Abstract and Applied Analysis
Volume 2017 (2017), Article ID 4013958, 8 pages
https://doi.org/10.1155/2017/4013958

Research Article

The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

Behar Baxhaku and Artan Berisha

Department of Mathematics, University of Prishtina, Mother Teresa, 10000 Prishtina, Kosovo

Correspondence should be addressed to Artan Berisha; artan.berisha@uni-pr.edu

Received 23 March 2017; Accepted 18 June 2017; Published 26 July 2017

Academic Editor: Jaume Giné

Copyright © 2017 Behar Baxhaku and Artan Berisha. 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 the Szász and Chlodowsky operators based on Gould-Hopper polynomials and study the statistical convergence of these operators in a weighted space of functions on a positive semiaxis. Further, a Voronovskaja type result is obtained for the operators containing Gould-Hopper polynomials. Finally, some graphical examples for the convergence of this type of operator are given.

1. Introduction

In [1], Jakimovski and Leviatan introduced and investigated some approximation properties of the Favard-Szász type operator, by using Appell polynomials which satisfy the identity

where is an analytic function in the disc , and ,

Varma et al. constituted a link between orthogonal polynomials and the positive linear operators. In [1], they proposed Szász operators involving the Brenke polynomials defined by

where and

Recently, Büyükyazıcı et al. in [2] introduced the Chlodowsky variant of operators (3). Inspired by this work, we give the Szász-Chlodowsky type operators including Gould-Hopper polynomials. The generating functions for these Gould-Hopper polynomials are given by

and the explicit representations

where denotes the integer part. Now, let us define Szász-Chlodowsky type generalization of the Szász operators with the help of generating function (4), as follows:

where and is a positive increasing sequence with the properties

The generalization of Szász type operators has been studied in [3–10].

The aim of the this paper is to study some direct results in terms of the modulus of continuity of the second order, convergence of derivative operators to derivative functions, the weighted space, and the degree of approximation of by . We also study the statistical convergence. The rate of convergence of the operators to a certain function is also illustrated through graphics using Matlab.

2. Notations and Auxiliary Results

Let us denote , for some The following notations and lemmas are needed to prove the main results.

In what follows, let , , be the test functions.

Lemma 1. From (4), one has

(i);(ii);(iii);(iv);(v) + .

Lemma 2. For the operators , one has

(i);(ii);(iii);(iv);(v) + .

Proof. From Lemma 2 and by definition of , we have

Now, we consider the case as follows:

For , we have

and, finally,

Lemma 3. Let , and then, for every , one has

(i);(ii);(iii)

Theorem 4. Let Then, , uniformly on each compact subset of

Proof. By Lemma 2, , , uniformly on every compact subset of So, by Bohman-Korovkin theorem, the result follows.

Example 5. For , , and , , and , the convergence of to is illustrated in Figure 1(a).

(a)

(b)

Figure 1

Example 6. For , , and , , and , the convergence of to is illustrated in Figure 1(b).

Theorem 7. Let , and then for any one has

Proof. By using Lemma 3 case (ii) and the well-known properties of the modulus of continuity, we have

Recalling the Cauchy-Schwarz inequality, we obtain the formula below:

By means of Lemma 3 case (ii), for , one gets

Using (22) and taking in (19), we obtain the desired result.

The theorem below shows that the derivative is also an approximation process for

Theorem 8. Let If exists at a point , then one has

where is the modulus of continuity of

Proof. By simple calculations, the following formula is obtained:

where is the difference of order of corresponding to the increment . Using the relation between finite difference and divided difference, the derivative of order of the operators is represented as follows:

where Then, using Theorem 4, we have

By using the mean value theorem and some known classical properties of the modulus of continuity [11], one has

where Hence, we obtain

On the other hand,

where Using the estimates in (15), we have the desired result.

3. Weighted Approximation

Let be the space of all functions defined on satisfying the condition , where is a positive constant depending only on and is a weight function. By , we denote the subspace of all continuous functions with the norm and

Theorem 9. Let , and let be a weighted function; then, the inequality

is satisfied.

Proof. Using Lemma 2, one has

Since , there exists constant such that

This proof is complete.

From [12], for , the weighted modulus of continuity of is defined by

where is a continuously differentiable function on , , and Now, we define following sequence of positive linear operators defined with the help of defined in (6):

Theorem 10 (see [12]). Let be the sequences of linear positive operators and , If

where and , , and tend to zero as , then, for all functions , the inequality

holds for large enough.

Theorem 11. Let be the sequences of linear positive operators defined by (27) and If , then the inequality

is satisfied, where and

Proof. By simple calculation, we have

From Lemma 2, we have

Using Lemma 2 and (31), we obtain

By means of Lemma 2 and (32), one gets

Finally, from (33), we obtain

If we apply Theorem 10, we obtain the desired result.

