp., (1.7)) reduce to the operators given by (1.3) (resp., (1.2)).

Remark 1.5. Let and . We meet again the Szasz operators (1.1).

In this paper, our aim is to construct linear positive operators by using Boas-Buck-type polynomials including the Brenke-type polynomials, Sheffer polynomials, and Appell polynomials with special cases. Boas-Buck-type polynomials [7] have generating functions of the type

where , , and are analytic functions

We will restrict ourselves to the Boas-Buck-type polynomials satisfying

Now, given the above restrictions, we present a new form of linear positive operators with Boas-Buck-type polynomials as follows:

where and .

Remark 1.6. Let . The operators (1.15) (resp., (1.12)) reduce to the operators given by (1.11) (resp., (1.7)).

Remark 1.7. Let . The operators (1.15) (resp., (1.12)) return to the operators given by (1.6) (resp., (1.4)).

Remark 1.8. Let and . It is obvious that one can get the operators (1.3) from the operators (1.15). In addition, if we choose , we meet again the well-known Szasz operators (1.1).

The paper is divided into three sections. Following the introduction, Section 2 is devoted to obtain qualitative and quantitative results for the operators (1.15). In the last section, we give some significant illustrations with the help of Laguerre, Charlier, and Gould-Hopper polynomials for the operators (1.15). Moreover, we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.

2. Approximation Properties of Operators

In this section, with the help of well-known Korovkin's theorem, we get approximation results by means of linear positive operators. Next, we present quantitative results for estimating the error of approximation using the classical approach and the second modulus of continuity.

Lemma 2.1. For the operators , one has

for any .

Proof. From the generating functions of the Boas-Buck-type polynomials given by (1.12), we obtain

With regard to these equalities, we get the assertions of the lemma.

Let us define the class of as follows:

Theorem 2.2. Let and assume that

Then,

uniformly on each compact subset of .

Proof. According to Lemma 2.1 and taking into account the assumptions (2.4), we find

The above-mentioned convergences are satisfied uniformly in each compact subset of . Applying the universal Korovkin-type property (vi) of Theorem from [8], we lead to the desired result.

In order to estimate the rate of convergence, we will give some definitions and lemmas.

Definition 2.3. Let and . The modulus of continuity of the function is defined by

where is the space of uniformly continuous functions on .

Definition 2.4. The second modulus of continuity of the function is defined by

where .

Lemma 2.5 (Gavrea and Raşa [9]). Let and be a sequence of linear positive operators with the property . Then,

Lemma 2.6 (Zhuk [10]). Let and . Let be the second-order Steklov function attached to the function . Then, the following inequalities are satisfied:

Lemma 2.7. For , one has

Proof. Using the linearity property of operators, one can write

Applying Lemma 2.1, we obtain the equality stated in the lemma.

Generally, we use the modulus of continuity and second modulus of continuity to obtain quantitative error estimation for convergence by linear positive operators. Now, we will calculate the rate of convergence in the following two theorems.

Theorem 2.8. Let . operators verify the following inequality:

where

Proof. Making use of Lemma 2.1 and the property of modulus of continuity, we deduce

Taking into account the Cauchy-Schwarz inequality and then by using Lemma 2.7, we get

Considering the last inequality in (2.15), we obtain

where is given by (2.14). In inequality (2.17), by choosing , we get the desired result.

Theorem 2.9. For , the following estimate

holds, where

Proof. Let be the second-order Steklov function attached to the function . With regard to the identity , we have

Taking account of the fact that , it follows from Lemma 2.5

If one combines Landau inequality with Lemma 2.6, we can write

From the last inequality and Lemma 2.6, (2.21) becomes by taking

Substituting (2.23) in (2.20), hence Lemma 2.6 gives the proof of the theorem.

Remark 2.10. In Theorem 2.9, we present the proof for . For the special case and , one can get from the equality . The inequality obtained in Theorem 2.9 still holds true when .

Remark 2.11. Note that in Theorems 2.8–2.9 when , respectively, and tend to zero under the assumptions (2.4).

3. Examples

Example 3.1. Laguerre polynomials are one of the most important classical orthogonal polynomials in the literature. Such polynomials are used in every area of mathematics. In addition, these polynomials have served with several interesting properties for physicists. For example, Laguerre polynomials arise as solutions of the Coulomb potential in quantum mechanics.

Laguerre polynomials have generating functions of the form

and explicit expressions

It is clear that Laguerre polynomials are Boas-Buck-type polynomials. Note that when , are positive. For ensuring the restrictions (1.14) and the assumptions (2.4), we have to modify the generating functions (3.1) as follows:

With the help of generating functions (3.3), we find the following linear positive operators including Laguerre polynomials from the operators (1.15)

where and .

Remark 3.2. It is worthy to note that we obtain new linear positive operators different from the one given in [4].

Example 3.3. Varma and Taşdelen [11] gave the following linear positive operators involving Charlier polynomials as a generalization of the Szasz operators:

where , and Charlier polynomials have the generating functions of the type

Note that one can get the operators (3.5) as an example of the operators given by the equality (1.6) defined in [4].

On the other hand, Charlier polynomials are also the Boas-Buck-type polynomials by choosing

For ensuring the restrictions (1.14) and the assumptions (2.4), we have to change the generating functions (3.6) by

In view of the generating functions (3.8), we have the linear positive operators (3.5) investigated in [11] from the operators (1.15).

Example 3.4. Gould-Hopper polynomials [12] have the generating functions of the form

and the explicit representations

where, as usual, denotes the integer part. Gould-Hopper polynomials are -orthogonal polynomial set of Hermite type [13]. Van Iseghem [14] and Maroni [15] discovered the notion of -orthogonality. Gould-Hopper polynomials are Boas-Buck-type polynomials with

Under the assumption ; the restrictions (1.14) and condition (2.4) for the operators given by (1.15) are satisfied. With the help of the generating functions (3.9), we obtain the explicit form of operators including Gould-Hopper polynomials by

where .

Remark 3.5. First time, the operators are given in [5] as an explicit example of the operators (1.11). For , we obtain that

Substituting and in the operators (3.12), we get the well-known Szasz operators. By the help of operators, we present an attractive generalization of the Szasz operators with Gould-Hopper polynomials.

Next, we give a Voronovskaya-type theorem for the operators . In order to prove this theorem, we need some auxiliary results.

Lemma 3.6. For the operators , one has

Proof. By virtue of the generating functions (3.9) for Gould-Hopper polynomials, we obtain the above equalities.

Lemma 3.7. For , the following equalities hold:

Proof. According to Lemma 3.6, it is easy to get the above equalities.

Theorem 3.8. Let . Then, one has

Proof. In view of Taylor formula for the function , we find

where and . Applying to the both sides of (3.17), we get

According to Lemmas 3.6–3.7, (3.18) becomes

where

Now, we consider the sum as follows:

From the continuity of function , it results that for all , there exists a positive such that if , then . Furthermore, since the function is bounded, we can write for . In view of these facts, (3.21) leads to

Taking into account the fact

in the last inequality, we have

Substituting the inequality (3.24) in the equality (3.19), then from Lemma 3.7, we obtain

Equivalently, we can write

Taking limits for , (3.26) becomes

which completes the proof.

Remark 3.9. Theorem 3.8 is an explicit example of Gonska's result given in [16]. It is worthy to note that this Voronovskaya-type result is given for the operators which contain Gould-Hopper polynomials.

Remark 3.10. Taking in Theorem 3.8, we get a Voronovskaya-type result for the Szasz operators.

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