Posts by Octavian Agratini

Abstract


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 \(\triangle(L_{n},f,g)\).

Authors

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

Keywords

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

Paper coordinates

O. Agratini, Properties of discrete non-multiplicative operators, Analysis and Mathematical Physics, 9 (2020), pp. 131-141. https://doi.org/10.1007/s13324-017-0186-4

PDF

About this paper

Print ISSN

?

Online ISSN

google scholar link

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)

Related Posts

Inequalities and approximation theory

Abstract The purpose of this paper is twofold. Firstly, we present an equivalence property involving isotonic linear functionals. Secondly, by…