Previous |  Up |  Next

Article

Keywords:
statistical convergence; set of the first category; Hausdorff dimension; homogeneous set
Summary:
In this paper we use the notion of statistical convergence of infinite series naturally introduced as the statistical convergence of the sequence of the partial sums of the series. We will discuss some questions related to the convergence of subseries of a given series.
References:
[1] R. C. Buck and H. Pollard: Convergence and summability properties of subsequences. Bull. Amer. Math. Soc. 49 (1943), 924–931. DOI 10.1090/S0002-9904-1943-08061-5 | MR 0009209
[2] R. G. Cooke: Infinite Matrices and Sequence Spaces (Russian Edition). Moscow, 1960. MR 0126098
[3] M. Dindoš, I. Martišovitš and T. Šalát: Remarks on infinite series in linear normed spaces. Tatra Mt. Math. Publ. 19 (2000), 31–46. MR 1771020
[4] R. Durret: Probability: Theory and Examples. Duxbury Press, 1995. MR 1609153
[5] H. Fast: Sur la convergence statistique. Coll. Math. 2 (1951), 241–244. MR 0048548 | Zbl 0044.33605
[6] J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313. MR 0816582 | Zbl 0588.40001
[7] P. Kostyrko, M. Mačaj and T. Šalát: Statistical convergence and I-convergence. To appear in Real Anal. Exch. MR 1844385
[8] H. H. Ostmann: Additive Zahlentheorie I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956. MR 0098721 | Zbl 0072.03101
[9] T. Šalát: On Hausdorff measure of linear sets. Czechoslovak Math. J. 11 (86) (1980), 24–56. (Russian) MR 0153802
[10] T. Šalát: On subseries. Math. Zeit. 85 (1964), 209–225. DOI 10.1007/BF01112142 | MR 0179507
[11] T. Šalát: Über die Cantorsche Reihen. Czechoslovak Math. J. 18 (93) (1968), 25–56.
[12] T. Šalát: On subseries of divergent series. Mat. Čas. SAV 18 (1968), 312–338. MR 0252890
[13] T. Šalát: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150. MR 0587239
[14] I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361–375. DOI 10.2307/2308747 | MR 0104946 | Zbl 0089.04002
[15] C. Visser: The law of nought-or-one. Studia Math. 7 (1938), 143–159. Zbl 0019.22501
[16] B. C. Tripathy: On statistically convergence series. Punjab University J. Math. XXXXII (1999), 1–8. MR 1778259
[17] B. C. Tripathy: On statistical convergence. Proc. Estonian Acad. Sci. Phys. Math. 47 (1998), 299–303. MR 1671445 | Zbl 0961.40002
[18] A. Zygmund: Trigonometric series I. MIR, Moscow, 1965. (Russian) MR 0178296
Partner of
EuDML logo