Quantitative approximation by nonlinear Angheluta-Choquet singular integrals
DOI:
https://doi.org/10.33993/jnaat491-1217Keywords:
Submodular set function, nonlinear Choquet integral, nonlinear Angheluta-Choquet operatorsAbstract
By using the concept of nonlinear Choquet integral with respect to a capacity and as a generalization of the Poisson-Cauchy-Choquet operators, we introduce the nonlinear Angheluta-Choquet singular integrals with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures and with respect to the Choquet measure \(\mu(A)=\sqrt{M(A)}\), where \(M\) represents the Lebesgue measure. For some subclasses of functions we prove that these Choquet type operators can have essentially better approximation properties than their classical correspondents. The paper ends with the important, independent remark that for Choquet-type operators which are comonotone additive too, like Kantorovich-Choquet operators, Szasz-Mirakjan-Kantorovich-Choquet operators and Baskakov-Kantorovich-Choquet operators studied in previous papers, the approximation results remain identically valid not only for non-negative functions, but also for all functions which take negative values too, if they are lower bounded.
Downloads
References
O. Agratini, Approximation by Linear Operators (in Romanian), Cluj University Press, Cluj-Napoca, 2000.
F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter Studies in Mathematics17, New York, Berlin, 1994, https://doi.org/10.1515/9783110884586 DOI: https://doi.org/10.1515/9783110884586
Th. Angheluta, Une remark sur l’integrale de Poisson, Bull. Sci. Math. (Paris), XLVIII (1924), pp. 138–140.
P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, vol. 1, One-Dimensional Theory, Academic Press, New York, London, 1971, https://doi.org/10.1007/978-3-0348-7448-9 DOI: https://doi.org/10.1007/978-3-0348-7448-9
P.L. Butzer and W. Trebels, Operateurs de Gauss-Weiesrtrass et de Cauchy-Poissonet conditions lipschitzienne dans L1 (En), C. R. Acad. Sci. Paris ser. A-B.,268(1969), pp. 700–703.
G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1953-1954), pp. 131–292, https://doi.org/10.5802/aif.53 DOI: https://doi.org/10.5802/aif.53
D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, Boston, London, 2010, https://doi.org/10.1007/978-94-017-2434-0 DOI: https://doi.org/10.1007/978-94-017-2434-0
D. Dubois and H. Prade, Possibility Theory, Plenum Press, New York, 1988, https://doi.org/10.1007/978-1-4684-5287-7 DOI: https://doi.org/10.1007/978-1-4684-5287-7
W. Feller, An Introduction to Probability Theory and Its Applications, II, Wiley, New York, 1966.
S.G. Gal, Approximation by nonlinear Choquet integral operators, Annali di Mat. Pura Appl., 195 (3) (2016), pp. 881–896, https://doi.org/10.1007/s10231-015-0495-x DOI: https://doi.org/10.1007/s10231-015-0495-x
S.G. Gal, Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions, Mediterr. J. Math., 14 (5) (2017), pp. 205–216, https://doi.org/10.1007/s00009-017-1007-6 DOI: https://doi.org/10.1007/s00009-017-1007-6
S.G. Gal, Quantitative approximation by nonlinear Picard-Choquet, Gauss-Weierstrass-Choquet and Poisson-Cauchy-Choquet singular integrals, Results Math., 73 (3) (2018), Art. 92, 23 pp., https://doi.org/10.1007/s00025-018-0852-3 DOI: https://doi.org/10.1007/s00025-018-0852-3
S.G. Gal, Correction to: Quantitative Approximation by Nonlinear Picard-Choquet, Gauss-Weierstrass-Choquet and Poisson-Cauchy-Choquet Singular Integrals, Results Math., 75 (1) (2020), Art. 31, 3 pp., https://doi.org/10.1007/s00025-020-1155-z DOI: https://doi.org/10.1007/s00025-020-1155-z
S.G. Gal, Shape preserving properties and monotonicity properties of the sequences of Choquet type integral operators, J. Numer. Anal. Approx. Theory, 47(2) (2018), pp.135–149, https://ictp.acad.ro/jnaat/journal/article/view/1154
S.G. Gal, Quantitative approximation by Stancu-Durrmeyer-Choquet-Sipos operators, Math. Slovaca, 69(3) (2019), pp. 625–638, https://doi.org/10.1515/ms-2017-0252 DOI: https://doi.org/10.1515/ms-2017-0252
S.G. Galand I.T. Iancu, Quantitative approximation by nonlinear convolution operators of Landau-Choquet type, Carpath. J. Math., to appear in vol.37(1) (2021).
S.G. Gal and B.D. Opris, Uniform and pointwise convergence of Bernstein-Durrmeyer operators with respect to monotone and submodular set functions, J. Math. Anal. Appl., 424 (2015), pp. 1374–1379, https://doi.org/10.1016/j.jmaa.2014.12.012 DOI: https://doi.org/10.1016/j.jmaa.2014.12.012
S.G. Gal and S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators, Carpath. J. Math., 33 (2017), pp. 49–58. DOI: https://doi.org/10.37193/CJM.2017.01.06
S.G. Gal and S. Trifa, Quantitative estimates in Lp-approximation by Bernstein-Durrmeyer-Choquet operators with respect to distorted Borel measures, Results Math., 72 (3) (2017), pp. 1405–1315, https://doi.org/10.1007/s00025-017-0759-4 DOI: https://doi.org/10.1007/s00025-017-0759-4
S. Wang and G.J. Klir, Generalized Measure Theory, Springer, New York, 2009, https://doi.org/10.1007/978-0-387-76852-6 DOI: https://doi.org/10.1007/978-0-387-76852-6
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
