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Some sequence spaces defined by Orlicz functions. (English) Zbl 0966.46002
A lacunary sequence \(\theta= (k_r)\), \(r= 0,1,2,\dots\) with \(k_0= 0\), \(k_r-k_{r-1}\to \infty\) is given. The intervals determined by \(\theta\) are \(I_r= (k_{r-1}, k_r]\). Let \(h_r= k_r-k_{r-1}\). Define \[ [N_\theta, M,p]= \Biggl\{(x_k): \lim_{r\to\infty} h^{-1}_r \sum_k\Biggl[M\Biggl({|x_k- \ell|\over\rho}\Biggr)\Biggr]^{p_k}= 0\text{ for some }\ell\text{ and }\rho>0\Biggr\}. \] Similarly we can define \([N_\theta, M,p]_0\) and \([N_\theta, M,p]_\infty\). The authors show that these three spaces are linear spaces over \(\mathbb{C}\); \([N_\theta, M,p]_0\) is a topological linear space; \(0< p_k\leq q_k\) and \((q_k/p_k)\) is a bounded sequence \(\Rightarrow [N_\theta, M,q]\subseteq [N_\theta, M,p]\).
There are two more results in this paper.

46A45 Sequence spaces (including Köthe sequence spaces)
40H05 Functional analytic methods in summability
40A05 Convergence and divergence of series and sequences
40D25 Inclusion and equivalence theorems in summability theory
40F05 Absolute and strong summability
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