Lacunary strong \(\sigma\)-convergence and \(\sigma\)-core (Q1769317)

The core of a complex number sequence was introduced by \textit{K. Knopp} [Math. Z. 31, 97--121 (1929; JFM 55.0730.07)]. A lacunary sequence \(\theta= (k_ r)\) is an increasing sequence of integers with \(k_ 0 =0,\) \(h_ r= k_ r- k_{r-1}\to \infty\) as \(r\to\infty\) and \(I_ r\) is the interval \((k_{r-1}, k_ r]\). These ideas are attributed to \textit{A. R. Freedman}, \textit{J. J. Sember} and \textit{M. Raphael} [Proc. Lond. Math. Soc., III. Ser. 37, 508--520 (1978; Zbl 0424.40008)]. Let \(V_\sigma= \{x\in m\mid\lim_p t_{pn}(x)=\sigma\)-\(\lim x\), uniformly in \(n\}\), where \(t_{pn}(x):= (x_n+ (Tx)_n+\cdots+(T^px)_n)/(p+ 1)\) for \(p= 0,1,\dots, n= 1,2,\dots\), and \(Tx:=(x_{\sigma(n)})\). In this paper, the author defines a new sublinear functional involving lacunary sequences to characterize \(V_\sigma\) and investigates some inequalities which are very similar to the Knopp core theorem.