Abstract
For the multivariate Bernstein–Durrmeyer operator \(D_{n, \mu }\), written in terms of the Choquet integral with respect to a distorted probability Borel measure \(\mu \) on the standard d-dimensional simplex \(S^{d}\), quantitative \(L^{p}\)-approximation results, \(1\le p <\infty \), in terms of a K functional are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berdysheva, E.E.: Uniform convergence of Bernstein–Durrmeyer operators with respect to arbitrary measure. J. Math. Anal. Appl. 394, 324–336 (2012)
Berdysheva, E.E.: Bernstein–Durrmeyer operators with respect to arbitrary measure II: pointwise convergence. J. Math. Anal. Appl. 418, 734–752 (2014)
Berdysheva, E.E., Li, B.-Z.: On \(L^{p}\)-convergence of Bernstein–Durrmeyer operators with respect to arbitrary measure. Publ. Inst. Math. (Beograd) 96(110), 23–29 (2014)
Cerdà, J., Martín, J., Silvestre, P.: Capacitary function spaces. Collect. Math. 62, 95–118 (2011)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)
Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1994)
Gal, S.G.: Approximation by Choquet integral operators. Ann. Matem Pure Appl. 195(3), 881–896 (2016)
Gal, S.G., Opris, B.D.: Uniform and pointwise convergence of Bernstein–Durrmeyer operators with respect to monotone and submodular set functions. J. Math. Anal. Appl. 424, 1374–1379 (2015)
Gal, S.G., Trifa, S.: Quantitative estimates in uniform and pointwise approximation by Bernstein–Durrmeyer–Choquet operators. Carpath. J. Math. 33(1), 49–58 (2017)
Li, B.-Z.: Approximation by multivariate Bernstein–Durrmeyer operators and learning rates of least-square regularized regression with multivariate polynomial kernel. J. Approx. Theory 173, 33–55 (2013)
Wang, R.S.: Some inequalities and convergence theorems for Choquet integrals. J. Appl. Math. Comput. 35, 305–321 (2011)
Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gal, S.G., Trifa, S. Quantitative Estimates in \(L^{p}\)-Approximation by Bernstein–Durrmeyer–Choquet Operators with Respect to Distorted Borel Measures. Results Math 72, 1405–1415 (2017). https://doi.org/10.1007/s00025-017-0759-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0759-4