WEIGHTED STATISTICAL CONVERGENCE OF REAL VALUED SEQUENCES
Abstract
Functions defined in the form ``$g:\mathbb{N}\to[0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\lim_{n\to\infty}\frac{n}{g(n)}=0$'' are called weight functions. Using the weight function, the concept of weighted density, which is a generalization of natural density, was defined by Balcerzak, Das, Filipczak and Swaczyna in the paper ``Generalized kinsd of density and the associated ideals'', Acta Mathematica Hungarica 147(1) (2015), 97-115.
In this study, the definitions of $g$-statistical convergence and $g$-statistical
Cauchy sequence for any weight function $g$ are given and it is proved that these two concepts are equivalent. Also some inclusions of the sets of all weight $g_1$-statistical convergent and weight $g_2$-statistical convergent sequences for $g_1,g_2$ which have the initial conditions are given.
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M. Balcerzak, P. Das, M. Filipczak and J. Swaczyna: Generalized kinds of density and the associated ideals. Acta Math. Hungar. 147(1) (2015), 97--115.
S. Bhunia, P. Das and S. K. Pal: Restricting statistical convergence. Acta Mathematica Hungarica, 134(1-2) (2012), 153--161.
R. Çolak: Statistical convergence of order . Modern Methods in Analysis and Its Applications, Anamaya Pub., New Delhi, India (2010), 121--129.
P. Das and E. Savaş: On generalized statistical and ideal convergence of metric-valued sequences. Reprinted in Ukrainian Math. J. 68(12) (2017), 1849--1859. Ukrain. Mat. Zh. 68(12) (2016), 1598--1606.
H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241--244.
J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301--313.
Ş. Konca, M. Küçükaslan and E. Genç: I-statistical convergence of double sequences dened by weight functions in a locally solid Riesz space. Konuralp J. Math. 7(1) (2019), 55--61.
M. Küçükaslan and M. Ylımaztürk: On deferred statistical convergence of sequences. Kyungpook Math. J., 56 (2006), 357--366.
M. Yılmaztürk, O. Mızrak and M. Küçükaslan: Deferred statistical cluster points of real valued sequences. Univ. J. Appl. Math., 1 (2013), 1--6.
E. Savaş: On some generalizd sequence spaces dened by modulus. Indian J. Pure Appl. Math., 30(5) (1999), 973--978.
E. Savaş: Strong almost convergence and almost-statistical convergence. Hokkaido Math., 29(3) (2000), 531--536.
E. Savaş and P. Das: On I-statistical and I-lacunary statistical convergence of weight g. Bull. Math. Anal. Appl., 11(2) (2019), 2--11.
E. Savaş: On I-lacunary statistical convergence of weight g of sequences of sets. Filomat 31(16) (2017), 5315--5322.
E. Savaş: I-statistical convergence of weight g in topological groups. Mathematics and computing, Springer Proc. Math. Stat., 253, Springer, Singapore, 2018, 43--51.
I. J. Schoenberg: The integrability of certain functions and related summability methods. The American Mathematical Monthly 66(5) (1959), 361--775.
DOI: https://doi.org/10.22190/FUMI2003887A
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