# zbMATH — the first resource for mathematics

Generalized statistical convergence and statistical core of double sequences. (English) Zbl 1219.40004
$$\lambda$$-statistical convergence was introduced in [Mursaleen, Math. Slovaca, 50, No. 1, 111–115 (2000; Zbl 0953.40002)] for single sequences as follows:
Let $$\lambda=(\lambda_n)$$ be a non-decreasing sequence of positive numbers tending to $$\infty$$ such that
$\lambda_{n+1}\leq \lambda_n+1, \qquad \lambda_1=0.$
A double sequence $$x=x_{jk}$$ is said to be $$(\lambda, \mu)$$-statistically convergent to $$l$$ if $$\delta_{\lambda\mu}(E)=0$$, where $$E=\{j\in J_m, k\in I_n:|x_{jk}-l|\geq\varepsilon\}$$, i.e., if for every $$\varepsilon>0$$,
$(P)\lim_{m,n} \frac{1}{\lambda_m\mu_n} \big|\{j\in J_m,\;k\in I_n:|x_{jk}-l|\geq\varepsilon\}\big|=0.$ In this case the authors write $$(st_{\lambda,\mu})\lim_{j,k}x_{j,k}=l$$ and they denote the set of all $$(\lambda, \mu)$$-statistically convergent double sequences by $$S_{\lambda,\mu}$$.
In this paper, they extended the notion of $$\lambda$$-statistical convergence to the $$(\lambda, \mu)$$-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 $$S_{\lambda,\mu}$$.

##### MSC:
 40A35 Ideal and statistical convergence 40B05 Multiple sequences and series (should also be assigned at least one other classification number in this section) 40C05 Matrix methods for summability
Full Text:
##### References:
 [1] Fast, H.: Sur la convergence statistique. Colloq. Math., 2, 241–244 (1951) · Zbl 0044.33605 [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 $$\sigma$$-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 $$\sigma$$-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) · Zbl 0588.40001 [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.: $$\lambda$$-Statistical convergence. Math. Slovaca, 50, 111–115 (2000) · Zbl 0953.40002 [10] ŞSalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca, 30, 139–150 (1980) · Zbl 0437.40003 [11] Pringsheim, A.: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Z., 53, 289–321(1900) · JFM 31.0249.01 [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) · Zbl 1041.40001 [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) · Zbl 1240.40021 [19] Çakan, C., Altay, B., Mursaleen, M.: The $$\sigma$$-convergence and $$\sigma$$-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) · Zbl 1132.40002 [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 $$\sigma$$-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 $$\sigma$$-conservative and $$\sigma$$-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) · Zbl 1050.40003 [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) · Zbl 0013.30301 · 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) · JFM 52.0223.01 · doi:10.1090/S0002-9947-1926-1501332-5 [32] Cooke, R. G.: Infinite Matrices and Sequence Spaces, Macmillan, London, 1950 · Zbl 0040.02501
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.