DOI QR코드

DOI QR Code

ON APPROXIMATION PROPERTIES OF STANCU VARIANT λ-SZÁSZ-MIRAKJAN-DURRMEYER OPERATORS

  • 투고 : 2022.08.11
  • 심사 : 2022.09.20
  • 발행 : 2022.09.30

초록

In the present paper, we aim to obtain several approximation properties of Stancu form Szász-Mirakjan-Durrmeyer operators based on Bézier basis functions with shape parameter λ ∈ [-1, 1]. We estimate some auxiliary results such as moments and central moments. Then, we obtain the order of convergence in terms of the Lipschitz-type class functions and Peetre's K-functional. Further, we prove weighted approximation theorem and also Voronovskaya-type asymptotic theorem. Finally, to see the accuracy and effectiveness of discussed operators, we present comparison of the convergence of constructed operators to certain functions with some graphical illustrations under certain parameters.

키워드

참고문헌

  1. A. M. Acu, N. Manav and D. F. Sofonea, Approximation properties of λ-Kantorovich operators, J. Inequal. Appl., 2018 (2018), 202. https://doi.org/10.1186/s13660-018-1795-7
  2. A. Alotaibi, F. Ozger, S. A. Mohiuddine and M. A. Alghamdi, Approximation of functions by a class of Durrmeyer-Stancu type operators which includes Euler's beta function, Adv. Differ. Equ., 2021 (2021), 1-14. https://doi.org/10.1186/s13662-020-03162-2
  3. F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, 17, Walter de Gruyter, 2011.
  4. K. J. Ansari, F. Ozger and Z. Odemis Ozger, Numerical and theoretical approximation results for Schurer-Stancu operators with shape parameter lambda, Comp. Appl. Math., 41 (2022), 1-18. https://doi.org/10.1007/s40314-021-01695-0
  5. R. Aslan, Some approximation results on λ-Szasz-Mirakjan-Kantorovich operators, FUJMA, 4 (2021), 150-158. https://doi.org/10.33401/fujma.903140
  6. R. Aslan, Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter λ, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71 (2022), pp, 407-421. https://doi.org/10.31801/cfsuasmas.941919
  7. M. Ayman Mursaleen, A. Kilicman and Md. Nasiruzzaman, Approximation by q-BernsteinStancu-Kantorovich operators with shifted knots of real parameters, Filomat, 36(4) (2022), 1179-1194. https://doi.org/10.2298/FIL2204179A
  8. M. Ayman Mursaleen and S. Serra-Capizzano, Statistical convergence via q-calculus and a korovkin's type Approximation theorem, Axioms, 11 (2022), 70. https://doi.org/10.3390/axioms11020070
  9. Q. B. Cai, K. J. Ansari, M. Temizer Ersoy and F. Ozger, Statistical blending-type approximation by a class of operators that includes shape parameters λ and α, Mathematics, 10(7) (2022), 1149. https://doi.org/10.3390/math10071149
  10. Q. B. Cai and R. Aslan, On a new construction of generalized q-Bernstein polynomials based on shape parameter λ, Symmetry, 13 (2021), 1919. https://doi.org/10.3390/sym13101919
  11. Q. B. Cai and R. Aslan, Note on a new construction of Kantorovich form q-Bernstein operators related to shape parameter λ, Computer Modeling in Engineering & Sciences, 130 (2022), 1479-1493. https://doi.org/10.32604/cmes.2022.018338
  12. Q. B. Cai and W. T. Cheng, Convergence of λ-Bernstein operators based on (p, q)-integers, J. Inequal. Appl., 2020 (2020), 35. https://doi.org/10.1186/s13660-020-2309-y
  13. Q. B. Cai, A. Kilicman and M. Ayman Mursaleen, Approximation Properties and q-Statistical Convergence of Stancu-Type Generalized Baskakov-Szasz Operators, J. Funct. Spaces, 2022 (2022).
  14. Q. B. Cai, B. Y. Lian and G. Zhou, Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 2018 (2018), 61. https://doi.org/10.1186/s13660-018-1653-7
  15. Q. B. Cai, G. Zhou and J. Li, Statistical approximation properties of λ-Bernstein operators based on q-integers, Open Math., 17 (2019), 487-498. https://doi.org/10.1515/math-2019-0039
  16. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Heidelberg, 1993.
  17. G. Farin, Curves and surfaces for computer-aided geometric design: a practical guide, Elsevier, 2014.
  18. A. D. Gadzhiev, 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 (1974), 1001-1004.
  19. V. Gupta, Simultaneous approximation by Szasz-Durrmeyer operators, Math. Stud., 64 (1995), 27-36.
