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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 935430, 9 pages
http://dx.doi.org/10.1155/2013/935430
Research Article

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.

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 [213]). 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, 1823].

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 [2427].

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 57, 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).

References

  1. O. Szasz, “Generalization of S. Bernstein's polynomials to the infinite interval,” Journal of Research of the National Bureau of Standards, vol. 45, pp. 239–245, 1950. View at MathSciNet
  2. A. Altın, O. Doğru, and F. Taşdelen, “The generalization of Meyer-König and Zeller operators by generating functions,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 181–194, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  3. O. Dogru, M. A. Özarslan, and F. Taşdelen, “On positive operators involving a certain class of generating functions,” Studia Scientiarum Mathematicarum Hungarica, vol. 41, no. 4, pp. 415–429, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  4. G. İçöz, F. Taşdelen Yeşildal, and O. Doğru, “Kantrorovich process of linear positive operators via biorthogonal polynomials,” Journal of Inequalities and Special Functions, vol. 3, pp. 77–84, 2012.
  5. G. İçöz, F. T. Yeşildal, and S. Varma, “On linear positive operators involving biorthogonal polynomial,” Ars Combinatoria, vol. 105, pp. 319–331, 2012. View at MathSciNet
  6. M. A. Özarslan, “q-Laguerre type linear positive operators,” Studia Scientiarum Mathematicarum Hungarica, vol. 44, no. 1, pp. 65–80, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. A. Özarslan and O. Duman, “Approximation properties of Poisson integrals for orthogonal expansions,” Taiwanese Journal of Mathematics, vol. 12, no. 5, pp. 1147–1163, 2008. View at MathSciNet
  8. F. Taşdelen, R. Aktaş, and A. Altın, “A Kantorovich type of szasz operators including brenke-type polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 867203, 13 pages, 2012. View at Publisher · View at Google Scholar
  9. F. Taşdelen and A. Erençin, “The generalization of bivariate MKZ operators by multiple generating functions,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 727–735, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. Varma, G. İçöz, and S. Sucu, “On some extensions of Szasz operators including Boas-Buck-type polynomials,” Abstract and Applied Analysis, vol. 2012, Article ID 680340, 15 pages, 2012. View at Publisher · View at Google Scholar
  11. S. Varma, S. Sucu, and G. İçöz, “Generalization of Szasz operators involving Brenke type polynomials,” Computers & Mathematics with Applications, vol. 64, no. 2, pp. 121–127, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Varma and F. Taşdelen, “Szasz type operators involving Charlier polynomials,” Mathematical and Computer Modelling, vol. 56, no. 5-6, pp. 118–122, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Varma and F. Taşdelen, “On a generalization of Szasz-Durrmeyer operators with some orthogonal polynomials,” Studia Universitatis Babeş-Bolyai, vol. 58, no. 2, pp. 225–232, 2013.
  14. A. Jakimovski and D. Leviatan, “Generalized Szasz operators for the approximation in the infinite interval,” Mathematica, vol. 11, no. 34, pp. 97–103, 1969. View at MathSciNet
  15. M. E.-H. Ismail, “On a generalization of Szasz operators,” Mathematica, vol. 39, no. 2, pp. 259–267, 1974. View at MathSciNet
  16. T. S. Chihara, An Introduction to Orthogonal Polynomials, vol. 13, Gordon and Breach Science Publishers, New York, NY, USA, 1978. View at MathSciNet
  17. P. L. Butzer, “On the extensions of Bernstein polynomials to the infinite interval,” Proceedings of the American Mathematical Society, vol. 5, pp. 547–553, 1954. View at MathSciNet
  18. O. Duman, M. A. Özarslan, and B. D. Vecchia, “Modified Szasz-Mirakjan-Kantorovich operators preserving linear functions,” Turkish Journal of Mathematics, vol. 33, no. 2, pp. 151–158, 2009. View at MathSciNet
  19. 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.
  20. G. Nowak and A. Sikorska-Nowak, “Some approximation properties of modified Szasz-Mirakyan-Kantorovich operators,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 38, no. 1, pp. 73–82, 2009. View at MathSciNet
  21. V. Totik, “Approximation by Szasz-Mirakjan-Kantorovich operators in Lp  (p>1),” Analysis Mathematica, vol. 9, no. 2, pp. 147–167, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. Başcanbaz-Tunca and A. Erençin, “A Voronovskaya type theorem for q-Szasz-Mirakyan-Kantorovich operators,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 40, no. 1, pp. 14–23, 2011. View at MathSciNet
  23. Z. Walczak, “On approximation by modified Szasz-Mirakyan operators,” Glasnik Matematički, vol. 37, no. 2, pp. 303–319, 2002. View at MathSciNet
  24. Ç. Atakut and İ. Büyükyazıcı, “Stancu type generalization of the Favard-Szàsz operators,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1479–1482, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  25. D. Bărbosu, “Kantorovich-Stancu type operators,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 3, article 53, 2004. View at MathSciNet
  26. İ. Büyükyazıcı, “Approximation by Stancu-Chlodowsky polynomials,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 274–282, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  27. A. Erençin, G. Başcanbaz-Tunca, and F. Taşdelen, “Kantorovich type q-Bernstein-Stancu operators,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 57, no. 1, pp. 89–105, 2012. View at MathSciNet
  28. F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  30. H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Mathematical Journal, vol. 29, pp. 51–63, 1962. View at MathSciNet