Some new insights into ideal convergence and subsequences

Year 2022, Volume: 51 Issue: 5, 1379 - 1384, 01.10.2022

Leila Miller-van Wieren , Emre Taş , Tuğba Yurdakadim

https://doi.org/10.15672/hujms.1087633

Cited By: 1

Abstract

Some results on the sets of almost convergent, statistically convergent, uniformly statistically convergent, $I$-convergent subsequences of $(s_{n})$ have been obtained by many authors via establishing a one-to-one correspondence between the interval $(0,1]$ and the collection of all subsequences of a given sequence $s=(s_{n})$. However, there are still some gaps in the existing literature. In this paper we plan to fill some of the gaps with new results. Some of them are easily derived from earlier results but they offer some new deeper insights.

Keywords

ideal convergence, subsequences, I-cluster and I-limit points

References

• [1] V. Balaz and T. Šalát, Uniform density $u$ and corresponding $I_{u}$- convergence, Math. Commun. 11, 1-7, 2006.
• [2] M. Balcerzak, S. Glab and A. Wachowicz, Qualitative properties of ideal convergent subsequences and rearrangements Acta Math. Hungar. 150, 312-323, 2016.
• [3] M. Balcerzak, S. Glab and P. Leonetti, Another characterization of meager ideals, submitted for publication, 2021.
• [4] M. Balcerzak and P. Leonetti, On the relationship between ideal cluster points and ideal limit points, Topology and Appl. 252, 178-190, 2019.
• [5] M. Balcerzak and P. Leonetti, The Baire category of subsequences and permutations which preserve limit points, Results Math. 121, 2020.
• [6] P. Billingsley, Probability and measure, Wiley, New york, 1979.
• [7] R.C. Buck and H. Pollard, Convergence and summability properties of subsequences, Bull. Amer. Math. Soc. 49, 924-931, 1943.
• [8] K. Demirci, I-limit superior and inferior, Math. Commun. 6, 165-172, 2001.
• [9] I. Farah, Analytic quotients. Theory of lifting for quotients over analytic ideals on integers, Mem. Amer. Math. Soc. 148, xvi+177 pp, 2000.
• [10] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241-244, 1951.
• [11] J.A. Fridy, Statistical limit points, Proc. Amer. Math. Soc. 118, 1187-1192, 1993.
• [12] X. He, H. Zhang, S. Zhang The Borel complexity of ideal limit points, Topology and Appl. 312, 2022.
• [13] T.A. Keagy, Summability of certain category two classes, Houston J. Math. 3, 61-65, 1977.
• [14] P. Kostyrko, T. Šalát and W. Wilczyński, $I$-convergence, Real Anal. Exchange, 26, 669-686, 2000/2001.
• [15] P. Leonetti, H.I. Miller and L. Miller-Van Wieren, Duality between measure and category of almost all subsequences of a given sequences, Period. Math. Hungar. 78, 152-156, 2019.
• [16] P. Leonetti, Thinnable ideals and invariance of cluster points, Rocky Mount. J. Math. 48(6), 2018.
• [17] P. Leonetti, Invariance of ideal limit points, Topology and Appl. 252, 169-177, 2019.
• [18] L. Miller-Van Wieren, E. Taş and T. Yurdakadim, Category theoretical view of Icluster and I-limit points of subsequences, Acta Comment. Univ. Tartu. Math. 24, 103-108, 2020.
• [19] H.I. Miller and C. Orhan, On almost convergence and statistically convergent subsequences, Acta. Math. Hungar. 93, 135-151, 2001.
• [20] H.I. Miller and L. Miller-Van Wieren, Some statistical cluster point theorems, Hacet. J. Math. Stat. 44, 1405-1409, 2015.
• [21] 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. 49, 494-497, 2020.
• [22] T. Yurdakadim and L. Miller-Van Wieren, Subsequential results on uniform statistical convergence, Sarajevo J. Math. 12, 1-9, 2016.
• [23] T. Yurdakadim and L. Miller-Van Wieren, Some results on uniform statistical cluster points, Turk. J. Math. 41, 1133-1139, 2017.
• [24] J. Zeager, Buck-type theorems for statistical convergence, Radovi Math. 9, 59-69, 1999.