Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we first introduce the notions of $\mathcal{F-}$relative modular convergence and $\mathcal{F-}$relative strong convergence for double sequences of functions.
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Key Findings & Data:Also, we present a non-trivial application such that our Korovkin-type approximation results in modular spaces are stronger than the classical ones and we present some estimates of rates of convergence for abstract Korovkin-type theorems.
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Conclusion & Impact:Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.
In this paper, we first introduce the notions of $\mathcal{F-}$relative modular convergence and $\mathcal{F-}$relative strong convergence for double sequences of functions. Then we prove some Korovkin-type approximation theorems via $\mathcal{F-}$relative $\mathcal{A}-$summation process on modular spaces for double sequences of positive linear operators. Also, we present a non-trivial application such that our Korovkin-type approximation results in modular spaces are stronger than the classical ones and we present some estimates of rates of convergence for abstract Korovkin-type theorems. Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.
Yıldız, S. (2022). $\mathcal{F-}$relative $\mathcal{A-}$summation process for double sequences and abstract Korovkin type theorems. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 4, 1047–1062. https://ajindex.com/articles/529973
G.A. Anastassiou and O. Duman, Towads intelligent modeling: Statistical approximation theory, Intelligent System Reference Library 14, Springer-Verlag, Berlin, Heidelberg, New York, 2011
Ö.G. Atlıhan and C. Orhan, Matrix summability and positive linear operators, Positivity 11, 387–389, 2007
Ö.G. Atlıhan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56, 1188–1195, 2008
C. Bardaro and I. Mantellini, Korovkin’s theorem in modular spaces, Comment. Math. 47, 239–253, 2007
C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, Walter de Gruyter, Berlin, Germany, 2003
C. Bardaro, A. Boccuto, X. Dimitriou and I. Mantellini, Abstract Korovkin type theorems in modular spaces and applications, Cent. Eur. J. Math. 11 (10), 1774–1784, 2013
C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan, Korovkin-Type Theorems for Modular $\Psi -A-$Statistical Convergence, J. Funct. Spaces, 2015, 1–11, 2015
A. Boccuto and X. Dimitriou, Korovkin-type theorems for abstract modular convergence, Results Math. 69 (3-4), 477–495, 2016
K. Demirci, $I-$limit superior and limit inferior, Math. Commun. 6, 165-172, 2001
K. Demirci, S. Orhan and B. Kolay, Statistical Relative $A-$Summation Process for Double Sequences on Modular Spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 112, 1249–1264, 2018
F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turk. J. Math. 34, 73–83, 2010
J. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (12), 3625–3631, 1997
A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129–138, 2002
U. Kadak, Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (3), 2201–2220, 2019
U. Kadak and S.A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the $(p,q)$-gamma function and related approximation theorems, Results Math. 73 (1), 1–31, 2018
S. Karakuş, K. Demirci and O. Duman, Statistical approximation by positive linear operators on modular spaces, Positivity 14, 321–334, 2010
B. Kolay, S. Orhan, and K. Demirci, Statistical Relative A−Summati