Abstract
In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy–Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Kharkov Math. Soc. 13, 1–2 (1912)
Burenkov, V., Ghorbanalizadeh, A., Sawano, Y.: Uniform boundedness of Kantorovich operators in Morrey spaces. Positivity 22, 1097–1107 (2018)
Herzog, F., Hill, J.D.: The Bernstein polynomials for discontinuous functions. Am. J. Math. 68, 109–124 (1946)
Kantorovich, L.V.: Sur certains developpements suivant les polynomes de la forme de S. Bernstein. I, II. C. R. Acad. Sci. USSR A 563–568, 595–600 (1930)
Lorentz, G.G.: Bernstein Polynomials. University Press, Toronto (1953)
Maier, V.: \(L_p\)-approximation by Kantorovich operators. Anal. Math. 4, 289–295 (1978)
Mathé, P.: Approximation of Hölder continuous functions by Bernstein polynomials. Am. Math. Mon. 106, 568–574 (1999)
Sawano, Y., Di Fazio, G. , Hakim, D.I.: Morrey spaces: introduction and applications to integral operators and PDE’s, Volume I. Chapman and Hall/CRC, New York, NY, USA (2020)
Sawano, Y. , Di Fazio, G. , Hakim, D.I., Morrey spaces: introduction and applications to integral operators and PDE’s, Volume II., Chapman and Hall/CRC, New York, NY, USA (2020)
Sawano, Y., Tian, X.X., Xu, J.: Uniform boundedness of Sz’asz–Mirakjan–Kantorovich operators in Morrey spaces with variable exponents. Filomat 34(7), 2109–2121 (2020)
Zeren, Y., Ismailov, M., Karacam, C.: The analogs of the Korovkin theorems in Banach function spaces. Positivity 26, no. 2, Paper No. 28, (2022)
Acknowledgements
We would like to thank the reviewer for their careful consideration of our manuscript. The third author would like to thank Dr. Janny Lindiarni for her useful suggestions.
Funding
Denny Ivanal Hakim is supported by PPMI-ITB 2021 Program.
Author information
Ethics declarations
Conflict of interest
The authors have no conflict of interest to declare that are relevant to the content of this article.
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
Arsana, M.P., Gunadi, R., Hakim, D.I. et al. On the convergence of Kantorovich operators in Morrey spaces. Positivity 27, 53 (2023). https://doi.org/10.1007/s11117-023-01004-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11117-023-01004-5