Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$

Reşat Aslan

doi:10.31801/cfsuasmas.941919

Research Article

Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$

Year 2022, Volume: 71 Issue: 2, 407 - 421, 30.06.2022

Reşat Aslan

https://doi.org/10.31801/cfsuasmas.941919

Cited By: 7

Abstract

In this
paper, we study several approximation properties of
Szasz-Mirakjan-Durrmeyer operators with shape parameter $λ \in [- 1, 1]$ . Firstly, we obtain some preliminaries results such as moments and
central moments. Next, we estimate
the order of convergence in terms of the usual modulus of continuity, for the
functions belong to Lipschitz type class and Peetre's K-functional, respectively. Also, we prove a Korovkin type approximation theorem on weighted spaces and derive a Voronovskaya type asymptotic theorem for these operators. Finally, we give the comparison of the convergence of these newly defined operators to the certain functions with some graphics and error of approximation table.

Keywords

Modulus of continuity, Lipschitz type class, Voronovskaya type asymptotic theorem, Szasz-Mirakjan-Durrmeyer operators, weighted approximation

References

Acar, T., Ulusoy, G., Approximation by modified Szasz-Durrmeyer operators, Period Math. Hung., 72(1) (2016), 64–75. https://doi.org/10.1007/s10998-015-0091-2
Acu, A. M., Manav, N., Sofonea, D. F., Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., 2018(1) (2018), 202. https://doi.org/10.1186/s13660-018-1795-7
Alotaibi, A., Özger, F., Mohiuddine, S. A., Alghamdi, M. A., Approximation of functions by a class of Durrmeyer–Stancu type operators which includes Euler’s beta function, Adv. Differ. Equ., 2021(1) (2021). https://doi.org/10.1186/s13662-020-03164-0
Altomare, F., Campiti, M., Korovkin-type approximation theory and its applications, Walter de Gruyter, 1994. https://doi.org/10.1515/9783110884586
Aslan, R., Some approximation results on λ-Szasz-Mirakjan-Kantorovich operators, FUJMA, 4(3) (2021), 150–158. https://doi.org/10.33401/fujma.903140
Cai, Q. -B., Aslan, R., On a new construction of generalized q-Bernstein polynomials based on shape parameter λ, Symmetry, 13(10) (2021), 1919. https://doi.org/10.3390/sym13101919
Cai, Q. -B., Lian, B. Y., Zhou, G., Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 2018(1) (2018), 61. https://doi.org/10.1186/s13660-018-1653-7
Cai, Q. -B., Zhou, G., Li, J., Statistical approximation properties of λ-Bernstein operators based on q−integers, Open Math., 17(1) (2019), 487–498. https://doi.org/10.1515/math-2019-0039
Devore, R. A., Lorentz, G. G., Constructive Approximation, Springer, Berlin Heidelberg, 1993. https://doi.org/10.1007/978-3-662-02888-9
Gadzhiev, A. D., The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk., 218(5) (1974), 1001–1004.
Gupta, M. K., Beniwal, M. S., Goel, P., Rate of convergence for Szasz–Mirakyan–Durrmeyer operators with derivatives of bounded variation, Appl. Math. comput., 199(2) (2008), 828–832. https://doi.org/10.1016/j.amc.2007.10.036
Gupta, V., Simultaneous approximation by Sz´asz-Durrmeyer operators, Math. Stud., 64(1-4) (1995), 27—36.
Gupta, V., Noor, M. A., Beniwal, M. S., Rate of convergence in simultaneous approximation for Szasz–Mirakyan–Durrmeyer operators, J. Math. Anal. Appl., 322(2) (2006), 964–970. https://doi.org/10.1016/j.jmaa.2005.09.063
Gupta, V., Pant, R. P., Rate of convergence for the modified Szasz–Mirakyan operators on functions of bounded variation, J. Math. Anal. Appl., 233(2) (1999), 476–483. https://doi.org/10.1006/jmaa.1999.6289
İçöz, G., Mohapatra, R. N., Weighted approximation properties of Stancu type modification of q-Szasz-Durrmeyer operators, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 65(1) (2016), 87–104. https://doi.org/10.1501/commua10000000746
Korovkin, P. P., On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR., 90(953) (1953), 961–964.
Mazhar, S., Totik, V., Approximation by modified Szasz operators, Acta Sci. Math., 49(1-4) (1985), 257–269.
Mirakjan, G. M., Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk. SSSR., 31 (1941), 201–205.
Mursaleen, M., Al-Abied, A. A. H., Salman, M. A., Approximation by Stancu-Chlodowsky type λ-Bernstein operators, J. Appl. Anal., 26(1) (2020), 97–110. https://doi.org/10.1515/jaa-2020-2009
Mursaleen, M., Al-Abied, A. A. H., Salman, M. A., Chlodowsky type (λ, q)-Bernstein-Stancu operators, Azerb. J. Math., 10(1) (2020), 75–101.
Özger, F., Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41(16) (2020), 1990–2006. https://doi.org/10.1080/01630563.2020.1868503
Özger, F., Weighted statistical approximation properties of univariate and bivariate λ Kantorovich operators, Filomat, 33(11) (2019), 3473–3486. https://doi.org/10.2298/fil1911473o
Özger, F., On new Bezier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(1) (2020), 376–393. https://doi.org/10.31801/cfsuasmas.510382
Özger, F., Demirci, K., Yıldız, S., Approximation by Kantorovich variant of λ-Schurer operators and related numerical results, In: Topics in Contemporary Mathematical Analysis and Applications, pp. 77-94, CRC Press, Boca Raton, 2020. https://doi.org/10.1201/9781003081197-3
Qi, Q., Guo, D., Yang, G., Approximation properties of λ-Szasz-Mirakian operators, Int. J. Eng. Res., 12(5) (2019), 662–669.
Rahman, S., Mursaleen, M., Acu, A. M., Approximation properties of λ-Bernstein-Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42(11) (2019), 4042–4053. https://doi.org/10.1002/mma.5632
Srivastava, H. M., Ansari, K. J., Özger, F., Ödemiş, Özger, Z., A link between approximation theory and summability methods via four-dimensional infinite matrices. Mathematics, 9(16) (2021), 1895. https://doi.org/10.3390/math9161895
Srivastava, H. M., Özger, F., Mohiuddine, S. A., Construction of Stancu-type Bernstein operators based on B´ezier bases with shape parameter λ, Symmetry, 11(3) (2019), 316. https://doi.org/10.3390/sym11030316
Szasz, O., Generalization of the Bernstein polynomials to the infinite interval, J. Res. Nat. Bur. Stand., 45(3) (1950), 239–245. https://doi.org/10.6028/jres.045.024
Ye, Z., Long, X., Zeng, X. M., Adjustment algorithms for Bezier curve and surface, In: International Conference on 5th Computer Science and Education, (2010), 1712–1716. https://doi.org/10.1109/iccse.2010.5593563

