On some new sequence spaces of invariant means defined by Orlicz functions. (English) Zbl 1010.46006

By combining the concept of invariant means, lacunary convergence and Orlicz functions \(M\), the authors define some sequence spaces and study their properties. Let \(p=(p_k)\) be a sequence of positive numbers and \(\theta =(k_r)\) be a lacunary sequence with \(h_r:=k_r-k_{r-1} \to\infty\). Denote \[ [w,M,p]_\sigma: =\{x=(x_k) \mid\lim_n {1\over n+1} \sum^n_{k=0} \left[M\left( {\bigl |t_{km}(x-l) \bigr|\over\rho} \right)\right]^{p_k}=0 \]
\[ \text{for some }l\text{ and }\rho>0, \text{ uniformly in }m\} \] and \[ [w^\theta, M, p]_\sigma: =\{x=(x_k) \mid\lim_r {1\over h_r} \sum^{k_r}_{k=k_{r-1}+1}\left[ M \left({\bigl |t_{km}(x-l)\bigr |\over \rho}\right) \right]^{p_k}=0 \]
\[ \text{for some }l\text{ and }\rho>0, \text{ uniformly in }m\}, \] where \(t_{km} (x):=(x_m+x_{\sigma(m)} +x_{\sigma^2(m)} +\cdots+x_{\sigma^k(m)})/(k+1)\) and \(\sigma: \mathbb{N}\to \mathbb{N}\) is the translation mapping \(n\mapsto n+1\). The authors show that (a) \([w,M,p]_\sigma \subset[w^\theta,M,p]_\sigma\) if \(\lim \inf_r q_r>1\) and (b) \([w^\theta,M,p]_\sigma \subset[w,M,p]_\sigma\) if \(\lim\sup q_r<\infty\) (here \(q_r:= k_r/k_{r-1}\), \(r=0,1,\dots)\).


46A45 Sequence spaces (including Köthe sequence spaces)
40D25 Inclusion and equivalence theorems in summability theory
40H05 Functional analytic methods in summability
40A05 Convergence and divergence of series and sequences
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