SavedAdd your journal

Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2026 / Volume: 70 Issue: 1

Bivariate Bernstein polynomials that reproduce exponential functions

Kenan Bozkurt(NATIONAL DEFENSE UNIVERSITY)0000-0001-9714-4729
Fırat Özsaraç(KIRIKKALE UNIVERSITY)0000-0001-7170-9613
Ali Aral(KIRIKKALE UNIVERSITY)0000-0002-2024-8607
Pages
—
Publication Date
2021-06-30
🏛️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)

Finally, a Voronovskaya-type theorem is given and our results are supported by graphics.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we construct Bernstein type operators that reproduce exponential functions on simplex with one moved curved side.
📊
Key Findings & Data:The operator interpolates the function at the corner points of the simplex.
💡
Conclusion & Impact:Finally, a Voronovskaya-type theorem is given and our results are supported by graphics.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Bernstein operators · exponential functions · classical and exponential convexity

Abstract

In this paper, we construct Bernstein type operators that reproduce exponential functions on simplex with one moved curved side. The operator interpolates the function at the corner points of the simplex. Used function sequence with parameters α and β not only are gained more modeling flexibility to operator but also satisfied to preserve some exponential functions. We examine the convergence properties of the new approximation processes. Later, we also state its shape preserving properties by considering classical convexity. Finally, a Voronovskaya-type theorem is given and our results are supported by graphics.

🔖

Cite This Article

Bozkurt, K., Özsaraç, F., Aral, A. (2021). Bivariate Bernstein polynomials that reproduce exponential functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 70 Issue: 1. https://doi.org/10.31801/cfsuasmas.793968

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (15)

  1. Adell, J.A., De La Cal, J., San Miguel, M., On the property of monotonic convergence for multivariate Bernstein-type operators, J. Approx. Theory., 80 (1995), 132–137. https://doi.org/10.1006/jath.1995.1008
  2. Aral, A., Acar, T., Ozsarac, F., Differentiated Bernstein type operators, Dolomites Research Notes on Approximation., 13 (1) (2020), 47-54. https://doi.org/10.14658/PUPJ-DRNA-2020-1-6
  3. Aral, A., C´ardenas-Morales, D., Garrancho, P., Bernstein-type operators that reproduce exponential functions, J. of Math. Ineq., 12 (3) (2018), 861-872. https://doi.org/10.7153/jmi-2018-12-64
  4. Aral, A., Limmam, M. L., Ozsarac, F., Approximation properties of Szász-Mirakyan-Kantorovich type operators, Math. Meth. Appl. Sci., 42 (16) (2018), 5233-5240. https://doi.org/10.1002/mma.5280
  5. Bodur, M., Yilmaz, O. G., Aral, A., Approximation by Baskakov-Szász-Stancu operators preserving exponential function, Constr. Math. Anal., 1 (1) (2018), 1–8. https://doi.org/10.33205/cma.450708
  6. Blaga, P., Catinaş, T., Coman, Gh., Bernstein-type operators on triangle with one curved side. Mediterr. J. Math., 9 (4) (2012), 843–855. https://doi.org/10.1007/s00009-011-0156-2
  7. Blaga, P., Catinaş, T., Coman, Gh., Bernstein-type operators on a triangle with all curved sides, Applied Mathematics and Computation., 218 (2011), 3072–3082. https://doi.org/10.1016/j.amc.2011.08.027
  8. Cárdenas-Morales, D., Garrancho, P., Munoz-Delgado, F.J., Shape preserving approximation by Bernstein-type operators which fix polynomials, Appl. Math. Comput., 182 (2) (2006), 1615–1622. https://doi.org/10.1016/j.amc.2006.05.046
  9. Cárdenas-Morales, D., Munoz-Delgado, F.J., Improving certain Bernstein-type approximation processes, Math. and Comp. in Simulation., 77 (2008), 170-178. https://doi.org/10. 1016/j.matcom.2007.08.009
  10. Censor, E., Quantitative results for positive linear approximation operators, J. Approx. Theory., 4 (1971), 442–450. https://doi.org/10.1016/0021-9045(71)90009-8
  11. Ditzian, Z., Inverse theorems for multidimensional Bernstein operators, Pac. J. Math., 121 (2) (1986), 293–319. https://doi.org/10.2140/pjm.1986.121.293
  12. Karlin, S., Studden, W.J., Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience, New York, 1966. https://doi.org/10.1137/1009050
  13. King, J.P., Positive linear operators which preserve x^2, Acta Math. Hungar., 99 (3) (2003), 203–208. https://doi.org/10.1023/A:1024571126455
  14. Ozsarac, F., Acar, T., Reconstruction of Baskakov operators preserving some exponential functions, Math. Meth. Appl. Sci., 42 (16) (2018), 5124-5132. https://doi.org/10.1002/ mma.5228
  15. Ozsarac, F., Aral, A., Karsli, H., On Bernstein–Chlodowsky type operators preserving exponential functions, Mathematical Analysis I: Approximation Theory-Springer., (2018), 121-138. https://doi.org/10.1007/978-981-15-1153-0_11