Journal of Function Spaces and Applications

Volume 2013, Article ID 935430, 9 pages

http://dx.doi.org/10.1155/2013/935430

## A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials

^{1}Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, 06100 Ankara, Turkey^{2}Department of Mathematics, Faculty of Science, Gazi University, Beşevler, 06500 Ankara, Turkey

Received 11 September 2013; Accepted 18 November 2013

Academic Editor: Anna Kaminska

Copyright © 2013 Rabia Aktaş 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

We introduce a Kantorovich-Stancu type modification of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain approximation properties of these operators. Also, we give a Voronovskaya type theorem for Kantorovich-Stancu type operators including Gould-Hopper polynomials.

#### 1. Introduction

For each positive and or , the Szasz-Mirakyan operators defined by
have an important role in the approximation theory [1]. Their Korovkin type approximation properties and rates of convergence have been investigated by many researchers. Recently, there is a growing interest in defining linear positive operators via special functions (see [2–13]). In particular, many authors have studied various generalizations of Szasz operators via special functions. In [14], Jakimovski and Leviatan constructed a generalization of Szasz operators by means of the Appell polynomials. Then, Ismail [15] presented another generalization of Szasz operators by means of Sheffer polynomials, which involves the operators (1) defined by Jakimovski and Leviatan in [14]. In [11],Varma et al*. *considered the following generalization of Szasz operators by means of the Brenke type polynomials, which are motivated by the operators defined by Jakimovski and Leviatanand Ismail, for and :
under the following assumptions:
where
are analytic functions and the Brenke type polynomials [16] have generating functions of the form
where

The Kantorovich type of Szasz-Mirakyan operators is defined by [17] The approximation properties of the Szasz-Mirakyan-Kantorovich operators and their various iterates were studied by many authors in [12, 18–23].

Recently, in [8], the Kantorovich type of the operators given by (2) under the assumptions (3) has been defined as where , and , and some of its properties have been investigated.

The purpose of this study is to introduce a Kantorovich-Stancu type modification of the operators given by (8) and to examine the approximation properties of these operators. We also present a Kantorovich-Stancu type of the operators including Gould-Hopper polynomials and then we prove a Voronovskaya type theorem for these operators including Gould-Hopper polynomials.

#### 2. Construction of the Operators

For each positive integer , and , or , let us consider the following operators: where and parameters satisfy the condition . For the approximation properties of Stancu type operators, we refer to [24–27].

It is clear that for , reduces to the operators defined by (8).

In the case of and with the help of (5) it follows that . So the operator gives the Kantorovich-Stancu type of Szasz-Mirakyan operators as follows: where and parameters satisfy the condition .

In the case of , the operator (10) turns out to be the Szasz-Mirakyan-Kantorovich operators given by (7).

For , gives the Kantorovich-Stancu type of the operators proposed by Jakimovski and Leviatan in [14].

Now, for the operators given by (9), we give some results which are necessary to prove the main theorem.

Lemma 1. *Kantorovich-Stancu type operators, defined by (9), are linear and positive.*

Lemma 2. *For each , the Kantorovich-Stancu type operators (9) have the following properties:
*

*Proof. *From the generating function of the Brenke type polynomials given by (5), a few calculations reveal that
By using these equalities, we obtain the assertions of the lemma by simple calculation.

Lemma 3. *For each , one has
*

Theorem 4. *Let
**
If , then
**
and the operators converge uniformly in each compact subset of .*

*Proof. *According to Lemma 2, by considering the equality (16), we get
This convergence is satisfied uniformly in each compact subset of . Then, the proof follows from the universal Korovkin-type property (vi) of Theorem in [28].

#### 3. Rates of Convergence

In this section, we compute the rates of convergence of the operators to by means of a classical approach, the second modulus of continuity, and Peetre’s -functional.

Let . Then for , the modulus of continuity of denoted by is defined to be where denotes the space of uniformly continuous functions on Then, for any and each , it is well known that one can write

The next result gives the rate of convergence of the sequence to by means of the modulus of continuity.

Theorem 5. *For , one has
**
where
*

*Proof. *Using linearity of the operators , (11) and (20), we get
According to the Cauchy-Schwarz inequality for integration, we obtain that
from which, it follows that
By using the Cauchy-Schwarz inequality for summation on the right hand side of (25), we may write
where is given by (22). Considering this inequality in (23), we find that
If we set , the proof is completed.

Now, we will study the rates of convergence of the operators to by means of the second modulus of continuity and Peetre’s -functional.

Recall that the second modulus of continuity of is defined by where is the class of real valued functions defined on which are bounded and uniformly continuous with the norm .

Peetre’s -functional of the function is defined by where and the norm (see [29]). It is clear that the following inequality: holds for all . The constant is independent of and .

Theorem 6. *Let . If is defined by (9), then one has
**
where
*

*Proof. *We can write from the Taylor expansion of , the linearity of the operators , and (11)
From Lemma 2, it is obvious that
for . Thus, by considering Lemmas 2 and 3 in (34), one can write
which completes the proof.

Theorem 7. *If , then one has
**
where
**
and is a constant which is independent of the function and . Also, is the same as in Theorem 6.*

*Proof. *Suppose that . From Theorem 6, we have
Since the left-hand side of inequality (39) does not depend on the function , we get
where is Peetre’s -functional defined by (29). By using the relation (31) in (39), the inequality
holds.

*Remark 8. *In Theorems 5–7, when under the assumption (16).

#### 4. Special Cases of the Operators and Further Properties

Gould-Hopper polynomials are defined through the identity and satisfy the generating function where, as usual, denotes the integer part [30].

The Gould-Hopper polynomials are Brenke-type polynomials for the special case of and in (5). From (2), the operators including the Gould-Hopper polynomials are as follows: where and (see [11]).

Similarly, the special case and of (9) gives the following Kantorovich-Stancu type operators including the Gould-Hopper polynomials: under the assumption .

*Remark 9. *For , we have and the operators given by (45) reduce to the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).

*Remark 10. *For , the operators (45) give the Kantorovich type operators including the Gould-Hopper polynomials given by
in [8].

*Remark 11. *For in Remark 10, we get and then the operators given by (46) reduce to the Szasz-Mirakyan-Kantorovich operators given by (7).

Now, in order to prove a Voronovskaya type theorem for the operators given by (45), let us prove the following lemmas.

Lemma 12. *For the operators , one has
*

*Proof. *The proof follows from the generating function (43) for the Gould-Hopper polynomials.

Lemma 13. *For each , one has
*

*Proof. *From Lemma 12, the proof is obvious.

Theorem 14. *Let . Then one has
*

*Proof. *By Taylor’s theorem for , we have
where and . By applying the operator to the both sides of (50), we have
According to Lemmas 12 and 13, the equality (51) can be written as follows:
where
By applying Cauchy-Schwarz inequality, we can write
If we consider Cauchy-Schwarz inequality again on the right-hand side of inequality above, then we arrive at

From Lemma 13, we have
On the other hand, since and , then it follows from Theorem 4 that
Therefore, we conclude from (55), (56), and (57) that
and then, by taking limit as in (52) and using (58), we find
which completes the proof.

*Remark 15. *For , Theorem 14 represents the Voronovskaya type theorem for the operators given by (46) (see [8]).

*Remark 16. *For , it yields a Voronovskaya type theorem for the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).

*Remark 17. *Getting in Theorem 14 gives the Voronovskaya type result for the Szasz-Mirakyan-Kantorovich operators given by (7).

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