Statistical convergence of double sequences. (English) Zbl 1032.40001

Summary: The idea of statistical convergence was first introduced by H. Fast [Colloq. Math. 2, 241-244 (1951; Zbl 0044.33605)] but the rapid developments were started after the papers of J. A. Šalát [Math. Slovaca 30, 139-150 (1980; Zbl 0437.40003)] and J. A. Fridy [Analysis 5, 301-313 (1985; Zbl 0588.40001)]. Nowadays it has become one of the most active areas of research in the field of summability. In this paper we define and study statistical analogue of convergence and Cauchy for double sequences. We also establish the relation between statistical convergence and strongly Cesàro summable double sequences.


40A05 Convergence and divergence of series and sequences
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[1] Buck, R. C., Generalized asymptotic density, Amer. J. Math., 75, 335-346 (1953) · Zbl 0050.05901
[2] Christopher, J., The asymptotic density of some \(k\)-dimensional sets, Amer. Math. Monthly, 63, 399-401 (1956) · Zbl 0070.04101
[3] Connor, J. S., The statistical and strong \(p\)-Cesàro convergence of sequences, Analysis, 8, 47-63 (1988) · Zbl 0653.40001
[4] Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951) · Zbl 0044.33605
[5] Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985) · Zbl 0588.40001
[6] Moricz, F., Tauberian theorems for Cesàro summable double sequences, Studia Math., 110, 83-96 (1994) · Zbl 0833.40003
[7] Neubrum, T.; Smital, J.; Salat, T., On the structure of the space \(M(0,1)\), Rev. Roumaine Math. Pures Appl., 13, 337-386 (1968)
[8] Pringsheim, A., Zur Ttheorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53, 289-321 (1900) · JFM 31.0249.01
[9] Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150 (1980) · Zbl 0437.40003
[10] Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, 361-375 (1959) · Zbl 0089.04002
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