Year 2022, Volume: 71 Issue: 2, 407 - 421, 30.06.2022

Reşat Aslan

https://doi.org/10.31801/cfsuasmas.941919

Cited By: 7

Abstract

References

Acar, T., Ulusoy, G., Approximation by modified Szasz-Durrmeyer operators, Period Math. Hung., 72(1) (2016), 64–75. https://doi.org/10.1007/s10998-015-0091-2
Acu, A. M., Manav, N., Sofonea, D. F., Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., 2018(1) (2018), 202. https://doi.org/10.1186/s13660-018-1795-7
Alotaibi, A., Özger, F., Mohiuddine, S. A., Alghamdi, M. A., Approximation of functions by a class of Durrmeyer–Stancu type operators which includes Euler’s beta function, Adv. Differ. Equ., 2021(1) (2021). https://doi.org/10.1186/s13662-020-03164-0
Altomare, F., Campiti, M., Korovkin-type approximation theory and its applications, Walter de Gruyter, 1994. https://doi.org/10.1515/9783110884586
Aslan, R., Some approximation results on λ-Szasz-Mirakjan-Kantorovich operators, FUJMA, 4(3) (2021), 150–158. https://doi.org/10.33401/fujma.903140
Cai, Q. -B., Aslan, R., On a new construction of generalized q-Bernstein polynomials based on shape parameter λ, Symmetry, 13(10) (2021), 1919. https://doi.org/10.3390/sym13101919
Cai, Q. -B., Lian, B. Y., Zhou, G., Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 2018(1) (2018), 61. https://doi.org/10.1186/s13660-018-1653-7
Cai, Q. -B., Zhou, G., Li, J., Statistical approximation properties of λ-Bernstein operators based on q−integers, Open Math., 17(1) (2019), 487–498. https://doi.org/10.1515/math-2019-0039
Devore, R. A., Lorentz, G. G., Constructive Approximation, Springer, Berlin Heidelberg, 1993. https://doi.org/10.1007/978-3-662-02888-9
Gadzhiev, A. D., The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, Dokl. Akad. Nauk., 218(5) (1974), 1001–1004.
Gupta, M. K., Beniwal, M. S., Goel, P., Rate of convergence for Szasz–Mirakyan–Durrmeyer operators with derivatives of bounded variation, Appl. Math. comput., 199(2) (2008), 828–832. https://doi.org/10.1016/j.amc.2007.10.036
Gupta, V., Simultaneous approximation by Sz´asz-Durrmeyer operators, Math. Stud., 64(1-4) (1995), 27—36.
Gupta, V., Noor, M. A., Beniwal, M. S., Rate of convergence in simultaneous approximation for Szasz–Mirakyan–Durrmeyer operators, J. Math. Anal. Appl., 322(2) (2006), 964–970. https://doi.org/10.1016/j.jmaa.2005.09.063
Gupta, V., Pant, R. P., Rate of convergence for the modified Szasz–Mirakyan operators on functions of bounded variation, J. Math. Anal. Appl., 233(2) (1999), 476–483. https://doi.org/10.1006/jmaa.1999.6289
İçöz, G., Mohapatra, R. N., Weighted approximation properties of Stancu type modification of q-Szasz-Durrmeyer operators, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 65(1) (2016), 87–104. https://doi.org/10.1501/commua10000000746
Korovkin, P. P., On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR., 90(953) (1953), 961–964.
Mazhar, S., Totik, V., Approximation by modified Szasz operators, Acta Sci. Math., 49(1-4) (1985), 257–269.
Mirakjan, G. M., Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk. SSSR., 31 (1941), 201–205.
Mursaleen, M., Al-Abied, A. A. H., Salman, M. A., Approximation by Stancu-Chlodowsky type λ-Bernstein operators, J. Appl. Anal., 26(1) (2020), 97–110. https://doi.org/10.1515/jaa-2020-2009
Mursaleen, M., Al-Abied, A. A. H., Salman, M. A., Chlodowsky type (λ, q)-Bernstein-Stancu operators, Azerb. J. Math., 10(1) (2020), 75–101.
