A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials
We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.
The Szasz operators (also called Szasz-Mirakyan operators) which are defined by  where , , and have an important role in the approximation theory, and their approximation properties have been investigated by many researchers.
In , Jakimovski and Leviatan proposed a generalization of Szasz operators by means of the Appell polynomials which have the generating functions of the form: where is an analytic function in the disc , and . Under the assumption that for , Jakimovski and Leviatan , defined the following linear positive operators:
After that, Ismail  defined another generalization of Szasz operators involving the operators (1.1) and (1.3) by means of Sheffer polynomials. Let and be analytic functions in the disc . Here, and are real. The Sheffer polynomials are generated by With the help of these polynomials, Ismail constructed the following linear positive operators: under the assumptions(i)for , ,(ii) and .
Later, Varma et al.  defined another generalization of Szasz operators by means of the Brenke-type polynomials. Suppose that are analytic functions. The Brenke-type polynomials  have generating functions of the form from which the explicit form of is as follows:
Under the assumptions
Varma et al. introduced the linear positive operators via where and .
The aim of this paper is to present a Kantorovich type of the operators given by (1.10) and to give their some approximation properties. We consider the Kantorovich version of the operators (1.10) under the assumptions (1.9) as follows: where , , and . It is easy to see that defined by (1.11) is linear and positive.
In the case of and , with the help of (1.7), it follows that , so the operators (1.11) reduce to the Szasz-Mirakyan-Kantorovich operators defined by  Various approximation properties of the Szasz-Mirakyan-Kantorovich operators and their iterates may be found in [7–13].
The case of gives the Kantorovich version of the operators (1.3).
The structure of the paper is as follows. In Section 2, the convergence of the operators (1.11) is given by means of Korovkin's theorem. The order of approximation is obtained with the help of a classical approach, the second modulus of continuity, and Peetre's -functional in Section 3. Finally, as an example, we present a Kantorovich type of the operators including Gould-Hopper polynomials and then we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.
2. Approximation Properties of Operators
In this section, we give our main theorem with the help of Korovkin theorem. We begin with the following lemma which is necessary to prove the main result.
Lemma 2.1. For all , the operators defined by (1.11) verify
Proof. Using the generating function of the Brenke-type polynomials given by (1.7), we can write From these equalities, the assertions of the lemma are obtained.
Lemma 2.2. For , one has
Proof. From the linearity of , we get Next, we apply Lemma 2.1.
Theorem 2.3. Let If , then and the operators converge uniformly in each compact subset of .
Proof. Using Lemma 2.1 and taking into account the equality (2.8) we get The above convergence is satisfied uniformly in each compact subset of . We can then apply the universal Korovkin-type property (vi) of Theorem 4.1.4 in  to obtain the desired result.
3. The Order of Approximation
In this section, we deal with the rates of convergence of the to by means of a classical approach, the second modulus of continuity, and Peetre's -functional.
Let . If , the modulus of continuity of is defined by where denotes the space of uniformly continuous functions on . It is also well known that, for any and each ,
The next result gives the rate of convergence of the sequence to by means of the modulus of continuity.
Theorem 3.1. Let . The operators satisfy the following inequality: where
Proof. Using (2.1), (3.2), and the linearity property of operators, we can write By using the Cauchy-Schwarz inequality for integration, we get which holds that By applying the Cauchy-Schwarz inequality for summation on the right-hand side of (3.7), we have where is given by (3.4). If we use this in (3.5), we obtain On choosing , we arrive at the desired result.
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 3.2. Let . The following holds, where
Theorem 3.3. Let . Then where and is a constant which is independent of the functions and . Also, is the same as in Theorem 3.2.
Proof. Suppose that . From Theorem 3.2, we can write The left-hand side of inequality (3.21) does not depend on the function , so where is Peetre's -functional defined by (3.11). By the relation between Peetre's -functional and the second modulus of smoothness given by (3.13), inequality (3.21) becomes whence we have the result.
4. Special Cases and Further Properties
In , the authors showed that the Gould-Hopper polynomials are Brenke-type polynomials with and , and the restrictions (1.9) and condition (2.8) for the operators given by (1.10) are satisfied under the assumption . These operators including the Gould-Hopper polynomials are as follows: where .
The special case and of (1.11) gives the following Kantorovich version of including the Gould-Hopper polynomials: under the assumption .
Now, we give a Voronovskaya-type theorem for the operators (4.4). In order to prove this theorem, we need the following lemmas.
Lemma 4.2. For the operators , one has
Proof. From the generating function (4.1) for the Gould-Hopper polynomials, one can easily find the above equalities.
Lemma 4.3. For , one has
Proof. It is enough to use Lemma 4.2 to obtain above equalities.
Theorem 4.4. Let . Then one has
Proof. By Taylor's theorem, we get where and . If we apply the operator to the both sides of (4.8), we obtain In view of Lemmas 4.2 and 4.3, the equality (4.9) can be written in the form where Applying Cauchy-Schwarz inequality, we get If we use Cauchy-Schwarz inequality again on the right-hand side of the inequality above, then we conclude that In view of Lemma 4.3, holds. On the other hand, since and , then it follows from Theorem 2.3 that Considering (4.13), (4.14), and (4.15), we immediately see that Then, taking limit as in (4.10) and using (4.16), we have which completes the proof.
A. Jakimovski and D. Leviatan, “Generalized Szasz operators for the approximation in the infinite interval,” Mathematica, vol. 11, pp. 97–103, 1969.View at: Google Scholar
T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, NY, USA, 1978.
D. Miclaus, “The voronovskaja type theorem for the Szász-Mirakjan-Kantorovich operators,” Journal of Science and Arts, vol. 2, no. 13, pp. 257–260, 2010.View at: Google Scholar
G. Başcanbaz-Tunca and A. Erençin, “A voronovskaya type theorem for q-Szasz-Mirakyan-Kantorovich operators,” Revue d'Analyse Numerique et de Theorie de l'Approximation, vol. 40, no. 1, pp. 14–23, 2011.View at: Google Scholar
Z. Walczak, “On approximation by modified Szasz-Mirakyan operators,” Glasnik Matematicki, vol. 37, no. 57, pp. 303–319, 2002.View at: Google Scholar
Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, NY, USA, 1987.