4. -Statistical Convergence

Now, let ,  , be an infinite summability matrix. For a given sequence , the -transform of , denoted by , is given by

which provides the converging series for each We say that is a regular if whenever A sequence is called -statistically convergent to if, for every , This limit is denoted by Replacing by , the Cesàro matrix of order one in (6), from -statistical convergence, is reduced to statistical convergence. Similarly, if we take , the identity matrix, then -statistical convergence coincides with the ordinary convergence. Kolk [13] proved that, in the case of , -statistical convergence is stronger than ordinary convergence.

Further, we will first obtain the following weighted Korovkin theorem via -statistical convergence.

Theorem 12. Let be a nonnegative regular infinite summability matrix and Let be a continuous function such that

Then, for all , we have

Proof. From [14], for any , it is enough to prove that

So, by Lemma 2, we can easily get

Again, by using Lemma 2, we have

Now, for every given , let us define the following sets:

It is clear that Hence, for all , we get

Therefore,
Similarly, we have

Now, we define the following sets:

In view of (46), it is clear that , which yields

Thus, we get
Similarly, from Lemma 3, we have

Now, we give a Voronovskaja type relation for the operators

Theorem 13. Let be a nonnegative regular infinite summability matrix. Then, for every such that , one has

uniformly with respect to ,

Proof. Let , and Define the function by

Then, by assumption, we get and
By the linearity of applied to the last equality, we obtain

In view of Lemma 2, we get

For the last term of the right hand side in (52), using Cauchy-Schwarz inequality, we are led to

We observe that , and In view of Theorem 4,

Then, it follows from (32) that

uniformly in Combining (27), (31), and (46), we get the desired result.
Let denote the space of all real-valued bounded and uniformly continuous functions on , endowed with the norm Further, the appropriate Peetre -functional is given by

where . By [15, Theorem ] we can find a constant such that

where is the second-order modulus of continuity of Now, we obtain the rate of -statistical convergence for the operators with the help of Peetre’s -functional.

Theorem 14. Let Then, one has

Proof. By the linearity property of operators and using the local Taylor formula to the function , we have

where
Thus, we get

In view of (49) for , we have

From (61), one can write the following:

Hence, taking the limit as , we get the desired result.

The following theorem contains quantitative estimates by means of Peetre’s -functional.

Theorem 15. Letting , one has the estimate

where

Proof. Letting , by (61), we have

Using the last inequality for , and , we get

Taking the infimum on the right sides of the above inequality for all functions , so

By using the relation between Peetre’s -functional and the second modulus of smoothness given in [15], we get

From (49), we get , and hence Therefore, we get the rate of -statistical convergence of the sequence to in the space

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

A. Jakimovski and D. Leviatan, “Generalized Szász operators for the approximation in the infinite interval,” Mathematica—Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 11, no. 34, pp. 97–103, 1969.

İ. Büyükyazıcı, H. Tanberkan, S. K. Serenbay, and Ç. Atakut, “Approximation by Chlodowsky type Jakimovski–Leviatan operators,” Journal of Computational and Applied Mathematics, vol. 259, pp. 153–163, 2014.

A. Aral and V. Gupta, “The -derivative and applications to -Szász Mirakyan operators,” Calcolo. A Quarterly on Numerical Analysis and Theory of Computation, vol. 43, no. 3, pp. 151–170, 2006.

A. Kajla and P. N. Agrawal, “Approximation properties of Szász type operators based on Charlier polynomials,” Turkish Journal of Mathematics, vol. 39, no. 6, pp. 990–1003, 2015.

A. Kajla and P. N. Agrawal, “Szász-Durrmeyer type operators based on Charlier polynomials,” Applied Mathematics and Computation, vol. 268, pp. 1001–1014, 2015.

A. Kajla and P. N. Agrawal, “Szász-Kantorovich type operators based on Charlier polynomials,” Kyungpook Mathematical Journal, vol. 56, no. 3, pp. 877–897, 2016.

F. Taşdelen, R. Aktaş, and A. Altin, “A Kantorovich type of Szasz operators including Brenke-type polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 867203, 13 pages, 2012.

N. I. Mahmudov, “Approximation by the -Szász-Mirakjan operators,” Abstract and Applied Analysis, vol. 2012, Article ID 754217, 16 pages, 2012.

S. Sucu, G. Içöz, and S. Varma, “On some extensions of szasz operators including boas-buck-type polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 680340, 15 pages, 2012.

V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, New York, NY, USA, 2014.

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, Walter de Gruyter, New York, NY, USA, 1994.

A. D. Gadjiev and A. Aral, “The estimates of approximation by using a new type of weighted modulus of continuity,” Computers & Mathematics with Applications. An International Journal, vol. 54, no. 1, pp. 127–135, 2007.

E. Kolk, “Matrix summability of statistically convergent sequences,” Analysis, vol. 13, no. 1-2, pp. 77–84, 1993.

A. D. Gadjiev, “Theorems of the type of P. P. Korovkin's theorems,” Akademiya Nauk SSSR. Matematicheskie Zametki, vol. 20, no. 5, pp. 781–786, 1976.

R. A. DeVore and G. G. Lorentz, Constructive approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, Germany, 1993.