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Generalized statistical convergence and statistical core of double sequences. (English) Zbl 1219.40004

λ-statistical convergence was introduced in [Mursaleen, Math. Slovaca, 50, No. 1, 111–115 (2000; Zbl 0953.40002)] for single sequences as follows:

Let λ=(λ n ) be a non-decreasing sequence of positive numbers tending to such that

λ n+1 λ n +1,λ 1 =0·

A double sequence x=x jk is said to be (λ,μ)-statistically convergent to l if δ λμ (E)=0, where E={jJ m ,kI n :|x jk -l|ε}, i.e., if for every ε>0,

(P)lim m,n 1 λ m μ n |{jJ m ,kI n :|x jk -l|ε}|=0·

In this case the authors write (st λ,μ )lim j,k x j,k =l and they denote the set of all (λ,μ)-statistically convergent double sequences by S λ,μ .

In this paper, they extended the notion of λ-statistical convergence to the (λ,μ)-statistical convergence for double sequences x=(x k ). They also determine some matrix transformations and establish some core theorems related to their new space of double sequences S λ,μ .

MSC:
40A35Ideal and statistical convergence
40B05Multiple sequences and series
40C05Matrix methods in summability
References:
[1]Fast, H.: Sur la convergence statistique. Colloq. Math., 2, 241–244 (1951)
[2]Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 2, 73–34 (1951)
[3]Aizpuru, A., Nicasio-Llach, M.: About the statistical uniform convergence. Bull. Braz. Math. Soc. (N.S.), 39(2), 173–182 (2008) · Zbl 1170.40001 · doi:10.1007/s00574-008-0078-1
[4]Çoşkun, H., Çakan, C., Mursaleen, M.: Statistical and σ-cores. Studia Math., 154, 29–35 (2003) · Zbl 1006.40006 · doi:10.4064/sm154-1-3
[5]Çoşkun, H., Çakan, C.: A class of statistical and σ-conservative matrices. Czechoslovak Math. J., 55(130), 791–801 (2005) · Zbl 1081.40003 · doi:10.1007/s10587-005-0065-2
[6]Fridy, J. A.: On statistical convergence. Analysis (Munich), 5, 301–313 (1985)
[7]de Malafosse, B., Rakocević, V.: Matrix transformation and statistical convergence. Linear Algebra Appl., 420, 377–387 (2007) · Zbl 1128.40003 · doi:10.1016/j.laa.2006.07.021
[8]Mursaleen, M., Mohiuddine, S. A.: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals, 41, 2414–2421 (2009) · Zbl 1198.40007 · doi:10.1016/j.chaos.2008.09.018
[9]Mursaleen, M.: λ-Statistical convergence. Math. Slovaca, 50, 111–115 (2000)
[10]ŞSalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca, 30, 139–150 (1980)
[11]Pringsheim, A.: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Z., 53, 289–321(1900)
[12]Christopher, J.: The asymptotic density of some k-dimensional sets. Amer. Math. Monthly, 63, 399–401 (1956) · Zbl 0070.04101 · doi:10.2307/2309400
[13]Moricz, F.: Statistical convergence of multiple sequences. Arch. Math. (Basel), 81, 82–89 (2003)
[14]Mursaleen, M., Edely, O. H. H.: Statistical convergence of double sequences. J. Math. Anal. Appl., 288, 223–231 (2003) · Zbl 1032.40001 · doi:10.1016/j.jmaa.2003.08.004
[15]Tripathy, B. C., Sarma, B.: Statistically convergent difference double sequence spaces. Acta Mathematica Sinica, English Series, 24(5), 737–742 (2008) · Zbl 1160.46003 · doi:10.1007/s10114-007-6648-0
[16]Leindler, L.: Über die de la Vallée-Pousinsche summierbarkeit allgemeiner orthogonalreihen. Acta Math. Acad. Sci. Hungar., 16, 375–387 (1965) · Zbl 0138.28802 · doi:10.1007/BF01904844
[17]Fridy, J. A., Orhan, C.: Statistical limit superior and limit inferior. Proc. Amer. Math. Soc., 125, 3625–3613 (1997) · Zbl 0883.40003 · doi:10.1090/S0002-9939-97-04000-8
[18]Çakan, C, Altay, B., Çoşkun, H.: Double lacunary density and lacunary statistical convergence of double sequences. Studia Sci. Math. Hungar., 47(1), 35–45 (2010)
[19]Çakan, C., Altay, B., Mursaleen, M.: The σ-convergence and σ-core of double sequences. Appl. Math. Lett., 19, 1122–1128 (2006) · Zbl 1122.40004 · doi:10.1016/j.aml.2005.12.003
[20]Çakan, C., Altay, B.: Statistically boundedness and statistical core of double sequences. J. Math. Anal. Appl., 317, 690–697 (2006) · Zbl 1084.40001 · doi:10.1016/j.jmaa.2005.06.006
[21]Çakan, C., Altay, B.: A class of conservative four-dimensional matrices. J. Inequal. Appl., Vol. 2006, Article ID 14721, 8 pages (2006)
[22]Mursaleen, M., Edely, O. H. H.: Almost convergence and a core theorem for double sequences. J. Math. Anal. Appl., 293, 532–540 (2004) · Zbl 1043.40003 · doi:10.1016/j.jmaa.2004.01.015
[23]Mursaleen, M., Mohiuddine, S. A.: Double σ-multiplicative matrices. J. Math. Anal. Appl., 327 991–996 (2007) · Zbl 1107.40004 · doi:10.1016/j.jmaa.2006.04.081
[24]Mursaleen, M., Mohiuddine, S. A.: Regularly σ-conservative and σ-coercive four-dimensional matrices. Comput. Math. Appl., 56, 1580–1586 (2008) · Zbl 1155.40303 · doi:10.1016/j.camwa.2008.03.007
[25]Mursaleen, M., Savaş, E.: Almost regular matrices for double sequences. Studia Sci. Math. Hungar., 40, 205–212 (2003)
[26]Mursaleen, M.: Almost strongly regular matrices and a core theorem for double sequences. J. Math. Anal. Appl., 293, 523–531 (2004) · Zbl 1043.40002 · doi:10.1016/j.jmaa.2004.01.014
[27]Patterson, R. F., Lemma, M.: Four dimensional matrix characterization of double oscillation via RH-conservative and RH-multiplicative matrices. Cent. Eur. J. Math., 6(4), 581–594 (2008) · Zbl 1165.40004 · doi:10.2478/s11533-008-0043-7
[28]Patterson, R. F.: Double sequence core theorems. Int. J. Math. Sci., 22, 785–793 (1999) · Zbl 0949.40007 · doi:10.1155/S0161171299227858
[29]Zeltser, M.: On conservative matrix methods for double sequence spaces. Acta Math. Hungar., 95, 225–242 (2002) · Zbl 0997.40003 · doi:10.1023/A:1015636905885
[30]Hamilton, H. J.: Transformations of multiple sequences. Duke Math. J., 2, 29–60 (1936) · doi:10.1215/S0012-7094-36-00204-1
[31]Robinson, G. M.: Divergent double sequences and series. Trans. Amer. Math. Soc., 28, 50–73 (1926) · doi:10.1090/S0002-9947-1926-1501332-5
[32]Cooke, R. G.: Infinite Matrices and Sequence Spaces, Macmillan, London, 1950