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

Year 2022, Volume: 71 Issue: 2, 407 - 421, 30.06.2022
https://doi.org/10.31801/cfsuasmas.941919

Abstract

In this
paper, we study several approximation properties of
Szasz-Mirakjan-Durrmeyer operators with shape parameter λ[1,1]λ∈[−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.

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

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.

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Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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