Previous |  Up |  Next


Title: Statistical convergence of sequences of functions with values in semi-uniform spaces (English)
Author: Georgiou, Dimitrios N.
Author: Megaritis, Athanasios C.
Author: Özçağ, Selma
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 59
Issue: 1
Year: 2018
Pages: 103-117
Summary lang: English
Category: math
Summary: We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzelà, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kočinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kočinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett. 25 (2012), no. 10, 1447--1451]. (English)
Keyword: statistical convergence
Keyword: semi-uniform space
Keyword: sequence
Keyword: function
Keyword: continuity
MSC: 40A30
MSC: 40A35
MSC: 54A20
MSC: 54E15
idZBL: Zbl 06890399
idMR: MR3783811
DOI: 10.14712/1213-7243.2015.231
Date available: 2018-04-17T13:50:59Z
Last updated: 2020-04-06
Stable URL:
Reference: [1] Alexandroff P. S.: Einführung in die Mengenlehre und die Theorie der reellen Funktionen.Zweite Auflage. Übersetzung aus dem Russischen: Manfred Peschel und Wolfgang Richter. Hochschulbücher für Mathematik, Band 23 VEB Deutscher Verlag der Wissenschaften, Berlin, 1964 (German).
Reference: [2] Arzelà C.: Intorno alla continuità della somma d'infinità di funzioni continue.Rend. dell'Accad. di Bologna (1883–1884), 79–84 (Italian).
Reference: [3] Balcerzak M., Dems K., Komisarski A.: Statistical convergence and ideal convergence for sequences of functions.J. Math. Anal. Appl. 328 (2007), no. 1, 715–729. 10.1016/j.jmaa.2006.05.040
Reference: [4] Bînzar T.: On some convergences for nets of functions with values in generalized uniform spaces.Novi Sad J. Math. 39 (2009), no. 1, 69–80.
Reference: [5] Caserta A., Di Maio G.: Convergences characterizing the continuity of the limits of functions: a survey from Arzelà's theorem (1883) to the present.Proceedings ICTA2011, Islamabad, Pakistan, July 4–10, 2011; Cambridge Scientific Publishers, 2012, pp. 75–103.
Reference: [6] Caserta A., Di Maio G., Holá L'.: Arzelà's theorem and strong uniform convergence on bornologies.J. Math. Anal. Appl. 371 (2010), no. 1, 384–392. 10.1016/j.jmaa.2010.05.042
Reference: [7] Caserta A., Di Maio G., Kočinac L. D. R.: Statistical convergence in function spaces.. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.
Reference: [8] Caserta A., Kočinac L. D. R.: On statistical exhaustiveness.Appl. Math. Lett. 25 (2012), no. 10, 1447–1451. 10.1016/j.aml.2011.12.022
Reference: [9] Engelking R.: General Topology.translated from the Polish by the author, Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, 1989. Zbl 0684.54001
Reference: [10] Ewert J.: Generalized uniform spaces and almost uniform convergence.Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(90) (1999), no. 4, 315–329.
Reference: [11] Fast H.: Sur la convergence statistique.Colloquium Math. 2 (1951), 241–244 (French). Zbl 0044.33605, 10.4064/cm-2-3-4-241-244
Reference: [12] Fridy J. A.: On statistical convergence.Analysis 5 (1985), no. 4, 301–313. Zbl 0588.40001, 10.1524/anly.1985.5.4.301
Reference: [13] Kelley J. L.: General Topology.reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, 27, Springer, New York-Berlin, 1975. Zbl 0518.54001
Reference: [14] Di Maio G., Kočinac L. D. R.: Statistical convergence in topology.Topology Appl. 156 (2008), no. 1, 28–45. Zbl 1155.54004, 10.1016/j.topol.2008.01.015
Reference: [15] Marjanović M.: A note on uniform convergence.Publ. Inst. Math. (Beograd) (N.S.) 1(15) (1961), 109–110.
Reference: [16] Megaritis A. C.: Ideal convergence of nets of functions with values in uniform spaces.Filomat 31 (2017), no. 20, 6281–6292. 10.2298/FIL1720281M
Reference: [17] Morita K.: On the simple extension of a space with respect to a uniformity.I.–IV., Proc. Japan Acad. 27 (1951), 65–72, 130–137, 166–171, 632–636. 10.3792/pja/1195571178
Reference: [18] Morita K., Nagata J. (eds.): Topics in General Topology.North-Holland Mathematical Library, 41, North-Holland Publishing Co., Amsterdam, 1989.
Reference: [19] Šalát T.: On statistically convergent sequences of real numbers.Math. Slovaca 30 (1980), no. 2, 139–150. Zbl 0437.40003
Reference: [20] Schoenberg I. J.: The integrability of certain functions and related summability methods.Amer. Math. Monthly 66 (1959), 361–375. Zbl 0089.04002, 10.1080/00029890.1959.11989303
Reference: [21] Steinhaus H.: Sur la convergence ordinaire et la convergence asymptotique.Colloq. Math. 2 (1951), 73–74 (French).
Reference: [22] Tukey J. W.: Convergence and Uniformity in Topology.Annals of Mathematics Studies, 2, Princeton University Press, Princeton, N.J., 1940.
Reference: [23] Zygmund A.: Trigonometric Series.. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. Zbl 1084.42003


Files Size Format View
CommentatMathUnivCarolRetro_59-2018-1_8.pdf 301.2Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo