Abstract
We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order (α,β).
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P. Das, P. Malik and E. Savas, On statistical limit points of double sequences, Appl. Math. Computation, 215 (2009), 1030–1034.
P. Das and S. Bhunia, Two valued measure and summability of double sequences, Czechoslovak Math. J., 59 (134) (2009), 1141–1155.
R. Colac, Statistical convergence of order α, preprint.
J. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1988), 47–63.
J. Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32 (1989), 194–198.
J. Connor, R type summability methods, Cauchy criterion, P-sets and statistical convergence, Proc. Amer. Math. Soc., 115 (1992), 319–327.
H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118 (1993), 1187–1192.
P. Kostyrko, M. Macaj, T. Šalát and O. Strauch, On statistical limit points, Proc. Amer. Math. Soc., 120 (2000), 2647–2654.
H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1995), 1811–1819.
F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel), 81 (2003), 82–89.
Mursaleen and O. H. H. Eedely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223–231.
A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289–321.
T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), 139–150.
I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
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Bhunia, S., Das, P. & Pal, S.K. Restricting statistical convergence. Acta Math Hung 134, 153–161 (2012). https://doi.org/10.1007/s10474-011-0122-2
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DOI: https://doi.org/10.1007/s10474-011-0122-2
Keywords
- natural density of order α
- statistical convergence of order α
- double density of order (α,β)
- statistical convergence of order (α,β)
- convergence
- closed linear subspace