We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained in the setting of general metric spaces.
This is a preview of subscription content, log in via an institution to check access.
References
F. G. Abdullayev, O. Dovgoshey, and M. Küçükaslan, “Metric spaces with unique pretangent spaces. Conditions of the uniqueness,” Ann. Acad. Sci. Fenn. Math., 36, 353–392 (2011).
G. S. Baranenkov, B. P. Demidovich, V. A. Efimenko, etc., Problems in Mathematical Analysis, Mir, Moscow (1976).
V. Bilet and O. Dovgoshey, “Isometric embeddings of pretangent spaces in E n,” Bull. Belg. Math. Soc. Simon Stevin, 20, 91–110 (2013).
V. Bilet, “Geodesic spaces tangent to metric spaces,” Ukr. Math. J., 62, No. 11, 1448–1456 (2013).
J. Cervanansky, “Statistical convergence and statistical continuity,” Zb. Ved. Pr. MtF STU, 6, 924–931 (1943).
J. Connor, “The statistical and strong p-Cesaro convergence of sequences,” Analysis, 8, 207–212 (1998).
O. Dovgoshey, “Tangent spaces to metric spaces and to their subspaces,” Ukr. Mat. Visn., 5, 468–485 (2008).
O. Dovgoshey, F. G. Abdullayev, and M. Küçükaslan, “Compactness and boundedness of tangent spaces to metric spaces,” Beitr. Algebra Geom., 51, 547–576 (2010).
O. Dovgoshey and D. Dordovskyi, “Ultrametricity and metric betweenness in tangent spaces to metric spaces,” P-Adic Numbers Ultrametric Anal. Appl., 2, 100–113 (2010).
O. Dovgoshey and O. Martio, “Tangent spaces to metric spaces,” Repts Math. Helsinki Univ., 480 (2008).
H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241–244 (1951).
J. A. Fridy, “On statistical convergence,” Analysis, 5, 301–313 (1995).
J. A. Fridy and M. K. Khan, “Tauberian theorems via statistical convergence,” J. Math. Anal. Appl., 228, 73–95 (1998).
J. A. Fridy and H. I. Miller, “A matrix characterization of statistical convergence,” Analysis, 11, 59–66 (1991).
J. Heinonen, Lectures on Analysis on Metric Spaces, Springer (2001).
M. Măcaj and T. Šalát, “Statistical convergence of subsequence of a given sequence,” Math. Bohemica, 126, 191–208 (2001).
H. I. Miller, “A measure theoretical subsequence characterization of statistical convergence,” Trans. Amer. Math. Soc., 347, 1811–1819 (1995).
A. Papadopoulos, “Metric spaces, convexity and nonpositive curvature,” Eur. Math. Soc. (2005).
T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30, 139–150 (1980).
H. Steinhous, “Sur la convergence ordinaire et la convergence asymtotique,” Colloq. Math., 2, 73–74 (1951).
P. Teran, “A reduction principle for obtaining Tauberian theorems for statistical convergence in metric spaces,” Bull. Belg. Math. Soc., 12, 295–299 (2005).
A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, UK (1979).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 5, pp. 712–720, May, 2014.
Rights and permissions
About this article
Cite this article
Küçükaslan, M., Değer, U. & Dovgoshey, O. On the Statistical Convergence of Metric-Valued Sequences. Ukr Math J 66, 796–805 (2014). https://doi.org/10.1007/s11253-014-0974-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0974-z