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Gal, Sorin G.

Semi-discrete quantitative Voronovskaya-type theorems for positive linear operators. (English) Zbl 1448.41019

Result. Math. 75, No. 3, Paper No. 117, 14 p. (2020).
Semidiscrete quantitative Voronovskaya type theorems are established using three particular cases of Lagrange-Hermite interpolation formula. Applications to Kantorovich operators and Bernstein operators are obtained.
Reviewer: Zoltán Finta (Cluj-Napoca)

Cited in 1 Review
Cited in 1 Document

MSC:

41A36 Approximation by positive operators
41A05 Interpolation in approximation theory
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)

Keywords:

Lagrange-Hermite interpolation; positive linear operators; quantitative semi-discrete Voronovskaya results; modulus of continuity; Bernstein-Kantorovich polynomials; Bernstein polynomials
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References:

[1] Acu, A-M; Gonska, H., Classical Kantorovich operators revisited, Ukr. Mat. Zh., 71, 6, 739-747 (2019) · Zbl 1435.41021 · doi:10.1007/s11253-019-01683-y
[2] Bernstein, SN, Complément à l’article de E. Voronovskaya “Détermination de la forme asymptotique de l’approximation des fonctions par les polynômes de M Bernstein”, C. R. Acad. Sci. URSS, 1932, 86-92 (1932) · Zbl 0005.01301
[3] Ditzian, Z.; Ivanov, KG, Strong converse inequalities, J. d’Analyse Math., 61, 61-111 (1993) · Zbl 0798.41009 · doi:10.1007/BF02788839
[4] Gavrea, I.; Ivan, M., An answer to a conjecture on Bernstein operators, J. Math. Anal. Appl., 390, 86-92 (2012) · Zbl 1254.41017 · doi:10.1016/j.jmaa.2012.01.021
[5] Gavrea, I.; Ivan, M., The Bernstein Voronovskaja-type theorems for positive linear approximation operators, J. Approx. Theory, 192, 291-296 (2015) · Zbl 1310.41003 · doi:10.1016/j.jat.2014.12.008
[6] Gonska, H.: On the degree of approximation in Voronovskaja’s theorem. In: Studia Universitatis Babes-Bolyai Mathematica, vol. 52, pp. 103-115 (2007) · Zbl 1199.41027
[7] Gonska, H., Pitul, P., Rasa, I.: On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators, In: Agratini, O., Blaga, P. (eds.) Numerical Analysis and Approximation Theory (Proceedings of the International Conference Cluj-Napoca), Casa Cartii de Stiinta, Cluj-Napoca, pp. 55-80 (2006) · Zbl 1123.41012
[8] Gonska, H.; Rasa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hung., 111, 119-130 (2006) · Zbl 1121.41004 · doi:10.1007/s10474-006-0038-4
[9] Hermite, C., Sur la formule d’interpolation de Lagrange, J. Reine Angew. Math., 84, 70-79 (1878) · JFM 09.0312.02 · doi:10.1515/crelle-1878-18788405
[10] Lorentz, GG, Bernstein Polynomials (1986), New York: Chelsea Publishing Company, New York · Zbl 0989.41504
[11] Mamedov, RG, The asymptotic value of the approximation of multiply differentiable functions by positive linear operators, Dokl. Akad. Nauk SSSR, 146, 1013-1016 (1962) · Zbl 0122.06206
[12] Sikkema, PC; van der Meer, PJC, The exact degree of local approximation by linear positive operators involving the modulus of continuity of the \(p\) th derivative, Nederl. Akad. Wetensch. Indag. Math., 41, 63-76 (1979) · Zbl 0399.41023 · doi:10.1016/1385-7258(79)90010-6
[13] Stancu, DD, Sur la formule d’interpolation d’Hermite et quelques applications de celle-ci (in Romanian), Acad. R. P. Romîne, Fil. Cluj, Stud. Cerc. Mat., 8, 339-355 (1957)
[14] Stancu, DD, Course (with Problems) in Numerical Analysis (in Romanian) (1977), Cluj-Napoca: Babes-Bolyai University Press, Cluj-Napoca
[15] Tachev, GT, The complete asymptotic expansion for Bernstein operators, J. Math. Anal. Appl., 385, 1179-1183 (2012) · Zbl 1232.41016 · doi:10.1016/j.jmaa.2011.07.042
[16] Videnskij, VS, Linear Positive Operators of Finite Rank, (Russian) (1985), Leningrad: A.I. Gerzen” State Pedagogical Institute, Leningrad
[17] Voronovskaja, E., Determination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein, C. R. Acad. Sci. URSS, 1932, 79-85 (1932) · JFM 58.1062.04
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