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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 935430, 9 pages
A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials
1Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, 06100 Ankara, Turkey
2Department 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.
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.
For each positive and or , the Szasz-Mirakyan operators defined by have an important role in the approximation theory . 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 , Jakimovski and Leviatan constructed a generalization of Szasz operators by means of the Appell polynomials. Then, Ismail  presented another generalization of Szasz operators by means of Sheffer polynomials, which involves the operators (1) defined by Jakimovski and Leviatan in . In ,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  have generating functions of the form where
The Kantorovich type of Szasz-Mirakyan operators is defined by  The approximation properties of the Szasz-Mirakyan-Kantorovich operators and their various iterates were studied by many authors in [12, 18–23].
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 .
For , gives the Kantorovich-Stancu type of the operators proposed by Jakimovski and Leviatan in .
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 .
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 ). 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.
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 .
Similarly, the special case and of (9) gives the following Kantorovich-Stancu type operators including the Gould-Hopper polynomials: under the assumption .
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:
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 16. For , it yields a Voronovskaya type theorem for the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).
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