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http://dx.doi.org/10.14317/jami.2022.327

ON ${\mathcal{I}}$-LACUNARY ARITHMETIC STATISTICAL CONVERGENCE  

KISI, OMER (Department of Mathematics, Faculty of Science, Bartin University)
Publication Information
Journal of applied mathematics & informatics / v.40, no.1_2, 2022 , pp. 327-339 More about this Journal
Abstract
In this paper, we introduce arithmetic ${\mathcal{I}}$-statistically convergent sequence space $A{\mathcal{I}}SC$, ${\mathcal{I}}$-lacunary arithmetic statistically convergent sequence space $A{\mathcal{I}}SC_{\theta}$, strongly ${\mathcal{I}}$-lacunary arithmetic convergent sequence space $AN_{\theta}[{\mathcal{I}}]$ and prove some inclusion relations between these spaces. Futhermore, we give ${\mathcal{I}}$-lacunary arithmetic statistical continuity. Finally, we define ${\mathcal{I}}$-Cesàro arithmetic summability, strongly ${\mathcal{I}}$-Cesàro arithmetic summability. Also, we investigate the relationship between the concepts of strongly ${\mathcal{I}}$-Cesàro arithmetic summability, strongly ${\mathcal{I}}$-lacunary arithmetic summability and arithmetic ${\mathcal{I}}$ -statistically convergence.
Keywords
Lacunary sequence; statistical convergence; ideal convergence; filter; arithmetic convergence;
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1 B. Hazarika, A. Alotaibi and S.A. Mohiuddine, Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Computing 24 (2020), 6613-6622.   DOI
2 T. Yaying and B. Hazarika, Lacunary Arithmetic Statistical Convergence, Natl. Acad. Sci. Lett. 43 (2020), 547-551.   DOI
3 T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
4 S.A. Mohiuddine and B.A.S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. 113 (2019), 1955-1973.   DOI
5 S.A. Mohiuddine, B. Hazarika and A. Alotaibi, On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Syst. 32 (2017), 4331-4342.   DOI
6 J. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 46-63.   DOI
7 J.A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), 43-51.   DOI
8 M. Gurdal and A. Sahiner, Extremal $\mathcal{I}$-limit points of double sequences, Appl. Math. E-Notes 8 (2008), 131-137.
9 M. Mursaleen and A. Alotaibi, On $\mathcal{I}$-convergence in random 2-normed spaces, Math. Slovaca 61 (2011), 933-940.   DOI
10 M. Mursaleen, S. Debnath and D. Rakshit, $\mathcal{I}$-statistical limit superior and $\mathcal{I}$-statistical limit inferior, Filomat 31 (2017), 2103-2108.   DOI
11 T. Yaying and B. Hazarika, On arithmetic continuity in metric spaces, Afr. Math. 28 (2017), 985-989.   DOI
12 H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.   DOI
13 J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313.   DOI
14 C. Belen and S.A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), 9821-9826.   DOI
15 U. Kadak and S.A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p,q)-Gamma function and related approximation theorems, Results Math. 73 (2018), 1-31.   DOI
16 S.A. Mohiuddine, A. Asiri and B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst. 48 (2019), 492-506.   DOI
17 S.A. Mohiuddine, B. Hazarika and M.A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat 33 (2019), 4549-4560.   DOI
18 A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. Lond. Math. Soc. 37 (1978), 508-520.
19 R. Colak, Lacunary strong convergence of difference sequences with respect to a modulus function, Filomat 17 (2003), 9-14.   DOI
20 J.A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl. 173 (1993), 497-504.   DOI
21 P. Kostyrko, M. Macaj, T. Salat and M. Sleziak, $\mathcal{I}$-convergence and extremal $\mathcal{I}$-limit points, Math. Slovaca 55 (2005), 443-464.
22 F. Nuray and W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513-527.   DOI
23 W.H. Ruckle, Arithmetical summability, J. Math. Anal. Appl. 396 (2012), 741-748.   DOI
24 T. Yaying and B. Hazarika, On arithmetical summability and multiplier sequences, Natl. Acad. Sci. Lett. 40 (2017), 43-46.   DOI
25 T. Yaying and B. Hazarika, On arithmetic continuity, Bol Soc Parana Mater. 35 (2017), 139-145.   DOI
26 T. Yaying, B. Hazarika and H. Cakalli, New results in quasi cone metric spaces, J Math Comput Sci. 16 (2016), 435-444.   DOI
27 E. Savas and M. Gurdal, Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Syst. 27 (2014), 1621-1629.   DOI
28 P. Das, E. Savas and S.K. Ghosal, On generalized of certain summability methods using ideals, Appl. Math. Letter 36 (2011), 1509-1514.
29 E. Savas and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), 826-830.   DOI
30 M. Gurdal and M.B. Huban, On $\mathcal{I}$-convergence of double sequences in the Topology induced by random 2-norms, Mat. Vesnik 66 (2014), 73-83.
31 E. Savas and M. Gurdal, A generalized statistical convergence in intuitionistic fuzzy normed spaces, Science Asia 41 (2015), 289-294.   DOI
32 M. Mursaleen, S.A. Mohiuddine and O.H.H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl. 59 (2010), 603-611.   DOI
33 M. Mursaleen and S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62 (2012), 49-62.   DOI
34 M. Mursaleen and S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Reports 12 (2010), 359-371.
35 E. Savas and M. Gurdal, $\mathcal{I}$-statistical convergence in probabilistic normed space, Sci. Bull. Series A Appl. Math. Physics 77 (2015), 195-204.
36 M. Mohiuddine and B. Hazarika, Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat 31 (2017), 1827-1834.   DOI
37 E. Savas and H. Gumus, A generalization on $\mathcal{I}$-asymptotically lacunary statistical equivalent sequences, J Inequal Appl. 2013 (2013), 1-9.   DOI