Year 2023,
Volume: 15 Issue: 1, 180 - 183, 30.06.2023
Tuğba Yurdakadim
,
Emre Taş
References
- Altomare, F., Campiti, M., Korovkin Type Approximation Theory and Its Applications, De Gruyter Studies in Math., vol.17, de Gruyter&Co, Berlin, 1994.
- Atlıhan, Ö.G., Orhan, C., Summation process of positive linear operators, Computers and Mathematics with Appl., 56(2008), 1188–1195.
- Atlıhan, Ö.G., Ünver, M., Duman, O., Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar., 75(2017), 201–209.
- Bardaro, C., Boccuto, A., Dimitriou, X., Mantellini, I., Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11(2013), 1774–1784.
- Connor, J.S., Two valued measures and summability, Analysis, 10(1990), 373–385.
- Connor, J.S., R-type summability methods, Cauchy criteria, P-sets and statistical convergence, Proc. Amer. Math. Soc., 115(1992), 319–327.
- Duman, O., Khan, M.K., Orhan, C., A-statistical convergence of approximating operators, Math. Inequal. Appl., 6(2003), 689–699.
- Duman, O., Orhan, C., Statistical approximation by positive linear operators, Studia Math., 161(2004), 187–197.
- Duman, O., Orhan, C., Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett., 18(2005), 1339–1344.
- Duman, O. Orhan, C., An abstract version of the Korovkin approximation theorem, Publ. Math. Debrecen, 69(2006), 33–46.
- Fast, H., Sur la convergence statistique, In Colloquium mathematicae, 2(1951), 241–244.
- Fridy, J.A., On statistical convergence, Analysis, 5(1985), 301–314.
- Fridy, J.A., Miller, H.I., Orhan, C., Statistical rates of convergence, Acta Sci. Math., 69(2003), 147–157.
- Gadziev, A.D., 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(1974), 1433–1436.
- Gadjiev, A.D., On P.P. Korovkin type theorems, Mat. Zametki, 20(1976), 781–786.
- Gadjiev, A.D., Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain Journal of Math., 32(2002), 129–137.
- Korovkin, P.P., On convergence of linear positive operators in the space of continuous functions, Doklady Akad. Nauk SSR., 90(1953), 961–964.
- Miller, H.I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(1995), 1811–1819.
- Popa, D., An operator version of the Korovkin theorem, J. Math. Anal. Appl., 515(2022), Article No: 126375.
- Salat, T., On statistically convergent sequences of real numbers, Mat. Slovaca., 30(1980), 139–150.
- Unver, M., Orhan, C., Statistical convergence with respect to power series methods and applications to approximation theory, Numerical Functional Analysis and Optimization, 40(2019), 535–547.
- Micchelli, C.A., Convergence of positive linear operators on C(X), J. Approx. Theory, 13(1975), 305–315.
On Relaxing the Identity Operator in Korovkin Theorem via Statistical Convergence
Year 2023,
Volume: 15 Issue: 1, 180 - 183, 30.06.2023
Tuğba Yurdakadim
,
Emre Taş
Abstract
An operator version of the Korovkin theorem has recently been obtained by D. Popa. With the motivation of this result, we have extended it by using a more powerful convergence which also includes ordinary convergence. We have also presented an example to illustrate the strength of our theorem.
References
- Altomare, F., Campiti, M., Korovkin Type Approximation Theory and Its Applications, De Gruyter Studies in Math., vol.17, de Gruyter&Co, Berlin, 1994.
- Atlıhan, Ö.G., Orhan, C., Summation process of positive linear operators, Computers and Mathematics with Appl., 56(2008), 1188–1195.
- Atlıhan, Ö.G., Ünver, M., Duman, O., Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar., 75(2017), 201–209.
- Bardaro, C., Boccuto, A., Dimitriou, X., Mantellini, I., Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11(2013), 1774–1784.
- Connor, J.S., Two valued measures and summability, Analysis, 10(1990), 373–385.
- Connor, J.S., R-type summability methods, Cauchy criteria, P-sets and statistical convergence, Proc. Amer. Math. Soc., 115(1992), 319–327.
- Duman, O., Khan, M.K., Orhan, C., A-statistical convergence of approximating operators, Math. Inequal. Appl., 6(2003), 689–699.
- Duman, O., Orhan, C., Statistical approximation by positive linear operators, Studia Math., 161(2004), 187–197.
- Duman, O., Orhan, C., Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett., 18(2005), 1339–1344.
- Duman, O. Orhan, C., An abstract version of the Korovkin approximation theorem, Publ. Math. Debrecen, 69(2006), 33–46.
- Fast, H., Sur la convergence statistique, In Colloquium mathematicae, 2(1951), 241–244.
- Fridy, J.A., On statistical convergence, Analysis, 5(1985), 301–314.
- Fridy, J.A., Miller, H.I., Orhan, C., Statistical rates of convergence, Acta Sci. Math., 69(2003), 147–157.
- Gadziev, A.D., 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(1974), 1433–1436.
- Gadjiev, A.D., On P.P. Korovkin type theorems, Mat. Zametki, 20(1976), 781–786.
- Gadjiev, A.D., Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain Journal of Math., 32(2002), 129–137.
- Korovkin, P.P., On convergence of linear positive operators in the space of continuous functions, Doklady Akad. Nauk SSR., 90(1953), 961–964.
- Miller, H.I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(1995), 1811–1819.
- Popa, D., An operator version of the Korovkin theorem, J. Math. Anal. Appl., 515(2022), Article No: 126375.
- Salat, T., On statistically convergent sequences of real numbers, Mat. Slovaca., 30(1980), 139–150.
- Unver, M., Orhan, C., Statistical convergence with respect to power series methods and applications to approximation theory, Numerical Functional Analysis and Optimization, 40(2019), 535–547.
- Micchelli, C.A., Convergence of positive linear operators on C(X), J. Approx. Theory, 13(1975), 305–315.