Deferred statistical order convergence in Riesz spaces

Year 2024, , 1368 - 1377, 15.10.2024

Mehmet Küçükaslan , Abdullah Aydın

https://doi.org/10.15672/hujms.1322652

Abstract

In recent years, researchers have focused on exploring different forms of statistical convergence in Riesz spaces, such as statistical order convergence and statistical unbounded order convergence. This study aims to present the concept of deferred statistical convergence within Riesz spaces, specifically concerning its relationship with order convergence. Furthermore, we delve into the interconnections between deferred statistical order convergence and various other types of statistical convergence. Moreover, we explore in depth the intricate connections between deferred statistical order convergence and other notable forms of statistical convergence. We provide valuable insights into the broader framework of statistical convergence theory in Riesz spaces.

Keywords

Riesz space, deferred statistical convergence, order convergence, deferred statistical order convergence

References

• [1] R.P. Agnew, On deferred Cesàro means, Anna. Math. 33 (3), 413-421, 1932.
• [2] C.D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs Centrum, 2003.
• [3] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, 2006.
• [4] A. Aydın, Multiplicative order convergence in f-algebras, Hacet. J. Math. Stat. 49 (3), 998-1005, 2020.
• [5] A. Aydın, The statistically unbounded $\tau$-convergence on locally solid vector lattices, Turk. J. Math. 44 (3), 949-956, 2020.
• [6] A. Aydın, The statistical multiplicative order convergence in vector lattice algebras, Fact. Univ. Ser.: Math. Infor. 36 (2), 409-417, 2021.
• [7] A. Aydın, M. Et, Statistically multiplicative convergence on locally solid Riesz algebras, Turk. J. Math. 45 (4), 1506-1516, 2021.
• [8] A. Aydın, E. Emelyanov and S. G. Gorokhova, Full lattice convergence on Riesz spaces, Indagat. Math. 32 (3), 658-690, 2021.
• [9] Z. Ercan, A characterization of u-uniformly completeness of Riesz spaces in terms of statistical u-uniformly pre-completeness, Demons. Math. 42 (2), 383-387, 2009.
• [10] M. Et, P. Baliarsingh, H. . Kandemir and M. Küçükaslan, On $\mu$-deferred statistical convergence and strongly deferred summable functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115(1), Paper No. 34, 14 pp, 2021.
• [11] M. Et, M. Cinar and H. S. Kandemir, Deferred statistical convergence of order α in metric spaces, AIMS Math. 5, Paper No. 4, 3731-3740, 2020.
• [12] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
• [13] J. Fridy, On statistical convergence, Analysis 5 (4), 301-313, 1985.
• [14] M. Küçükaslan and M. Yilmazturk, On deferred statistical convergence of sequences, Kyung. Math. J. 56 (2), 357-366, 2016.
• [15] M. Küçükaslan, U. Deer and U. Dovgoshey, On statistical convergence of metric valued sequences, Ukrain. Math. J. 66 (5), 796-805, 2014.
• [16] W.A.J. Luxemburg and A.C. Zaanen, Vector Lattices I, North-Holland Pub. Co., Amsterdam, 1971.
• [17] I.J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambr. Phil. Soc. 104 (1), 141-145, 1988.
• [18] F. Riesz, Sur la Décomposition des Opérations Fonctionelles Linéaires. Bologna, Atti Del Congresso Internazionale Dei Mathematics Press, 1928.
• [19] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2, 73-74, 1951.
• [20] B.C. Tripathy, On statistically convergent sequences, Bul. Calcut. Math. Soc. 90, 259262, 1998.
• [21] A.C. Zaanen, Riesz Spaces II, North-Holland Publishing C., Amsterdam, 1983.