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Zbl 0821.40001
Pehlivan, Serpil; Fisher, Brian
Lacunary strong convergence with respect to a sequence of modulus functions.
(English)
[J] Commentat. Math. Univ. Carol. 36, No.1, 69-76 (1995). ISSN 0010-2628

A lacunary sequence $\theta= (k\sb r)$ is an increasing sequence of integers with $k\sb 0 =0$ and $h\sb r= k\sb r- k\sb{r-1}\to \infty$ as $r\to\infty$. A sequence $(x\sb i)$ is lacunary strong convergent to $l$ with respect to $\theta$ if $$\lim\sb{r\to\infty} {\textstyle {1\over h\sb r}} \sum\sb{i\in I\sb r} \vert x\sb i- l\vert =0,$$ where $I\sb r$ is the interval $(k\sb{r-1}, k\sb r]$, and $N\sb 0$ denotes the linear space of all lacunary strongly convergent sequences. These ideas are attributed to {\it A. R. Freedman}, {\it J. J. Sember} and {\it M. Raphael} [Proc. Lond. Math. Soc., III. Ser. 37, 508-520 (1978; Zbl 0424.40008)]. A modulus function $f$ is a function on $(0, \infty]$ to $(0, \infty]$ with $f(x)= 0 \iff x=0$, $f(x+y)\leq f(x)+ f(y)$, $f$ is increasing and continuous from the right at 0. (It follows that $f$ is continuous on $(0, \infty]$.) Let $F= (f\sb i)$ be a sequence of modulus functions and $X$ be a Banach space. The authors introduce the following notion of lacunary strong convergence of a sequence $(x\sb i)$ in $X$ with respect to $F$ and $\theta$: $$\lim\sb{r\to\infty} {\textstyle {1\over h\sb r}} \sum\sb{i\in I\sb r} f\sb i (\Vert x\sb i- l\Vert)=0.$$ Properties of this convergence, and its relation to statistical convergence in a Banach space, as introduced by {\it H. Fast} [Colloq. Math. 2, 241-244 (1951; Zbl 0044.336)] are examined.
MSC 2000:
*40A05 Convergence of series and sequences
40F05 Special cases of summability

Keywords: lacunary sequence; modulus function; Banach space; statistical convergence

Citations: Zbl 0424.40008; Zbl 0044.336

Cited in: Zbl 1119.40300 Zbl 0946.40003

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