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Lacunary strong convergence with respect to a sequence of modulus functions. (English) Zbl 0821.40001

A lacunary sequence \(\theta= (k_ r)\) is an increasing sequence of integers with \(k_ 0 =0\) and \(h_ r= k_ r- k_{r-1}\to \infty\) as \(r\to\infty\). A sequence \((x_ i)\) is lacunary strong convergent to \(l\) with respect to \(\theta\) if \[ \lim_{r\to\infty} {\textstyle {1\over h_ r}} \sum_{i\in I_ r} | x_ i- l| =0, \] where \(I_ r\) is the interval \((k_{r-1}, k_ r]\), and \(N_ 0\) denotes the linear space of all lacunary strongly convergent sequences. These ideas are attributed to A. R. Freedman, J. J. Sember and 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 \(F= (f_ i)\) 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 \((x_ i)\) in \(X\) with respect to \(F\) and \(\theta\): \[ \lim_{r\to\infty} {\textstyle {1\over h_ r}} \sum_{i\in I_ r} f_ i (\| x_ i- l\|)=0. \] Properties of this convergence, and its relation to statistical convergence in a Banach space, as introduced by H. Fast [Colloq. Math. 2, 241-244 (1951; Zbl 0044.336)] are examined.

MSC:

40A05 Convergence and divergence of series and sequences
40F05 Absolute and strong summability
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