Abstract
The aim of this paper is to study variation detracting property and convergence in variation of the Bernstein–Durrmeyer modifications of the classical Bernstein operators in the space of functions of bounded variation. These problems are studied with respect to the variation seminorm. Moreover we also study the rate of convergence in terms of total variation.
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Durrmeyer, J.L.: Une formule d’inversion de la transformée de Laplace: Applications à la théorie des moments. Thèse de 3e Cycle, Faculté des Sciences del’Université de Paris (1967)
Lupas, A.: Die Folge der Betaoperatoren. Dissertation, Universitat Stuttgart (1972)
Derriennic, M.M.: Sur l’approximation de fonctions intégrables sur [0, I] par des polynômes de Bernstein modifies. J. Approx. Theory 31, 325–343 (1981)
Gonska, H.H.: A global inverse theorem on simultaneous approximation by Bernstein–Durrmeyer operators. J. Approx. Theory 67, 284–302 (1991)
Ditzian, Z., Ivanov, K.: Bernstein-type operators and their derivatives. J. Approx. Theory 56, 72–90 (1989)
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. Analysis (Munich) 23(4), 299–346 (2003)
Agratini, O.: On the variation detracting property of a class of operators. Appl. Math. Lett. 19, 1261–1264 (2006)
Pych-Taberska, P., Karsli, H.: On the rates of convergence of Bernstein–Chlodovsky polynomials and their Bézier-type variants. Appl. Anal. 90(3–4), 403–416 (2011)
Kivinukk, A., Metsmagi, T.: Approximation in variation by the Meyer–König and Zeller operators. Proc. Estonian Acad. Sci. 60(2), 88–97 (2011)
Öksüzer, Ö., Karsli, H., Tasdelen, F.: On convergence of Bernstein–Stancu polynomials in the variation seminorm. Numer. Funct. Anal. Optim. 37(4), 1–20 (2016)
İlarslan, H.G.İ, Başcanbaz-Tunca, G.: Convergence in variation for Bernstein-type operators. Mediterr. J. Math. 13(5), 2577 (2016). doi:10.1007/s00009-015-0640-1
Lorentz, G.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953)
Trigub, R.M., Belinsky, E.S.: Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, Dordrecht (2004)
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Karsli, H., Öksüzer Yılık, Ö. & Taşdelen Yeşildal, F. Convergence of the Bernstein–Durrmeyer Operators in Variation Seminorm. Results Math 72, 1257–1270 (2017). https://doi.org/10.1007/s00025-017-0653-0
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DOI: https://doi.org/10.1007/s00025-017-0653-0