Abstract
In this note Choquet type operators are introduced in connection with Choquet’s theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear Korovkin type approximation theorem associated to Bernstein–Kantorovich–Choquet operators is proved. The paper also includes a large generalization of Hölder’s inequality within the framework of monotone and sublinear operators acting on spaces of continuous functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agahi, H.: A refined Hölder’s inequality for Choquet integral by Cauchy–Schwarz’s inequality. Inf. Sci. 512, 929–934 (2020)
Bhatia, R.: Notes on Functional Analysis. Texts and Readings in Mathematics, vol. 50. Hindustan Book Agency, New Delhi (2009)
Cerdà, J., Martín, J., Silvestre, P.: Capacitary function spaces. Collect. Math. 62, 95–118 (2011)
Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Signed integral representations of comonotonic additive functionals. J. Math. Anal. Appl. 385(2), 895–912 (2012)
Choquet, G.: Theory of capacities. Annales de l’ Institut Fourier 5, 131–295 (1954)
Choquet, G.: La naissance de la théorie des capacités: réflexion sur une expérience personnelle. Comptes rendus de l’Académie des sciences, Série générale, La Vie des sciences 3, 385–397 (1986)
Dellacherie, C.: Quelques commentaires sur les prolongements de capacités. Séminaire Probabilités V, Strasbourg. Lecture Notes in Math., vol. 191. Springer, Berlin (1970)
Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publisher, Dordrecht (1994)
Gal, S.G.: Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions. Mediterr. J. Math. 14(5), 205–216 (2017)
Gal, S.G., Niculescu, C.P.: A nonlinear extension of Korovkin’s theorem. Mediterr. J. Math. 17(5), 1–14 (2020)
Gal, S.G., Niculescu, C.P.: Choquet operators associated to vector capacities. J. Math. Anal. Appl. 500(2), 125153 (2021). arXiv:2009.08946
Grabisch, M.: Set Functions. Games and Capacities in Decision Making. Springer, Berlin (2016)
Mesiar, R., Li, J., Pap, E.: The Choquet integral as Lebesgue integral and related inequalities. Kybernetika 46, 1098–1107 (2010)
Niculescu, C.P., Persson, L.-E: Convex Functions and their Applications. A Contemporary Approach, 2nd edn. CMS Books in Mathematics, Springer (2018)
Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)
Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009)
Zhou, L.: Integral representation of continuous comonotonically additive functionals. Trans. Am. Math. Soc. 350, 1811–1822 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gal, S.G., Niculescu, C.P. A note on the Choquet type operators. Aequat. Math. 95, 433–447 (2021). https://doi.org/10.1007/s00010-021-00803-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-021-00803-z
Keywords
- Choquet integral
- Monotone operator
- Sublinear operator
- Comonotone additive operator
- Hölder’s inequality
- Cauchy–Bunyakovsky–Schwarz inequality
- Bernstein–Kantorovich–Choquet operator