SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 Sayı: 3

Approximation by $\alpha$-Bernstein-Schurer operator

Nursel ÇETİN
Pages
732–743
Publication Date
2021-06-07
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Also, we present Voronovskaya and Grüss-Voronovskaya type results for these operators.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we introduce a new family of generalized Bernstein-Schurer operators and investigate some approximation properties of these operators.
📊
Key Findings & Data:We obtain a uniform approximation result using the well-known Korovkin theorem and give the degree of approximation via second modulus of smoothness.
💡
Conclusion & Impact:Also, we present Voronovskaya and Grüss-Voronovskaya type results for these operators.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Bernstein-Schurer operators · α-Bernstein operator · Modulus of continuity

Abstract

In this paper, we introduce a new family of generalized Bernstein-Schurer operators and investigate some approximation properties of these operators. We obtain a uniform approximation result using the well-known Korovkin theorem and give the degree of approximation via second modulus of smoothness. Also, we present Voronovskaya and Grüss-Voronovskaya type results for these operators.

🔖

Cite This Article

ÇETİN, N. (2021). Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 Sayı: 3, 732–743. https://ajindex.com/articles/929239

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  1. T. Acar and A. Kajla, Degree of approximation for bivariate generalized Bernstein type operators, Results Math. 73 Article number: 79, 2018
  2. T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine Bernstein- Durrmeyer type operators, J. Math. Inequal. 12 (4), 975–987, 2018
  3. A.M. Acu, H. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J. 63 (6), 843–864, 2011
  4. A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Com- mun. 24, 119–131, 2019
  5. F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, Walter de Gruyter, Berlin-New York, 1994
  6. S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1–2, 1912/1913
  7. Q. Cai and X. Xu, Shape-preserving properties of a new family of generalized Bern- stein operators, J. Inequal. Appl. 2018, Article number: 241, 2018
  8. D. Cárdenas-Morales, P. Garrancho and I. Raşa, Bernstein-type operators which pre- serve polynomials, Comput. Math. Appl. 62 (1), 158–163, 2011
  9. X. Chen, J. Tan and Z. Liu, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450, 244–261, 2017
  10. N. Çetin, Approximation and geometric properties of complex -Bernstein operator, Results Math. 74, Article number: 40, 2019
  11. N. Çetin and V.A. Radu, Approximation by generalized Bernstein-Stancu operators, Turkish J. Math. 43, 2032–2048, 2019
  12. R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993
  13. J.L. Durrmeyer, 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
  14. S.G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx. 7 (1), 97– 122, 2015
  15. H. Gonska and G. Tachev, Grüss-type inequalities for positive linear operators with second order moduli, Mat. Vesnik, 63 (4), 247–252, 2011
  16. G. Grüss, Uber das maximum des absoluten betrages von $\frac{1}{b-a}\int\limits_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{\left( b-a\right) ^{2}}\int\limits_{a}^{b}f\left( x\right)dx\int\limits_{a}^{b}g\left( x\right) dx,$, Math. Z. 39, 215–226, 1935
  17. A. Kajla and D. Miclăuş, Blending Type Approximation by GBS Operators of Gener- alized Bern