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Keywords:
statistical convergence; semi-uniform space; sequence; function; continuity
statistical convergence; semi-uniform space; sequence; function; continuity
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].
References:
[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).
[2] Arzelà C.: Intorno alla continuità della somma d'infinità di funzioni continue. Rend. dell'Accad. di Bologna (1883–1884), 79–84 (Italian).
[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. DOI 10.1016/j.jmaa.2006.05.040
[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.
[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.
[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. DOI 10.1016/j.jmaa.2010.05.042
[7] Caserta A., Di Maio G., Kočinac L. D. R.: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.
[8] Caserta A., Kočinac L. D. R.: On statistical exhaustiveness. Appl. Math. Lett. 25 (2012), no. 10, 1447–1451. DOI 10.1016/j.aml.2011.12.022
[9] Engelking R.: General Topology. translated from the Polish by the author, Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, 1989. Zbl 0684.54001
[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.
[11] Fast H.: Sur la convergence statistique. Colloquium Math. 2 (1951), 241–244 (French). DOI 10.4064/cm-2-3-4-241-244 | Zbl 0044.33605
[12] Fridy J. A.: On statistical convergence. Analysis 5 (1985), no. 4, 301–313. DOI 10.1524/anly.1985.5.4.301 | Zbl 0588.40001
[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
[14] Di Maio G., Kočinac L. D. R.: Statistical convergence in topology. Topology Appl. 156 (2008), no. 1, 28–45. DOI 10.1016/j.topol.2008.01.015 | Zbl 1155.54004
[15] Marjanović M.: A note on uniform convergence. Publ. Inst. Math. (Beograd) (N.S.) 1(15) (1961), 109–110.
[16] Megaritis A. C.: Ideal convergence of nets of functions with values in uniform spaces. Filomat 31 (2017), no. 20, 6281–6292. DOI 10.2298/FIL1720281M
[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. DOI 10.3792/pja/1195571178
[18] Morita K., Nagata J. (eds.): Topics in General Topology. North-Holland Mathematical Library, 41, North-Holland Publishing Co., Amsterdam, 1989.
[19] Šalát T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), no. 2, 139–150. Zbl 0437.40003
[20] Schoenberg I. J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361–375. DOI 10.1080/00029890.1959.11989303 | Zbl 0089.04002
[21] Steinhaus H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951), 73–74 (French).
[22] Tukey J. W.: Convergence and Uniformity in Topology. Annals of Mathematics Studies, 2, Princeton University Press, Princeton, N.J., 1940.
[23] Zygmund A.: Trigonometric Series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. Zbl 1084.42003
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