Korovkin type criterion for some Voronovskaja type formulae
DOI:
https://doi.org/10.24193/subbmath.2026.3.7Keywords:
Linear Positive Operators, Voronovskaja type theorem., criterion, test functionsAbstract
In this note we present a simple criterion to validate some Voronovskaja type formulae. We show that it is sufficient to check the validity of the Voronovskaja formula on the test functions \(\psi^i, \psi=0,\ldots,4\), where \(\psi\) is a function satisfying some conditions, to get the validity for the general case.
References
[1] U. Abel, A.M. Acu, M. Heilmann, I. Rasa, Genuine Bernstein-Durrmeyer operators preserving 1 and xj, Ann. Funct. Anal. (2024), 15:4. DOI: https://doi.org/10.1007/s43034-023-00305-w
[2] F. Altomare, M. Campitti, Korovkin type approximation theory and its applications. W. de Gruyter, Berlin-New York, 1994. DOI: https://doi.org/10.1515/9783110884586
[3] A. Aral, D. Cardenas-Morales, P. Garrancho, Bernstein type operators that reproduce exponential functions, J. Math. Ineq., 12(2018), no. 3, 861-872. DOI: https://doi.org/10.7153/jmi-2018-12-64
[4] Bernstein S.N., Demonstration du theoreme de Weierstrass, fondee sur la calcul des probabilites, Commun. Soc. Math., Kharkow, 13(1912), no. 2, 1-2.
[5] M.M. Birou, A class of Markov operators which preserve ej, j ≥ 1, Appl. Math. Comp., 250(2015), 1-15. DOI: https://doi.org/10.1016/j.amc.2014.10.078
[6] M. Birou, Discrete Bernstein type operators which preserve e0 and ej, j ≥ 1, Annals of Tiberiu Popoviciu Seminar, 13(2015), 3-12.
[7] M.M. Birou, A proof of a conjecture about the asymptotic formula of a Bernstein type operator, Results Math., 72(2017), no. 3, 1129-1138. DOI: https://doi.org/10.1007/s00025-016-0608-x
[8] Cardenas-Morales, D., Garrancho, P., Rasa, I.: Asymptotic formulae via a Korovkin-type result. Abstr. Appl. Anal., 12(2012), Article ID 217464. DOI: https://doi.org/10.1155/2012/217464
[9] W. Chen, On the modified Bernstein-Durrmeyer operators, Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China, 1987.
[10] I. Gavrea, M. Ivan, The Bernstein Voronovskaja-type theorem for positive linear approximation operators, J. Approx. Theory, 192(2015), 291-296. DOI: https://doi.org/10.1016/j.jat.2014.12.008
[11] I. Gavrea, M. Ivan, A simple proof of a generalized Bernstein Voronovskaja-type theorem, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, 12(2014), 33-36.
[12] T.N.T. Goodman, A. Sharma, A modified Bernstein-Schoenberg operator, in: Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed. by Sendov et al.), Sofia: Publ. House Bulg. Acad. of Sci. 1988, 166-173.
[13] S. Morigi, M. Neamtu, Some results for a class of generalized polynomials, Adv. Comput. Math., 12(2000), no. 1, 133-149. DOI: https://doi.org/10.1023/A:1018908917139
[14] R. P˘alt˘anea, Approximation theory using positive linear operators, Birkhauser, Boston, 2004. DOI: https://doi.org/10.1007/978-1-4612-2058-9
[15] E. Voronovskaja, Determination de la forme asymptotique d’approximation des functions par les polynomes de S. Bernstein. Doklady Akademii Nauk SSSR, (1932), 79-85.
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