×

Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces. (English) Zbl 1380.40004

Summary: The notion of statistical convergence was defined by H. Fast [Colloq. Math. 2, 241–244 (1951; Zbl 0044.33605)] and over the years was further studied by many authors in different setups. In this paper, we define and study statistical \(\tau\)-convergence, statistically \(\tau\)-Cauchy and \(S^*(\tau)\)-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.

MSC:

40A35 Ideal and statistical convergence
40G15 Summability methods using statistical convergence
46A40 Ordered topological linear spaces, vector lattices

Citations:

Zbl 0044.33605
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique.Colloq. Math. 1951, 2:73-74.
[2] Fast H: Sur la convergence statistique.Colloq. Math. 1951, 2:241-244. · Zbl 0044.33605
[3] Çakalli H: Lacunary statistical convergence in topological groups.Indian J. Pure Appl. Math. 1995,26(2):113-119. · Zbl 0835.43006
[4] Çakalli H: On statistical convergence in topological groups.Pure Appl. Math. Sci. 1996, 43:27-31. · Zbl 0876.40002
[5] Çakalli H, Khan MK: Summability in topological spaces.Appl. Math. Lett. 2011, 24:348-352. · Zbl 1216.40009 · doi:10.1016/j.aml.2010.10.021
[6] Çakalli H, Savaş E: Statistical convergence of double sequence in topological groups.J. Comput. Anal. Appl. 2010,12(2):421-426. · Zbl 1204.40006
[7] Edely OHH, Mursaleen M: On statisticalA-summability.Math. Comput. Model. 2009, 49:672-680. · Zbl 1182.40004 · doi:10.1016/j.mcm.2008.05.053
[8] Fridy JA: On statistical convergence.Analysis 1985, 5:301-313. · Zbl 0588.40001
[9] Karakuş, S.; Demirci, K., Statistical convergence of double sequences on probabilistic normed spaces, No. 2007 (2007) · Zbl 1147.54016
[10] Karakuş S, Demirci K, Duman O: Statistical convergence on intuitionistic fuzzy normed spaces.Chaos Solitons Fractals 2008, 35:763-769. · Zbl 1139.54006 · doi:10.1016/j.chaos.2006.05.046
[11] Maddox IJ: Statistical convergence in a locally convex space.Math. Proc. Camb. Philos. Soc. 1988, 104:141-145. · Zbl 0674.40008 · doi:10.1017/S0305004100065312
[12] Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces.Appl. Math. Inf. Sci. 2012,6(3):581-585.
[13] Mohuiddine, SA; Alotaibi, A.; Alsulami, SM, Ideal convergence of double sequences in random 2-normed spaces, No. 2012 (2012)
[14] Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space.Chaos Solitons Fractals 2009, 42:1731-1737. · Zbl 1200.46067 · doi:10.1016/j.chaos.2009.03.086
[15] Mohiuddine SA, Savaş E: Lacunary statistically convergent double sequences in probabilistic normed spaces.Ann. Univ. Ferrara 2012, 58:331-339. · Zbl 1309.40004 · doi:10.1007/s11565-012-0157-5
[16] Mohiuddine SA, Şevli H, Cancan M: Statistical convergence in fuzzy 2-normed space.J. Comput. Anal. Appl. 2010,12(4):787-798. · Zbl 1198.40002
[17] Mohiuddine SA, Şevli H, Cancan M: Statistical convergence of double sequences in fuzzy normed spaces.Filomat 2012,26(4):673-681. · Zbl 1289.40027 · doi:10.2298/FIL1204673M
[18] Mursaleen M: On statistical convergence in random 2-normed spaces.Acta Sci. Math. 2010, 76:101-109. · Zbl 1274.40015
[19] Mursaleen M, Alotaibi A: OnI-convergence in random 2-normed spaces.Math. Slovaca 2011,61(6):933-940. · Zbl 1289.40028 · doi:10.2478/s12175-011-0059-5
[20] Mursaleen M, Edely OHH: Generalized statistical convergence.Inf. Sci. 2004, 162:287-294. · Zbl 1062.40003 · doi:10.1016/j.ins.2003.09.011
[21] Mursaleen M, Edely OHH: Statistical convergence of double sequences.J. Math. Anal. Appl. 2003, 288:223-231. · Zbl 1032.40001 · doi:10.1016/j.jmaa.2003.08.004
[22] Mursaleen M, Edely OHH: On the invariant mean and statistical convergence.Appl. Math. Lett. 2009, 22:1700-1704. · Zbl 1183.40006 · doi:10.1016/j.aml.2009.06.005
[23] Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces.Chaos Solitons Fractals 2009, 41:2414-2421. · Zbl 1198.40007 · doi:10.1016/j.chaos.2008.09.018
[24] Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space.J. Comput. Appl. Math. 2009, 233:142-149. · Zbl 1183.46070 · doi:10.1016/j.cam.2009.07.005
