Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
pointwise and uniform convergence; $\mu $-statistical convergence; convergence in $\mu $-density; finitely additive measure; additive property for null sets
pointwise and uniform convergence; $\mu $-statistical convergence; convergence in $\mu $-density; finitely additive measure; additive property for null sets
Summary:
In the present paper we are concerned with convergence in $\mu $-density and $\mu $-statistical convergence of sequences of functions defined on a subset $D$ of real numbers, where $\mu $ is a finitely additive measure. Particularly, we introduce the concepts of $\mu $-statistical uniform convergence and $\mu $-statistical pointwise convergence, and observe that $\mu $-statistical uniform convergence inherits the basic properties of uniform convergence.
References:
[1] R. G. Bartle: Elements of Real Analysis. John Wiley & Sons, Inc., New York, 1964. MR 0393369
[2] J. Connor: The statistical and strong $p$-Cesàro convergence of sequences. Analysis 8 (1988), 47–63. DOI 10.1524/anly.1988.8.12.47 | MR 0954458 | Zbl 0653.40001
[3] J. Connor: Two valued measures and summability. Analysis 10 (1990), 373–385. DOI 10.1524/anly.1990.10.4.373 | MR 1085803 | Zbl 0726.40009
[4] J. Connor: $R$-type summability methods, Cauchy criteria, $P$-sets and statistical convergence. Proc. Amer. Math. Soc. 115 (1992), 319–327. MR 1095221 | Zbl 0765.40002
[5] J. Connor and M. A. Swardson: Strong integral summability and the Stone-Čech compactification of the half-line. Pacific J. Math. 157 (1993), 201–224. DOI 10.2140/pjm.1993.157.201 | MR 1197054
[6] J. Connor: A topological and functional analytic approach to statistical convergence. Analysis of Divergence, Birkhäuser-Verlag, Boston, 1999, pp. 403–413. MR 1734462 | Zbl 0915.40002
[7] J. Connor and J. Kline: On statistical limit points and the consistency of statistical convergence. J. Math. Anal. Appl. 197 (1996), 393–399. DOI 10.1006/jmaa.1996.0027 | MR 1372186
[8] J. Connor, M. Ganichev and V. Kadets: A characterization of Banach spaces with separable duals via weak statistical convergence. J. Math. Anal. Appl. 244 (2000), 251–261. DOI 10.1006/jmaa.2000.6725 | MR 1746802
[9] K. Demirci and C. Orhan: Bounded multipliers of bounded $A$-statistically convergent sequences. J. Math. Anal. Appl. 235 (1999), 122–129. DOI 10.1006/jmaa.1999.6371 | MR 1758671
[10] H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241–244. DOI 10.4064/cm-2-3-4-241-244 | MR 0048548 | Zbl 0044.33605
[11] J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313. DOI 10.1524/anly.1985.5.4.301 | MR 0816582 | Zbl 0588.40001
[12] J. A. Fridy and C. Orhan: Lacunary statistical convergence. Pacific J. Math. 160 (1993), 43–51. DOI 10.2140/pjm.1993.160.43 | MR 1227502
[13] J. A. Fridy and C. Orhan: Lacunary statistical summability. J. Math. Anal. Appl. 173 (1993), 497–503. DOI 10.1006/jmaa.1993.1082 | MR 1209334
[14] J. A. Fridy and M. K. Khan: Tauberian theorems via statistical convergence. J. Math. Anal. Appl. 228 (1998), 73–95. DOI 10.1006/jmaa.1998.6118 | MR 1659877
[15] E. Kolk: Convergence-preserving function sequences and uniform convergence. J. Math. Anal. Appl. 238 (1999), 599–603. DOI 10.1006/jmaa.1999.6533 | MR 1715505 | Zbl 0939.40001
[16] I. J. Maddox: Statistical convergence in a locally convex space. Math. Proc. Cambridge Phil. Soc. 104 (1988), 141–145. DOI 10.1017/S0305004100065312 | MR 0938459 | Zbl 0674.40008
[17] H. I. Miller: A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347 (1995), 1811–1819. DOI 10.1090/S0002-9947-1995-1260176-6 | MR 1260176 | Zbl 0830.40002
[18] T. Šálat: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150. MR 0587239
[19] H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951), 73–74.
[20] W. Wilczyński: Statistical convergence of sequences of functions. Real Anal. Exchange 25 (2000), 49–50. DOI 10.2307/44153029
[21] A. Zygmund: Trigonometric Series. Second edition. Cambridge Univ. Press, Cambridge, 1979.
Search
Partner of
