[1] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
[2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2(1) (1951), 73-74.
[3] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.
[4] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
[5] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.
[6] J. A. Fridy, H. I. Miller, A matrix characterization of statistical convergence, Analysis, 11 (1991), 59-66.
[7] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43-51.
[8] F. Moricz, C. Orhan, Tauberian conditions under which statistical convergence follows from statistical summability by weighted means, Studia Sci. Math. Hungar., 41 (2004), 391-403.
[9] J. S. Connor, The statistical and strong p-Ces`aro convergence of sequences, Analysis, 8 (1988), 47-63.
[10] P. Kostyrko, T. Salat, W. Wilczynki, I-convergence, Real Anal. Exchange, 26 (2) (2000-2001), 669-685.
[11] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24 (2011), 826-830.
[12] P. Das, E. Savas, S. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett., 24 (2011), 1509-1514.
[14] H. Aktuglu, Korovkin type approximation theorems proved via abstatistical convergence, J. Comput. Appl. Math., 259 (2014), 174-181.
[15] A. Karaisa, Statistical absummability and Korovkin type approximation theorem, Filomat, 30(13) (2016), 3483-3491.
[16] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289-321.
[17] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
[18] M. Mursaleen, C. Cakan, S. A. Mohiuddine, E. Savas, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2131-2144.
[19] E. Savas, R. F. Patterson, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett., 19 (2006), 527-534.
[20] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960.
[21] V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk. SSSR (N.S.), 115 (1957), 17-19.
[22] M. Mursaleen, V. Karakaya, M. Erturk, M. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132-9137.
[23] M. Mursaleen, A. Alotaibi, Statistical summability and approximation by de la Vall´ee-Poussin mean, Appl. Math. Lett., 24 (2011), 320-324.
[24] A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.
[25] E. Erkus, O. Duman, A Korovkin type approximation theorem in statistical sense, Studia Sci. Math. Hungar., 43 (2006), 285-294.
[26] O. Duman, A Korovkin type approximation theorems via I-convergence, Czechoslovak Math. J., 57 (2007), 367-375.
Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence
Year 2019,
Volume: 2 Issue: 3, 198 - 204, 26.12.2019
In this paper, we introduce the concepts of αβ−statistical convergence and strong αβ− summability of double sequences and investigate the relation between these two new concepts. Moreover, statistical convergence and αβ− statistical convergence of double sequences are compared under some certain assumptions. Finally, as an application, we prove Korovkin type approximation theorem for a function of two variables by using the notion of αβ−statistical convergence.
[1] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
[2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2(1) (1951), 73-74.
[3] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.
[4] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
[5] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.
[6] J. A. Fridy, H. I. Miller, A matrix characterization of statistical convergence, Analysis, 11 (1991), 59-66.
[7] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43-51.
[8] F. Moricz, C. Orhan, Tauberian conditions under which statistical convergence follows from statistical summability by weighted means, Studia Sci. Math. Hungar., 41 (2004), 391-403.
[9] J. S. Connor, The statistical and strong p-Ces`aro convergence of sequences, Analysis, 8 (1988), 47-63.
[10] P. Kostyrko, T. Salat, W. Wilczynki, I-convergence, Real Anal. Exchange, 26 (2) (2000-2001), 669-685.
[11] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24 (2011), 826-830.
[12] P. Das, E. Savas, S. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett., 24 (2011), 1509-1514.
[14] H. Aktuglu, Korovkin type approximation theorems proved via abstatistical convergence, J. Comput. Appl. Math., 259 (2014), 174-181.
[15] A. Karaisa, Statistical absummability and Korovkin type approximation theorem, Filomat, 30(13) (2016), 3483-3491.
[16] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289-321.
[17] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
[18] M. Mursaleen, C. Cakan, S. A. Mohiuddine, E. Savas, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2131-2144.
[19] E. Savas, R. F. Patterson, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett., 19 (2006), 527-534.
[20] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960.
[21] V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk. SSSR (N.S.), 115 (1957), 17-19.
[22] M. Mursaleen, V. Karakaya, M. Erturk, M. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132-9137.
[23] M. Mursaleen, A. Alotaibi, Statistical summability and approximation by de la Vall´ee-Poussin mean, Appl. Math. Lett., 24 (2011), 320-324.
[24] A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129-138.
[25] E. Erkus, O. Duman, A Korovkin type approximation theorem in statistical sense, Studia Sci. Math. Hungar., 43 (2006), 285-294.
[26] O. Duman, A Korovkin type approximation theorems via I-convergence, Czechoslovak Math. J., 57 (2007), 367-375.
Altundağ, S., & Sözbir, B. (2019). Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence. Journal of Mathematical Sciences and Modelling, 2(3), 198-204. https://doi.org/10.33187/jmsm.652626
AMA
Altundağ S, Sözbir B. Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence. Journal of Mathematical Sciences and Modelling. December 2019;2(3):198-204. doi:10.33187/jmsm.652626
Chicago
Altundağ, Selma, and Bayram Sözbir. “Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence”. Journal of Mathematical Sciences and Modelling 2, no. 3 (December 2019): 198-204. https://doi.org/10.33187/jmsm.652626.
EndNote
Altundağ S, Sözbir B (December 1, 2019) Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence. Journal of Mathematical Sciences and Modelling 2 3 198–204.
IEEE
S. Altundağ and B. Sözbir, “Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence”, Journal of Mathematical Sciences and Modelling, vol. 2, no. 3, pp. 198–204, 2019, doi: 10.33187/jmsm.652626.
ISNAD
Altundağ, Selma - Sözbir, Bayram. “Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence”. Journal of Mathematical Sciences and Modelling 2/3 (December 2019), 198-204. https://doi.org/10.33187/jmsm.652626.
JAMA
Altundağ S, Sözbir B. Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence. Journal of Mathematical Sciences and Modelling. 2019;2:198–204.
MLA
Altundağ, Selma and Bayram Sözbir. “Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence”. Journal of Mathematical Sciences and Modelling, vol. 2, no. 3, 2019, pp. 198-04, doi:10.33187/jmsm.652626.
Vancouver
Altundağ S, Sözbir B. Korovkin Type Approximation Theorem for Functions of Two Variables Through αβ−Statistical Convergence. Journal of Mathematical Sciences and Modelling. 2019;2(3):198-204.