[25] Mursaleen M, Mohiuddine SA: On ideal convergence of double sequences in probabilistic normed spaces.Math. Rep. 2010,12(64)(4):359-371. · Zbl 1240.40032
[26] Mursaleen M, Mohiuddine SA: On ideal convergence in probabilistic normed spaces.Math. Slovaca 2012, 62:49-62. · Zbl 1274.40034 · doi:10.2478/s12175-011-0071-9
[27] Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces.Comput. Math. Appl. 2010, 59:603-611. · Zbl 1189.40003 · doi:10.1016/j.camwa.2009.11.002
[28] Savaş E, Mohiuddine SA:-statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 2012,62(1):99-108. · Zbl 1274.40016 · doi:10.2478/s12175-011-0075-5
[29] Savaş E, Mursaleen M: On statistically convergent double sequences of fuzzy numbers.Inf. Sci. 2004, 162:183-192. · Zbl 1057.40002 · doi:10.1016/j.ins.2003.09.005
[30] Di Maio G, Kočinac LDR: Statistical convergence in topology.Topol. Appl. 2008, 156:28-45. · Zbl 1155.54004 · doi:10.1016/j.topol.2008.01.015
[31] Albayrak H, Pehlivan S: Statistical convergence and statistical continuity on locally solid Riesz spaces.Topol. Appl. 2012, 159:1887-1893. · Zbl 1244.40002 · doi:10.1016/j.topol.2011.04.026
[32] Mohiuddine, SA; Alghamdi, MA, Statistical summability through a lacunary sequence in locally solid Riesz spaces, No. 2012 (2012) · Zbl 1283.40005
[33] Mohiuddine, SA; Alotaibi, A.; Mursaleen, M., Statistical convergence of double sequences in locally solid Riesz spaces, No. 2012 (2012) · Zbl 1262.40005
[34] Zaanen AC: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin; 1997. · Zbl 0878.47022 · doi:10.1007/978-3-642-60637-3
[35] Aliprantis CD, Burkinshaw O: Locally Solid Riesz Spaces with Applications to Economics. 2nd edition. Am. Math. Soc., Providence; 2003. · Zbl 1043.46003
[36] Roberts GT: Topologies in vector lattices.Math. Proc. Camb. Philos. Soc. 1952, 48:533-546. · Zbl 0047.10503 · doi:10.1017/S0305004100076295
[37] Mursaleen M: λ-statistical convergence.Math. Slovaca 2000, 50:111-115. · Zbl 0953.40002
[38] Çolak R, Bektaş CA: λ-statistical convergence of orderα.Acta Math. Sci., Ser. B 2011,31(3):953-959. · Zbl 1240.40016
[39] Edely OHH, Mohiuddine SA, Noman AK: Korovkin type approximation theorems obtained through generalized statistical convergence.Appl. Math. Lett. 2010, 23:1382-1387. · Zbl 1206.40003 · doi:10.1016/j.aml.2010.07.004
[40] de Malafosse B, Rakočević V: Matrix transformation and statistical convergence.Linear Algebra Appl. 2007, 420:377-387. · Zbl 1128.40003 · doi:10.1016/j.laa.2006.07.021
[41] Mursaleen M, Çakan C, Mohiuddine SA, Savaş E: Generalized statistical convergence and statistical core of double sequences.Acta Math. Sin. Engl. Ser. 2010, 26:2131-2144. · Zbl 1219.40004 · doi:10.1007/s10114-010-9050-2
[42] Kumar V, Mursaleen M: On -statistical convergence of double sequences on intuitionistic fuzzy normed spaces.Filomat 2011,25(2):109-120. · Zbl 1299.40009 · doi:10.2298/FIL1102109K
[43] Farah I: Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers. 2000. [Mem. Amer. Math. Soc. 148] · Zbl 0966.03045
[44] Caserta A, Kočinac LDR: On statistical exhaustiveness.Appl. Math. Lett. 2012, 25:1447-1451. · Zbl 1255.54010 · doi:10.1016/j.aml.2011.12.022
[45] Caserta, A.; Di Maio, G.; Kočinac, LDR, Statistical convergence in function spaces, No. 2011 (2011) · Zbl 1242.40003
[46] Mohiuddine SA: An application of almost convergence in approximation theorems.Appl. Math. Lett. 2011, 24:1856-1860. · Zbl 1252.41022 · doi:10.1016/j.aml.2011.05.006
[47] Mohiuddine SA, Alotaibi A: Statistical convergence and approximation theorems for functions of two variables.J. Comput. Anal. Appl. 2013,15(2):218-223. · Zbl 1275.41011
[48] Mohiuddine, SA; Alotaibi, A.; Mursaleen, M., Statistical summability [InlineEquation not available: see fulltext.] and a Korovkin type approximation theorem, No. 2012 (2012)
[49] Srivastava HM, Mursaleen M, Khan A: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems.Math. Comput. Model. 2012, 55:2040-2051. · Zbl 1255.41013 · doi:10.1016/j.mcm.2011.12.011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.