Abstract
This paper gives an overview of the theory of statistical convergence and extends a result of Fridy and Orhan. A sequence x is said to be statistically convergent to L with respect to the finitely additive measure μ provided that ‘almost all’ of the values x are arbitarily close to L with respect to μ. One can also define what is meant by a μ-statistical cluster point of a sequence, μ-statistical limit superior of a sequence and so forth and thus create a theory of convergence that includes ordinary convergence. In this note we review some of the basic results of μ-statistical convergence, indicate either topological or functional analytic proofs of some basic results and provide a means of isolating the invariants of statistical convergence.
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References
Agnew, R.P., Cores of complex sequences and of their transforms, Amer. J. Math. 61 (1939), 178–186
Atalla, R. and Bustoz, J., On sequential cores and a theorem of R.R. Phelps, Proc. Amer. Math. Soc. 21 (1969), 36–42.
Atalla, R. On the multiplicative behavior of regular matrices, Proc. Amer. Math. Soc. 26 (1970), 437–446.
Atalla, R., On the consistency theorem in matrix summability, Proc. Amer. Math. Soc. 35 (1972) no. 2, 416–422.
Buck, R.C., Generalized Asymptotic Densities, American J. Math. 75 (1953), 335–46.
Chun, C.S. and Freedman, A.R., Theorems and examples for R-type summability methods, J. Korean Math. Soc. 25 (1988), 315–324.
Chun, C.S. and Freedman, A.R., A bounded consistency theorem for strong summabilities, Inter. J. Math. Sci. 12 (1989), 39–46.
Connor, J., The statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), 47–63.
Connor, J., Two valued measures and summability, Analysis 10 (1990), 373–385.
Connor, J., R-type summability methods, Cauchy criteria, P-sets and statistical convergence, Proc. Amer. Math. Soc. 115 (1992), no. 2, 319–327.
Connor, J., Gap tauberian theorems, Bull. Austral. Math. Soc. 47 (1993), no. 3, 385–393.
Connor, J. and Kline, J., On Statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl. 197 (1996), no. 2 392–399.
Connor, J. and Swardson, M.A., Measures and ideals of C* (X), Papers on general topology and applications, 80–91, Ann. New York Acad. Sci., 704, New York Acad. Sci., New York, 1993.
Fast, H., Sur la convergence statistique. Colloq. Math 2 (1951), 241–244.
Freedman, A. R. and Sember, J.J., Densities and summability. Pacific J. Math 95 (1981), 293–305.
Fridy, J., On statistical convergence, Analysis 5 (1985), 301–313.
Fridy, J., Statistical limit points, Proc. Amer. Math. Soc. 118 (1993), no. 4, 1187–1192.
Fridy, J. and Miller, H.I., A matrix characterization of statistical convergence, Analysis 11 (1991), 59–66.
Fridy, J. and Orhan, C., Lacunary Statistical Convergence, Pacific Journal of Math., 160 (1993), 43–51.
Fridy, J. and Orhan, C., Lacunary Statistical Summability, Journal of Mathematical Analysis and Applications, Vol. 173 (1991), no. 2, 497–504.
Fridy, J. and Orhan, C., Statistical Limit Superior and Limit Inferior, Proc. Amer. Math. Soc., to appear.
Fridy, J.;Orhan, C., Statistical core theorems, J. Math. Anal. Appl. 208 (1997), no. 2, 520–527.
Henriksen, M., Multiplicative summability methods and the Stone- Čech compactification, Math. Z. 71 (1959), 427–435.
Hill, J.D. and Sledd, W.T., Approximation in bounded summability fields, Canad. J. Math. 20 (1968), 410–415.
Kline, J., The T-statistically convergent sequences are not an FK space, Internat. J. Math. Math. Sci. 18 (1995), no. 4, 825–827.
Kline, J. Statistical convergence and densities generated by sequences of measures, Ph.D. dissertation, Ohio University, 1995.
Knopp, K., Zur Theorie de Limitierungsverfahren (Erste Mitteiiung), M. Zeit. 31 (1930), 115–127.
Kolk, E., Matrix summability of statistically convergent sequences. Analysis 13 (1993), no. 2, 497–504.
Kolk, E., The statistical convergence in Banach spaces, Tartu Ul. Toimetised No. 928 (1991), 41–52.
Maddox, I.J., Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), 141–145.
Maddox, I.J., A tauberian theorem for statistical convergence, Math. Proc. Camb. Phil. Soc. 106 (1989), 277–280.
Miller, H., A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347 (1995) no. 5, 1811–1819.
Phelps, R.R., The range of Tf for certain linear operators T, Proc. Amer. Math. Soc. 16 (1965), 381–382.
Rainwater, J., Regular matrices with nowhere dense support, Proc. Amer. Math. Soc. 29 (1971), 90–91.
Savac, E.; Faith, N, On σ-statistically convergence and lacunary σ statistically convergence. Math. Slovaca 43 (1993), no. 3, 309–315.
Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), No. 2, 139–150.
Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.
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Connor, J. (1999). A topological and functional analytic approach to statistical convergence. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_23
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DOI: https://doi.org/10.1007/978-1-4612-2236-1_23
Publisher Name: Birkhäuser Boston
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