Abstract
The Voronovskaja formula for Aldaz–Kounchev–Render operators was established in terms of pointwise convergence. For suitable functions we prove it with uniform convergence. Moreover, we establish the Voronovskaja formula of second order with uniform convergence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material.
No data were used to support this study.
References
Abel, U., Ivan, M.: Asymptotic expansion of the multivariate Bernstein polynomials on a simplex. Approx. Theory Appl. 16, 85–93 (2000)
Acar, T., Aral, A., Cárdenas-Morales, D., Garrancho, P.: Szász-Mirakyan type operators which fix exponentials. Results Math. 72, 1393–1404 (2017)
Acu, A.M., Gonska, H., Heilmann, M.: Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render. J. Numer. Anal. Approx. Theory 50(1), 3–11 (2021)
Acu, A.M., De Marchi, S., Raşa, I.: Aldaz-Kounchev-Render operators and their approximation properties. Results Math. 78(1), 21 (2023)
Acu, A.M., Rasa, I.: An asymptotic formula for Aldaz-Kounchev-Render operators on the hypercube, arXiv:2301.01280, (2023)
Acu, A.M., Măduţa, A.I., Raşa, I.: Voronovskaya type results and operators fixing two functions. Math. Model. Anal. 26(3), 395–410 (2021)
Aldaz, J.M., Kounchev, O., Render, H.: Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces. Numer. Math. 114(1), 1–25 (2009)
Birou, M.: A proof of a conjecture about the asymptotic formula of a Bernstein type operator. Results Math. 72, 1129–1138 (2017)
Cárdenas-Morales, D., Garrancho, P., Muñoz-Delgado, F.J.: Shape preserving approximation by Bernstein-type operators which fix polynomials. Appl. Math. Comput. 182, 1615–1622 (2006)
Cárdenas-Morales, D., Garrancho, P., Rasa, I.: Bernstein-type operators which preserve polynomials. Comput. Math. Appl. 62, 158–163 (2011)
Cárdenas-Morales, D., Garrancho, P., Rasa, I.: Asymptotic Formulae via a Korovkin-Type Result. Abstr. Appl. Anal. 2012, 12 (2012)
Finta, Z.: A quantitative variant of Voronovskaja’s theorem for King-type operators. Constr. Math. Anal. 2(3), 124–129 (2019)
Finta, Z.: Bernstein type operators having \(1\) and \(x^j\) as fixed points. Cent. Eur. J. Math. 11, 2257–2261 (2013)
Gavrea, I., Ivan, M.: Complete asymptotic expansions related to conjecture on a Voronovskaja-type theorem. J. Math. Anal. Appl. 458(1), 452–463 (2018)
Gonska, H., Piţul, P., Raşa, I.: General King-type operators. Result. Math. 53, 279–286 (2009)
King, J.P.: Positive linear operators which preserve \(x^2\). Acta Math. Hungar. 99(3), 203–208 (2003)
Nasaireh, F., Raşa, I.: Another look at Voronovskaja type formulas. J. Math. Ineq. 12(1), 95–105 (2018)
Yilmaz, O.G., Gupta, V., Aral, A.: A note on Baskakov-Kantorovich type operators preserving \(e^{-x}\). Math. Meth. Appl. Sci. 43(13), 7511–7517 (2020)
Acknowledgements
The authors are grateful to the referee for a thorough reading of the manuscript and useful comments. The recommendations and suggestions led to an improved exposition of the paper.
Funding
Project financed by Lucian Blaga University of Sibiu through the research grant LBUS-IRG-2023.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work.
Corresponding author
Ethics declarations
Conflicts of interest.
The authors declare no competing financial interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Abel, U., Acu, A.M., Heilmann, M. et al. Voronovskaja formula for Aldaz–Kounchev–Render operators: uniform convergence. Anal.Math.Phys. 14, 2 (2024). https://doi.org/10.1007/s13324-023-00861-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-023-00861-3