Lacunary statistical convergence and inclusion properties between lacunary methods. (English) Zbl 0952.40001

Li, Jinlu

Int. J. Math. Math. Sci. 23, No. 3, 175-180 (2000).

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 \[ \lim_{r\to \infty} \frac 1{k_r-k_{r-1}} \# \{k: k_{r-1}< k \leq k_r\text{ and } |x_k-L|\geq \varepsilon \} = 0. \] 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 J. A. Fridy and 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.

Reviewer: Laśzló Tóth (Pécs)

MSC:

40A05	Convergence and divergence of series and sequences
40D05	General theorems on summability
40C05	Matrix methods for summability
11B05	Density, gaps, topology

Keywords:

lacunary sequence; statistical convergence; Cauchy sequence