A new approach to the approximation  by positive linear operators in weighted spaces

Ö. G.  Atlihan; T.  Yurdakadim; E.  Taş

doi:10.37863/umzh.v74i11.6427

Ö. G. Atlihan Pamukkale Univ., Denizli, Turkey
T. Yurdakadim Bilecik Şeyh Edebali Univ., Turkey
E. Taş Kırşehir Ahi Evran Univ., Turkey

DOI: https://doi.org/10.37863/umzh.v74i11.6427

Keywords: Korovkin type approximation, Power series method, Statistical convergence, Weighted spaces

Abstract

UDC 517.5

In the present paper, we deal with the problem of approximating a function by positive linear operators in weighted spaces. In this case, our main tool is the $P_{p}$-statistical convergence recently defined by [M. Ünver, C. Orhan, Numer. Funct. Anal. and Optim., 40, 535–547 (2019)]. It is worth noting that the $P_{p}$-statistical convergence and the statistical convergence do not imply each other.

References

F. Altomare, M. Campiti, Korovkin type approximation theory and its application, de Gruyter Stud. Math., vol. 17, de Gruyter & Co., Berlin (1994). DOI: https://doi.org/10.1515/9783110884586

G. A. Anastassiou, O. Duman, Towards intelligent modelling: statistical approximation theory, Intell. Syst. Ref. Libr., vol. 14, Springer-Verlag, Berlin (2011).

Ö. G. Atlıhan, C. Orhan, Summation process of positive linear operators, Comput. and Math. Appl., 56, 1188–1195 (2008). DOI: https://doi.org/10.1016/j.camwa.2008.02.020

Ö. G. Atlıhan, M. Ünver, O. Duman, Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar., 75, 201–209 (2017). DOI: https://doi.org/10.1007/s10998-017-0187-y

C. Bardaro, A. Boccuto, X. Dimitriou, I. Mantellini, Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11, 1774–1784 (2013). DOI: https://doi.org/10.2478/s11533-013-0288-7

J. Boos, Classical and modern methods in summability, Oxford Univ. Press (2000).

O. Duman, M. K. Khan, C. Orhan, A-statistical convergence of approximating operators, Math. Inequal. Appl., 6, 689–699 (2003). DOI: https://doi.org/10.7153/mia-06-62

O. Duman, C. Orhan, Statistical approximation by positive linear operators, Studia Math., 161, 187–197 (2004). DOI: https://doi.org/10.4064/sm161-2-6

O. Duman, C. Orhan, Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett., 18, 1339–1344 (2005). DOI: https://doi.org/10.1016/j.aml.2005.02.029

O. Duman, C. Orhan, An abstract version of the Korovkin approximation theorem, Publ. Math. Debrecen, 69, 33–46 (2006). DOI: https://doi.org/10.5486/PMD.2006.3199

J. A. Fridy, H. I. Miller, C. Orhan, Statistical rates of convergence, Acta Sci. Math., 69, 147–157 (2003).

A. D. Gadžiev, The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.~P.~Korovkin, Soviet Math. Dokl., 15, 1433–1436 (1974).

A. D. Gadjiev, On P.~P.~Korovkin type theorems, Mat. Zametki, 20, 781–786 (1976).

A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32, 129–137 (2002). DOI: https://doi.org/10.1216/rmjm/1030539612

P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSR, 90, 961–964 (1953).

W. Kratz, U. Stadtmüller, Tauberian theorems for $J_{p}$-summability, J. Math. Anal. and Appl., 139, 362–371 (1989). DOI: https://doi.org/10.1016/0022-247X(89)90113-3

C. A. Micchelli, Convergence of positive linear operators on $C(X)$, J. Approx. Theory, 13, 305–315 (1975). DOI: https://doi.org/10.1016/0021-9045(75)90040-4

M. Ünver, C. Orhan, Statistical convergence with respect to power series methods and applications to approximation theory, Numer. Funct. Anal. and Optim., 40, 535–547 (2019). DOI: https://doi.org/10.1080/01630563.2018.1561467