ABSTRACT KOROVKIN THEOREMS VIA RELATIVE MODULAR CONVERGENCE FOR DOUBLE SEQUENCES OF LINEAR OPERATORS
Abstract
We will obtain an abstract version of the Korovkin type approximation theorems with respect to the concept of statistical relative convergence in modular spaces for double sequences of positive linear operators. We will give an application showing that our results are stronger than classical ones. We will also study an extension to non-positive operators.
Keywords
Keywords
Full Text:
PDFReferences
{sc G. A. Anastassiou {rm and} O. Duman}: textit{Towads intelligent modeling: Statistical approximation theory}. Intelligent System Reference Library textbf{14}, Springer-Verlag, Berlin, Heidelberg, New York, 2011.
bibitem{BARDARO}
{sc C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan}: textit{Triangular $A-$statistical approximation by double sequences of positive linear operators}. Results Math. {bf 68} (2015), 271--291.
bibitem{Bardaro2}
{sc C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan}: textit{Korovkin-type theorems for modular $Psi -A-$statistical convergence}. J. Funct. Spaces Article ID 160401, {bf 2015} (2015), p. 11.
bibitem{Bardaro}
{sc C. Bardaro, A. Boccuto, X. Dimitriou {rm and} I. Mantellini}: textit{Abstract Korovkin type theorems in modular spaces and applications}. Cent. Eur. J. Math. {bf 11}(10) (2013), 1774-1784.
bibitem{Bardaroandmantellini}
{sc C. Bardaro {rm and} I. Mantellini}: textit{Korovkin's theorem in modular spaces}. Commentationes Math. {bf 47} (2007), 239-253.
bibitem{Bardaro1}
{sc C. Bardaro, J. Musielak, G. Vinti}: textit{Nonlinear integral operators and applications}. de Gruyter Series in Nonlinear Analysis and Appl. Vol., textbf{9} Walter de Gruyter Publ., Berlin, 2003.
bibitem{BM2}
{sc C. Bardaro {rm and} I. Mantellini}: textit{A Korovkin Theorem in multivariate modular function spaces}. J. Funct. Spaces Appl. {bf 7} (2009), 105-120.
bibitem{Boccuto}
{sc A. Boccuto {rm and} X. Dimitriou}: textit{Korovkin-type theorems for abstract modular convergence}. Results in Mathematics {bf 69} (2016), 477--495.
bibitem{Demirci1}
{sc K. Demirci, S. Orhan}: textit{Statistically Relatively Uniform Convergence of Positive Linear Operators}. Results in Mathematics {bf 69} (2016), 359--367.
bibitem{Demirci}
{sc K. Demirci, S. Orhan}: textit{Statistical relative approximation on modular spaces}. Results in Mathematics {bf 71} (2017), 1167--1184.
bibitem{DK}
{sc K. Demirci, B. Kolay}: textit{$A-$Statistical Relative Modular Convergence of Positive Linear Operators}. Positivity {bf 21} (2017), 847--863.
bibitem{Dirik}
{sc F. Dirik, P. Okc{c}u c{S}ahin}: textit{Statistical Relatively Equal Convergence and Korovkin-Type Approximation Theorem}. Results in Mathematics {bf 72} (2017), 1613-1621.
bibitem{Kadak}
{sc U. Kadak}: textit{Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results}. Numerical Functional Analysis and Optimization {bf 39} (2018), 1181-1207.
bibitem{Karakus}
{sc S. Karakuc{s}, K. Demirci, O. Duman}: textit{Statistical approximation by positive linear operators on modular spaces}. Positivity {bf 14} (2010), 321--334.
bibitem{Korovkin}
{sc P.P. Korovkin}: textit{Linear Operators and Approximation Theory}. Hindustan Publ. Co., Delhi, 1960.
bibitem{Kozlowski}
{sc W. M. Kozlowski}: textit{Modular function spaces}. Pure Appl.
Math., Vol. textbf{122}, Marcel Dekker, Inc., New York, 1988.
bibitem{Kuratowski}
{sc K.. Kuratowski}: textit{Topology, Volls I and II}. Academic Press, New York-London, 1966/1968.
bibitem{Mantellini}
{sc I. Mantellini}: textit{Generalized sampling operators in modular spaces}. Commentationes Math. {bf 38} (1998), 77-92.
bibitem{Musielak1}
{sc J. Musielak}: textit{Orlicz spaces and modular spaces}. Lecture
Notes in Mathematics, Vol. textbf{1034} Springer-Verlag, Berlin, 1983.
bibitem{Musielak}
{sc J. Musielak}: textit{Nonlinear approximation in some modular function spaces I}. Math. Japon. {bf 38} (1993), 83-90.
bibitem{Okcu}
{sc P. Okc{c}u c{S}ahin, F. Dirik}: textit{Statistical relative uniform convergence of double sequences of positive linear operators}. Applied Mathematics E-Notes {bf 17} (2017), 207-220.
bibitem{Orhan}
{sc S. Orhan, K. Demirci}: textit{Statistical approximation by double sequences of positive linear operators on modular spaces}. Positivity {bf 19} (2015), 23--36.
bibitem{YDO}
{sc B. Yilmaz, K. Demirci, S. Orhan}: textit{Relative Modular Convergence of Positive Linear Operators}. Positivity {bf 20} (2016), 565--577.
DOI: https://doi.org/10.22190/FUMI2003561Y
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)