Özger, F., Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41(16) (2020), 1990–2006. https://doi.org/10.1080/01630563.2020.1868503
Özger, F., Weighted statistical approximation properties of univariate and bivariate λ Kantorovich operators, Filomat, 33(11) (2019), 3473–3486. https://doi.org/10.2298/fil1911473o
Özger, F., On new Bezier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(1) (2020), 376–393. https://doi.org/10.31801/cfsuasmas.510382
Özger, F., Demirci, K., Yıldız, S., Approximation by Kantorovich variant of λ-Schurer operators and related numerical results, In: Topics in Contemporary Mathematical Analysis and Applications, pp. 77-94, CRC Press, Boca Raton, 2020. https://doi.org/10.1201/9781003081197-3
Qi, Q., Guo, D., Yang, G., Approximation properties of λ-Szasz-Mirakian operators, Int. J. Eng. Res., 12(5) (2019), 662–669.
Rahman, S., Mursaleen, M., Acu, A. M., Approximation properties of λ-Bernstein-Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42(11) (2019), 4042–4053. https://doi.org/10.1002/mma.5632
Srivastava, H. M., Ansari, K. J., Özger, F., Ödemiş, Özger, Z., A link between approximation theory and summability methods via four-dimensional infinite matrices. Mathematics, 9(16) (2021), 1895. https://doi.org/10.3390/math9161895
Srivastava, H. M., Özger, F., Mohiuddine, S. A., Construction of Stancu-type Bernstein operators based on B´ezier bases with shape parameter λ, Symmetry, 11(3) (2019), 316. https://doi.org/10.3390/sym11030316
Szasz, O., Generalization of the Bernstein polynomials to the infinite interval, J. Res. Nat. Bur. Stand., 45(3) (1950), 239–245. https://doi.org/10.6028/jres.045.024
Ye, Z., Long, X., Zeng, X. M., Adjustment algorithms for Bezier curve and surface, In: International Conference on 5th Computer Science and Education, (2010), 1712–1716. https://doi.org/10.1109/iccse.2010.5593563

There are 30 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Articles
Authors	Reşat Aslan 0000-0002-8180-9199
Publication Date	June 30, 2022
Submission Date	May 24, 2021
Acceptance Date	November 16, 2021
Published in Issue	Year 2022 Volume: 71 Issue: 2

Cite

APA	Aslan, R. (2022). Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 407-421. https://doi.org/10.31801/cfsuasmas.941919
AMA	Aslan R. Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2022;71(2):407-421. doi:10.31801/cfsuasmas.941919
Chicago	Aslan, Reşat. “Approximation by Szasz-Mirakjan-Durrmeyer Operators Based on Shape Parameter $\lambda$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 407-21. https://doi.org/10.31801/cfsuasmas.941919.
EndNote	Aslan R (June 1, 2022) Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 407–421.
IEEE	R. Aslan, “Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 407–421, 2022, doi: 10.31801/cfsuasmas.941919.
ISNAD	Aslan, Reşat. “Approximation by Szasz-Mirakjan-Durrmeyer Operators Based on Shape Parameter $\lambda$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 407-421. https://doi.org/10.31801/cfsuasmas.941919.
JAMA	Aslan R. Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:407–421.
MLA	Aslan, Reşat. “Approximation by Szasz-Mirakjan-Durrmeyer Operators Based on Shape Parameter $\lambda$”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 407-21, doi:10.31801/cfsuasmas.941919.
Vancouver	Aslan R. Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(2):407-21.