The minimal $f^g$-statistical convergence and Cauchy degree of a sequence

Year 2025, Volume: 54 Issue: 6, 2206 - 2224, 30.12.2025

Tamim Aziz , Sanjoy Ghosal

https://doi.org/10.15672/hujms.1589077

Abstract

In this paper, we introduce and characterize the rough $f^g$-statistical limit set, minimal $f^g$-statistical convergence degree, and minimal $f^g$-statistical Cauchy degree of a sequence in an arbitrary normed space. We clarify these concepts for normed spaces of any dimension and explore their properties and relationships. Our findings offer a new perspective that differs from some established results.

Keywords

rough $f^g$-statistical convergence , rough $f^g$-statistical Cauchy sequence , rough $f^g$-statistical limit set , minimal $f^g$-statistical convergence degree , minimal $f^g$-statistical Cauchy degree

Project Number

Human Resource Development Group, CSIR, India, through NET-JRF and the grant number is 09/0285(12636)/2021-EMR-I.

References

• [1] A. Aizpuru, M.C. Listán-García and F. Rambla-Barreno, Density by moduli and statistical convergence, Quaest. Math. 37 (4), 525–530, 2014.
• [2] M. Arslan and E. Dündar, On rough convergence in 2-normed spaces and some properties, Filomat 33 (16), 5077–5086, 2019.
• [3] A. Aydın, Statistically order compact operators on Riesz spaces, Hacet. J. Math. Stat. 53 (3), 1–9, 2024.
• [4] S. Aytar, The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optim. 29 (3-4), 283-290, 2008.
• [5] S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optim. 29 (3-4), 291–303, 2008.
• [6] S. Aytar, Rough statistical cluster points, Filomat 31 (16), 5295-5304, 2017.
• [7] T. Aziz and S. Ghosal, f-rough Cauchy sequences, Quaest. Math. 48 (1), 2025, DOI: 10.2989/16073606.2025.2464951.
• [8] M. Balcerzak, P. Das, M. Filipczak and J. Swaczyna, Generalized kinds of density and the associated ideals, Acta Math. Hungar. 147 (1), 97–115, 2015.
• [9] K. Bose, P. Das and A. Kwela, Generating new ideals using weighted density via modulus functions, Indag. Math. 29 (5), 1196–1209, 2018.
• [10] P. Das and A. Ghosh, Generating subgroups of the circle using a generalized class of density functions, Indag. Math. 32 (3), 598–618, 2021.
• [11] E. Dündar, On rough $\mathcal{I}_2$-convergence of double sequences, Numer. Funct. Anal. Optim. 37 (4), 480-491, 2016.
• [12] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (3-4), 241–244, 1951.
• [13] J. A. Fridy, On statistical convergence, Analysis 5 (4), 301–314, 1985.
• [14] J. A. Fridy, Statistical limit points, Proc. Am. Math. Soc. 118 (4), 1187–1192, 1993.
• [15] A. L. Garkavi, On the optimal net and best cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1), 87–106, 1962.
• [16] S. Ghosal and M. Banerjee, Rough weighted statistical convergence on locally solid Riesz spaces, Positivity 25 (5), 1789-1804, 2021.
• [17] S. Ghosal and S. Mandal, The degree of roughness, Topol. Appl. 307, 107944, 2022.
• [18] S. Ghosal and S. Mandal, Rough weighted $\mathcal{I}$-$\alpha\beta$-statistical convergence in locally solid Riesz spaces, J. Math. Anal. Appl. 506 (2), 125681, 2022.
• [19] H. I. Miller and L. Miller-Van Wieren, Statistical cluster point and statistical limit point sets of subsequences of a given sequence, Hacet. J. Math. Stat. 49 (2), 494–497, 2020.
• [20] L. Miller-Van Wieren, E. Taş and T. Yurdakadim, Some new insights into ideal convergence and subsequences, Hacet. J. Math. Stat. 51 (5), 1379–1384, 2022.
• [21] P. Kostyrko, T. Šalát and W. Wilczyński, $\mathcal{I}$-convergence, Real Anal. Exchange 26, 669–685, 2000.
• [22] M. C. Listán-García, A characterization of uniform rotundity in every direction in terms of rough convergence, Numer. Funct. Anal. Optim. 22 (11), 1166– 1174, 2011.
• [23] M. C. Listán-García, f-statistical convergence, completeness and f-cluster points, Bull. Belg. Math. Soc. Simon Stevin 23 (2), 235–245, 2016.
• [24] M. C. Listán-García and F. Rambla-Barreno, Rough convergence and Chebyshev centers in Banach spaces, Numer. Funct. Anal. Optim. 35 (4), 432-442, 2014.
• [25] S. K. Pal, D. Chandra and S. Dutta, Rough ideal convergence, Hacet. J. Math. Stat. 42 (6), 633–640, 2013.
• [26] S. Pehlivan, A. Güncan and M. Mamedov, Statistical cluster points of sequences in finite dimensional spaces, Czechosl. Math. J. 54 (1), 95–102, 2004.
• [27] H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. Optim. 22 (1-2), 199-222, 2001 .
• [28] H. X. Phu, Rough convergence in infinite dimensional normed space, Numer. Funct. Anal. Optim. 24 (3-4), 285-301, 2003.
• [29] S. K. A. Rahaman and M. Mursaleen, On rough deferred statistical convergence of difference sequences in L-fuzzy normed spaces, J. Math. Anal. Appl. 530 (2), 127684, 2024.
• [30] M. H. M. Rashida, Rough statistical convergence and rough ideal convergence in random 2-normed spaces, Filomat 38 (3), 979–996, 2024.
• [31] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (2), 139-150, 1980.
• [32] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1), 73-74, 1951.
• [33] B. C. Tripathy, On statistically convergent and statistically bounded sequences, Bull. Malays. Math. Soc. (Second Series) 20 (1), 31-33, 1997.