Startseite Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Convergence of α-Bernstein-Durrmeyer operators about a collection of measures

  • Harmanjit Kaur und Meenu Rani Goyal EMAIL logo
Veröffentlicht/Copyright: 25. Februar 2025
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

This work offers a generalization of the α-Bernstein-Durrmeyer operators Dm,Zα about a collection Z of arbitrary measures, with the goal of building a bridge between approximation, probability, and measure theory. It has been discussed how to achieve pointwise convergence of Dm,Zα and constraints on measure collection have been developed for these operators’ convergence. With certain values of the measures considered, the generalization yields a range of Bernstein-like operators with their α variation. In order to handle a wider variety of p.l.o. and investigate their convergence, the paper comprises numerous operators, making it a useful tool.

Funding statement: This work is supported by National Board of Higher Mathematics – Department of Atomic Energy (NBHM-DAE), Government of India (Sanction No. 02011/25/2021/NBHM(RP)/R&D II/7997) and Department of Mathematics, Thapar Institute of Engineering and Technology, Patiala, India. We are also thankful to Ministry of Science & Technology, Department of Science & Technology, Government of India, for providing the optimal infrastructure facilities under FIST project.

  1. (Communicated by David Buhagiar)

References

[1] Acar, T.—Acu, A. M.—Manav, N.: Approximation of functions by genuine Bernstein-Durrmeyer type operators, J. Math. Inequal. 12(4) (2018), 975–987.10.7153/jmi-2018-12-74Suche in Google Scholar

[2] Acar, T.—Agrawal, P. N.—Neer, T.: Bézier variant of the BernsteinDurrmeyer type operators, Results Math. 72 (2017), 1341–1358.10.1007/s00025-016-0639-3Suche in Google Scholar

[3] Altomare, F.: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010), 92–164.Suche in Google Scholar

[4] Aral, A.: On a new approach in the space of measurable functions, Constr. Math. Anal. 6(4) (2023), 237–248.10.33205/cma.1381787Suche in Google Scholar

[5] Aral, A.: View of weighted approximation: Korovkin and quantitative type theorems, Modern Math. Methods 1(1) (2023), 1–21.Suche in Google Scholar

[6] Berdysheva, E. E.: Uniform convergence of BernsteinDurrmeyer operators with respect to arbitrary measure, J. Math. Anal. Appl. 394(1) (2012), 324–336.10.1016/j.jmaa.2012.03.004Suche in Google Scholar

[7] Berdysheva, E. E.: BernsteinDurrmeyer operators with respect to arbitrary measure, II: pointwise convergence, J. Math. Anal. Appl. 418(2) (2014), 734–752.10.1016/j.jmaa.2014.04.006Suche in Google Scholar

[8] Berdysheva, E. E.—Heilmann, M.—Hennings, K.: Pointwise convergence of the BernsteinDurrmeyer operators with respect to a collection of measures, J. Approx. Theory 251 (2020), Art. ID 105339.10.1016/j.jat.2019.105339Suche in Google Scholar

[9] Berdysheva, E. E.—Jetter, K.: Multivariate BernsteinDurrmeyer operators with arbitrary weight functions, J. Approx. Theory 162(3) (2010), 576–598.10.1016/j.jat.2009.11.005Suche in Google Scholar

[10] Berdysheva, E. E.—Li, B. Z.: On Lp-convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure, Publ. Inst. Math. 96(110) (2014), 23–29.10.2298/PIM1410023BSuche in Google Scholar

[11] Chen, X.—Tan, J.—Liu, Z.—Xie, J.: Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450(1) (2017), 244–261.10.1016/j.jmaa.2016.12.075Suche in Google Scholar

[12] Durrmeyer, J. L.: Une formule d’inversion de la transformée de Laplace Applications à la théorie des moments, Thèse de 3e cycle, Faculté des Sciences, Université de Paris, 1967.Suche in Google Scholar

[13] Gavrea, I.—Ivan, M.: The Bernstein Voronovskaja-type theorem for positive linear approximation operators, J. Approx. Theory 192 (2015), 291–296.10.1016/j.jat.2014.12.008Suche in Google Scholar

