A Survey on Estimates for the Differences of Positive Linear Operators

Year 2018, Volume: 1 Issue: 2, 113 - 127, 07.11.2018

Ana Maria Acu , Sever Hodış Ioan Raşa

https://doi.org/10.33205/cma.478408

Cited By: 12

Abstract

We survey some results concerning differences of positive linear operators from Approximation Theory, and present some new results in this direction.

Keywords

Positive linear operators, Inverses of operators, Moduli of smoothness, Estimations of differences of operators

References

• [1] U. Abel, M. Ivan, Asymptotic expansion of the multivariate Bernstein polynomials on a simplex, Approx. Theory Appl. 16 (2000), 85-93.
• [2] A. M. Acu, I. Raşa, New estimates for the differences of positive linear operators, Numerical Algorithms 73(3) (2016), 775-789.
• [3] A. M. Acu, I. Raşa, Estimates for the differences of positive linear operators and their derivatives, arXiv:1810.08839v1, submitted.
• [4] A. Aral, D. Inoan and I. Raşa, On differences of linear positive operators, Anal. Math. Phys.(2018). DOI https://doi.org/10.1007/s1332
• [5] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Communications de la Société Mathematique de Kharkov, 13 (1913), 1-2.
• [6] W. Chen, On the modified Bernstein-Durrmeyer operator, Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China, 1987.
• [7] H. Gonska, I. Ra¸sa, E.-D. Stanila, Beta operators with Jacobi weights, In: Constructive theory of functions, Sozopol 2013 (K. Ivanov, G. Nikolov and R. Uluchev, Eds.), 99–112. Prof. Marin Drinov Academic Publishing House, Sofia, 2014.
• [8] H. Gonska, I. Raşa, Differences of positive linear operators and the second order modulus, Carpathian J. Math. 24(3) (2008), 332-340.
• [9] H. Gonska, P. Pitul, I. Ra¸sa, On differences of positive linear operators, Carpathian J. Math. 22(1-2) (2006), 65-78.
• [10] H. Gonska, P. Pitul, I. Ra¸sa, On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators, in Numerical Analysis and Approximation Theory (Proc. Int. Conf. Cluj-Napoca 2006; ed. by O. Agratini and P. Blaga), Cluj-Napoca, Casa C˘ar¸tii de ¸Stiin¸ta, 2006, 55-80.
• [11] H. Gonska, R. Paltanea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czechoslovak Math. J. 60(135) (2010), 783-799.
• [12] H. Gonska, R. Paltanea, Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions, Ukrainian Math. J. 62 (2010), 913-922.
• [13] T. N. T. Goodman, A. Sharma, A modified Bernstein-Schoenberg operator, Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed. by Bl. Sendov et al.). Sofia: Publ. House Bulg. Acad. of Sci., 1988, 166-173.
• [14] M. Heilmann, F. Nasaireh, I. Ra¸sa, Complements to Voronovskaja’s formula, Chapter 11 In: D. Ghosh et al. (eds.), Mathematics and Computing, Springer Proceedings in Mathematics & Statistics 253, Springer Nature Singapore Pte Ltd. 2018, https://doi.org/10.1007/978-981-13-2095-8 11.
• [15] M. Heilmann, F. Nasaireh, I. Ra¸sa, Discrete Operators associated with Linking Operators, arXiv:1808.07239v1.
• [16] N. Ispir, On modified Baskakov operators on weighted spaces Turk. J. Math. 26(3) (2001), 355-365.
• [17] L.V. Kantorovich, Sur certain développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. URSS 563-568 (1930), 595-600.
• [18] A. Lupaş, The approximation by means of some linear positive operators, in Approximation Theory (M.W. Müller et al., eds), Akademie-Verlag, Berlin, 1995, 201-227.
• [19] A. Lupaş, Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart, 1972.
• [20] L. Lupaş, A. Lupa¸s, Polynomials of Binomial Type and Approximation Operators, Studia Univ. Babe¸s-Bolyai, Mathematica, XXXII, 4 (1987), 61-70.
• [21] D.H. Mache, Gewichtete Simultanapproximation in der Lp-Metrik durch das Verfahren der Kantorovic Operatoren, Dissertation, Univ. Dortmund, 1991.
• [22] D.H. Mache, A link between Bernstein polynomials and Durrmeyer polynomials with Jacobi weights, In: Approx. Theory VIII, Vol. 1: Approximation and Interpolation, Ch.K. Chui and L.L. Schmaker (Eds.), 403-410, World Scientific Publ., 1995.
• [23] S.M. Mazhar, V. Totik, Approximation by modified Szász operators, Acta Sci. Math. 49 (1985), 257-269.
• [24] F. Nasaireh, I. Ra¸sa, Another look at Voronovskaya type formulas, Journal of Mathematical Inequalities, 12(1) (2018), 95-105.
• [25] F. Nasaireh, Voronovskaja-type formulas and applications, General Mathematics, 25(1-2) (2017), 37-43.
• [26] R. Paltanea, A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca) 5 (2007), 109-117.
• [27] I. Raşa, Discrete operators associated with certain integral operators, Stud. Univ. Babe¸s- Bolyai Math., 56(2) (2011), 537-544.
• [28] I. Raşa, E. Stanila, On some operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, J. Appl. Funct. Anal. 9 (2014), 369-378.
• [29] D. D. Stancu, Asupra unei generalizari a polinoamelor lui Bernstein, Stud. Univ. Babes-Bolyai Math. 14 (1969), 31-45.
• [30] E. D. Stanila, On Bernstein-Euler-Jacobi Operators, Ph D Thesis, Duisburg-Essen University, July, 2014.
• [31] O. Szász, Generalization of S. Bernstein’s polynomials to the infinite interval J. Res. Natl. Bur. Stand. 45 (1950), 239-245.