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Zbl 0952.40001
Li, Jinlu
Lacunary statistical convergence and inclusion properties between lacunary methods. (English)
[J] Int. J. Math. Math. Sci. 23, No.3, 175-180 (2000). ISSN 0161-1712; ISSN 1687-0425/e

The integer sequence $\theta=\{k_r\}$ is called a lacunary sequence if it is increasing and $\lim_{r\to \infty} (k_r-k_{r-1})= \infty$. A complex number sequence $x=\{x_k\}$ is said to be $s_{\theta}$-convergent to $L$ if for each $\varepsilon >0$ one has $$ \lim_{r\to \infty} \frac 1{k_r-k_{r-1}} \# \{k: k_{r-1}< k \le k_r\text{ and } |x_k-L|\ge \varepsilon \} = 0. $$ Let $S_{\theta}$ be the family of all sequences $x$ which are $s_{\theta}$-convergent to some $L$. In this paper, which continues the work of {\it J. A. Fridy} and {\it C. Orhan} [Pac. J. Math. 160, No. 1, 43-51 (1993; Zbl 0794.60012)], the author studies inclusion properties between $S_{\theta}$ and $S_{\beta}$, where $\theta$ and $\beta$ are two arbitrary lacunary sequences.
[Laśzló Tóth (Pécs)]

MSC 2000:: *40A05 Convergence of series and sequences
40D05 General summability theorems
40C05 Matrix methods in summability
11B05 Topology etc. of sets of numbers

Keywords: lacunary sequence; statistical convergence; Cauchy sequence

Citations: Zbl 0794.60012

Cited in: Zbl 1106.40002

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