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

Savas, E.; Rhoades, B. E.

Math. Inequal. Appl. 5, No. 2, 271-281 (2002).

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 \(q_r:= k_r/k_{r-1}\), \(r=0,1,\dots)\).

Reviewer: Toivo Leiger (Tartu)

MSC:

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

Keywords:

invariant means; Orlicz sequence spaces; inclusion of sequences spaces