[14] Gupta, V.—Tachev, G.: Approximation with Positive Linear Operators and Linear Combinations, Springer, Berlin, 2017.10.1007/978-3-319-58795-0Suche in Google Scholar

[15] Kajla, A.—Acar, T.: Blending type approximation by generalized Bernstein-Durrmeyer type operators, Miskolc Math. Notes 19(1) (2018), 319–336.10.18514/MMN.2018.2216Suche in Google Scholar

[16] Kajla, A.—Acar, T.: Modified α-Bernstein operators with better approximation properties, Ann. Funct. Anal. 10(4) (2019), 570–582.10.1215/20088752-2019-0015Suche in Google Scholar

[17] Kajla, A.—Acar, T.: Bézier-Bernstein-Durrmeyer type operators, Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. RACSAM 114 (2020), 1–11.10.1007/s13398-019-00759-5Suche in Google Scholar

[18] Kajla, A.—Mursaleen, M.—Acar, T.: Durrmeyer-type generalization of parametric Bernstein operators, Symmetry 12(7) (2020), Art. No. 1141.10.3390/sym12071141Suche in Google Scholar

[19] Korovkin, P. P.: Linear Operators and Approximation Theory, Hindustan Publishing Corporation, 1960.Suche in Google Scholar

[20] Li, B. Z.: Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels, J. Approx. Theory 173 (2013), 33–55.10.1016/j.jat.2013.04.007Suche in Google Scholar

[21] Lorentz, G. G.: Bernstein Polynomials, American Mathematical Society, United States, 2013.Suche in Google Scholar

[22] Mohiuddine, S. A.—Acar, T.—Alghamdi, M. A.: Genuine modified Bernstein-Durrmeyer operators, J. Inequal. Appl. 2018(1) (2018), 1–13.10.1186/s13660-018-1693-zSuche in Google Scholar PubMed PubMed Central

[23] Mohiuddine, S. A.—Acar, T.—Alotaibi, A.: Construction of a new family of Bernstein-Kantorovich operators, Math. Methods Appl. Sci. 40(18) (2017), 7749–7759.10.1002/mma.4559Suche in Google Scholar

[24] Mohiuddine, S. A.—Özger, F.: Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α, Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. RACSAM 114(2) (2020), 1–17.10.1007/s13398-020-00802-wSuche in Google Scholar

[25] Occorsio, D.—Russo, M. G.—Themistoclakis, W.: Some numerical applications of generalized Bernstein operators, Constr. Math. Anal. 4(2) (2021), 186–214.10.33205/cma.868272Suche in Google Scholar

[26] Paltanea, R.: Durrmeyer type operators on a simplex, Constr. Math. Anal. 4(2) (2021), 215–228.10.33205/cma.862942Suche in Google Scholar

[27] Taylor, M. E.: Measure Theory and Integration, American Mathematical Society, United States, 2006.Suche in Google Scholar

[28] Wang, Z.—Klir, G. J.: Generalized Measure Theory, Springer, New York, 2009.10.1007/978-0-387-76852-6Suche in Google Scholar

Received: 2024-05-29
Accepted: 2024-08-23
Published Online: 2025-02-25
Published in Print: 2025-02-25

© 2025 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Generalized Sasaki mappings in d0-Algebras
  2. On a theorem of Nathanson on Diophantine approximation
  3. Constructing infinite families of number fields with given indices from quintinomials
  4. Partitions into two Lehmer numbers in ℤq
  5. Fundamental systems of solutions of some linear differential equations of higher order
  6. On k-Circulant matrices involving the Lucas numbers of even index
  7. Explicit formulae for the Drazin inverse of the sum of two matrices
  8. On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
  9. Novel generalized tempered fractional integral inequalities for convexity property and applications
  10. Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
  11. The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
  12. Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
  13. Improvements on the Leighton oscillation theorem for second-order dynamic equations
  14. Topogenous orders on forms
  15. Comparison of topologies on fundamental groups with subgroup topology viewpoint
  16. An elementary proof of the generalized Itô formula
  17. Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
  18. Advancing reliability and medical data analysis through novel statistical distribution exploration
  19. Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
Heruntergeladen am 25.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0010/html
Button zum nach oben scrollen