  20. M. K. Gupta, M. S. Beniwal and P. Goel, Rate of convergence for Szasz-Mirakyan-Durrmeyer operators with derivatives of bounded variation, Appl. Math. comput., 199 (2008), 828-832. https://doi.org/10.1016/j.amc.2007.10.036
  21. V. Gupta, M. A. Noor and M. S. Beniwal, Rate of convergence in simultaneous approximation for Szasz-Mirakyan-Durrmeyer operators, J. Math. Anal. Appl., 322 (2006), 964-970. https://doi.org/10.1016/j.jmaa.2005.09.063
  22. K. Khan, D. K. Lobiyal and A. Kilicman, Bezier curves and surfaces based on modified Bernstein polynomials, Azerb. J. Math., 9 (2019), 3-21.
  23. P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 90 (1953), 961-964.
  24. A. Kumar, Approximation properties of generalized λ-Bernstein-Kantorovich type operators, Rend. Circ. Mat. Palermo (2), 70 (2020), 505-520. https://doi.org/10.1007/s12215-020-00509-2
  25. S. Mazhar and V. Totik, Approximation by modified Szasz operators, Acta Sci. Math., 49 (1985), 257-269.
  26. G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk SSSR, 31 (1941), 201-205.
  27. V. N. Mishra and R. B. Gandhi, A summation-integral type modification of Szasz-Mirakjan operators, Math. Methods Appl. Sci., 40 (2017), 175-182. https://doi.org/10.1002/mma.3977
  28. V. N. Mishra, R. B. Gandhi and R. N. Mohapatra, A summation-integral type modification of Szasz-Mirakjan-Stancu operators, J. Numer. Anal. Approx. Theory, 45 (2016), 27-36. https://doi.org/10.33993/jnaat451-1075
  29. V. N. Mishra, R. B. Gandhi and F. Nasaireh, Simultaneous approximation by Szasz-MirakjanDurrmeyer-type operators, Bollettino dell'Unione Matematica Italiana, 8 (2016), 297-305. https://doi.org/10.1007/s40574-015-0045-x
  30. M. Mursaleen, A. A. H. Al-Abied and M. A. Salman, Chlodowsky type (λ, q)-Bernstein-Stancu operators, Azerb. J. Math., 10 (2020), 75-101.
  31. M. Mursaleen, A. Alotaibi and K. J. Ansari, On a Kantorovich variant of-Szasz-Mirakjan operators, J. Funct. Spaces, 2016 (2016).
  32. H. Oruc and G. M. Phillips, q-Bernstein polynomials and Bezier curves, J. Comput. Appl. Math., 151 (2003), 1-12. https://doi.org/10.1016/S0377-0427(02)00733-1
  33. F. Ozger, Weighted statistical approximation properties of univariate and bivariate λKantorovich operators, Filomat, 33 (2019), 3473-3486. https://doi.org/10.2298/fil1911473o
  34. F. Ozger, Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41 (2020), 1990-2006. https://doi.org/10.1080/01630563.2020.1868503
  35. F. Ozger, On new Bezier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376-393.
  36. F. Ozger, E. Aljimi and M. Temizer Ersoy, Rate of weighted statistical convergence for generalized blending-type Bernstein-Kantorovich operators, Mathematics, 10(12) (2022), 2027. https://doi.org/10.3390/math10122027
  37. F. Ozger, K. Demirci and S. Yildiz, 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.
  38. Q. Qi, D. Guo and G. Yang, Approximation properties of λ-Szasz-Mirakian operators, Int. J. Eng. Res., 12 (2019), 662-669.
  39. S. Rahman, M. Mursaleen and A. M. Acu, Approximation properties of λ-Bernstein- Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42 (2019), 4042-4053. https://doi.org/10.1002/mma.5632
  40. T. W. Sederberg, Computer Aided Geometric Design Course Notes, Department of Computer Science Brigham Young University, October 9, 2014.
  41. H. M. Srivastava, K. J. Ansari, F. Ozger and Z. Odemis Ozger, A link between approximation theory and summability methods via four-dimensional infinite matrices, Mathematics, 9 (2021), 1895. https://doi.org/10.3390/math9161895
  42. H. M. Srivastava, F. Ozger and S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on Bezier bases with shape parameter λ, Symmetry, 11 (2019), 316. https://doi.org/10.3390/sym11030316
  43. D. D. Stancu, Asupra unei generalizari a polinoamelor lui Bernstein, Studia Univ. Babes-Bolyai Ser. Math.-Phys., 14 (1969), 31-45.
  44. O. Szasz, Generalization of the Bernstein polynomials to the infinite interval, J. Res. Nat. Bur. Stand., 45 (1950) 239-245. https://doi.org/10.6028/jres.045.024
  45. Z. Ye, X. Long and X. M. Zeng, Adjustment algorithms for Bezier curve and surface, In: International Conference on Computer Science and Education, pp, 1712-1716, 2010.