Posts by Octavian Agratini


The paper is focused on general sequences of discrete linear operators, say \((L_{n})_{n}\geq1\). The special case of positive operators is also to our attention. Concerning the quantity \(\triangle(L_{n},f,g):=L_{n}(f_{g})-(L_{n}f)(L_{n}g)\),\(\ f\) and g belonging to some certain spaces, we propose different estimates. Firstly, we study its asymptotic behavior in Voronovskaja’s sense. Examples are presented. Secondly, we prove an extension of Chebyshev-Gruss type inequality for the above quantity. Special cases are investigated separately. Finally we establish sufficient conditions that ensure statistical convergence of the sequence \(\triangle(L_{n},f,g)\).


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


Linear operator ·Voronovskaja formula · Grüss-type inequality · Bernstein operator · Jain operator · Generalized sample operator · Statistical convergence

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O. Agratini, Properties of discrete non-multiplicative operators, Analysis and Mathematical Physics, 9 (2020), pp. 131-141.


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1. Acu, A.M., Gonska, H., Rasa, I.: Grüss- and Ostrowski-type in approximation theory. Ukr. Math. J. 63(6), 843–864 (2011)
2. Bardaro, C., Mantellini, I.: A Voronovskaja-type theorem for a general class of discrete operators. Rocky Mt. J. Math. 39(3), 1411–1442 (2009)
3. Butzer, P.L., Stens, R.L.: Sampling theory for not necessarily band-limited functions: an historical overview. SIAM Rev. 34, 40–53 (1992)
4. Chebyshev, P.L.: Sur les expressions approximatives des integrales définies par les autres prises entre les même limites. Proc. Math. Soc. Kharkov 2, 93–98 (1882)
5. Farcas, A.: An asymptotic formula for Jain’s operators. Stud. Univ. Babe¸s-Bolyai Math. 57, 511–517 (2012)
6. Gadjiev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32(1), 129–138 (2002)
7. Gavrea, I., Tachev, G.: On the multiplicity of linear operators. J. Math. Anal. Appl. 408, 203–208 (2013)
8. Gonska, H., Rasa, I., Rusu,M.D.: C˘ ebys˘ev–Grüss-type inequalities revisited.Math. Slov. 63(5), 1007–1024 (2013)
9. Gonska, H., Rasa, I., Rusu, M.D.: Chebyshev–Grüss-type inequalities via discrete oscillations. Buletinul Academiei de Stiinte a Republicii Moldova, Matematica 1(74), 63–89 (2014)
10. Grüss, G.: Über das Maximum des absoluten Betrages von 1 b−a b a f (x)g(x)dx − 1 (b−a)2 b a f (x)dx b a g(x)dx. Math. Z. 39, 215–226 (1935)
11. Jain, G.C.: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13(3), 271–276 (1972)
12. Mirakjan, G.M.: Approximation of continuous functions with the aid of polynomials. Dokl. Akad. Nauk SSSR 31, 201–205 (1941). (in Russian)
13. Ries, S., Stens, R.L.: Approximation by generalized sampling series. In: Sendov, B., Petrushev, P., Maalev, R., Tashev, S. (eds.) Constructive Theory of Functions: Proc. Conf. Varna, Bulgaria, May/June 1984, pp. 746–756. Publ. House Bulgarian Academy of Sciences, Sofia (1984)
14. Rusu, M.D.: On Grüss-type inequalities for positive linear operators. Stud. Univ. Babes-Bolyai Math. 56(2), 551–565 (2011)
15. Szász, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Natl. Bureau Stand. 45, 239–245 (1950)
16. Uchiyama,M.: Proofs of Korovkin’s theorems via inequalities. Am. Math.Mon. 110, 334–336 (2003)
17. Voronovskaja, E.: Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein, pp. 79–85. C.R. Acad. Sci. URSS (1932)

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