Lacunary Statistically Convergence via Modulus Function Sequences

Yıl 2024, Cilt: 12 Sayı: 4, 187 - 195

Erdal Bayram , Çiğdem Bektaş

https://doi.org/10.36753/mathenot.1477450

Öz

Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order $\alpha \in \left( 0,1\right] $ to introduce a new density. Based on this density framework, we define strong $(f_k)$-lacunary summability of order $\alpha $ and $(f_k)$-lacunary statistical convergence of order $\alpha $ for a sequence of modulus functions $(f_k)$. This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.

Anahtar Kelimeler

Lacunary statistical convergence, Lacunary summability, Modulus function, Weighted density

Kaynakça

• [1] Zygmund, A.: Trigonometric Series, Cambridge University Press, Cambridge, (1979).
• [2] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum. 2, 73-74 (1951).
• [3] Fast, H.: Sur la convergence statistique. Colloquium Mathematicum. 2, 241-244 (1951).
• [4] Schoenberg, I.J.: The integrability of certain functions and related summability methods. The American Mathematical Monthly. 66, 361-375 (1959).
• [5] Salat, T.: On statistically convergent sequences of real numbers. Mathematica Slovaca. 30, 139-150 (1980).
• [6] Fridy, J.: On statistical convergence. Analysis. 5, 301-313 (1985).
• [7] Çolak, R.: Statistical convergence of order α. In:Modern Methods in Analysis and Its Applications. Anamaya Publishers, New Delhi (2010).
• [8] Çolak, R., Bektaş, Ç.A.: λ− Statistical convergence of order α. Acta Mathematica Scientia. 31(3), 953-959 (2011).
• [9] Fridy, J., Orhan, C.: Lacunary statistical convergence. Pacific Journal of Mathematics. 160(1), 43-51 (1993).
• [10] Fridy, J., Orhan, C.: Lacunary statistical summability. Journal of Mathematical Analysis and Applications. 173(2), 497-504 (1993).
• [11] Connor, J.: On strong matrix summability with respect to a modulus and statistical convergence. Canadian Mathematical Bulletin. 32(2), 194-198 (1989).
• [12] Çolak, R.: Lacunary strong convergence of difference sequences with respect to a modulus function. Filomat. 17, 9-14 (2003).
• [13] Şengül, H., Et, M.: On lacunary statistical convergence of order α. Acta Mathematica Scientia. 34(2), 473-482 (2014).
• [14] Et, M., Şengül, H.: Some Cesàro-Type Summability of order and Lacunary Statistical Convergence of order α. Filomat. 28, 1593-1602 (2014).
• [15] Pehlivan, S., Fisher, B., Lacunary strong convergence with respect to a sequence of modulus functions. Commentationes Mathematicae Universitatis Carolinae. 36(1), 69-76 (1995).
• [16] Nakano, H.: Concave modulars. Journal of the Mathematical Society of Japan. 5, 29-49 (1953).
• [17] Ruckle, W.H.: FK-Spaces in which the sequence of coordinate vectors is bounded. Canadian Journal of Mathematics. 25, 973-978 (1973).
• [18] Bhardwaj, V.K., Singh, N.: On some sequence spaces defined by a modulus. Indian Journal of Pure and Applied Mathematics. 30, 809-817 (1999).
• [19] Ghosh, D., Srivastava, P.D.: On some vector valued sequence spaces defined using a modulus function. Indian Journal of Pure and Applied Mathematics. 30, 819-826 (1999).
• [20] Maddox, I.J.: Sequence spaces defined by a modulus. Mathematical Proceedings of the Cambridge Philosophical Society. 100, 161-166 (1986).
• [21] Pehlivan, S., Fisher, B.: Some sequence spaces defined by a modulus function. Mathematica Slovaca. 45(3), 275-280 (1995).
• [22] Aizpuru, A., Listán-Garcia, M.C., Rambla-Barreno, F.: Density by moduli and statistical convergence. Quaestiones Mathematicae. 37, 525-530 (2014).
• [23] Bhardwaj, V.K., Dhawan,S.: f-Statistical convergence of order α and strong Cesàro summability of order with respect to a modulus. Journal of Inequalities and Applications. 332 (2015).
• [24] León-Saavedra, F., Listán-García, M.d.C., Pérez Fernández, F.J., de la Rosa, M.P.R.: On statistical convergence and strong Cesàro convergence by moduli. Journal of Inequalities and Applications. 298 (2019).
• [25] İbrahim, İ.S., Çolak, R.: On strong Lacunary summability of order with respect to modulus functions. Math. Comp. Sci. Series. 48(1), 127-136 (2021).
• [26] Freedman, A.R., Sember, J.J., Raphael, M.: Some Cesàro-Type summability spaces. Proceedings of the London Mathematical Society. 37(3), 508-520 (1978).
• [27] Şengül, H., Et, M.: f-Lacunary statistical convergence and strong f-Lacunary summability of order α. Filomat. 32(13), 4513-4521 (2018).