Fridli, S.; Ivanov, V.; Simon, P. Representation of functions in the space \(\phi\) (L) by Vilenkin series. (English) Zbl 0589.42018 Acta Sci. Math. 48, 143-154 (1985). Let \(\phi\) be an even real function non-decreasing \([0,+\infty)\) for which i) \(\phi (0)=\phi (+0)=0\), ii) \(\phi (x)>0\) \((x>0)\), iii) \(\phi (2x)=O(\phi (x))\) (x\(\to \infty)\) hold. \(\phi\) (L) denotes the set of measurable and a.e. finite functions defined on [0,1] with \(\int^{1}_{0}\phi (f(x))dx<\infty\). We solve the so-called Uljanov’s problem for Vilenkin systems, namely we prove that the Vilenkin systems are systems of representation in \(\phi\) (L) if and only if either \(\phi (L)\not\subset L_ 1[0,1]\) or \(\phi\) (L) is equivalent to a separable Orlicz space. On the other hand we characterize the Orlicz spaces in which the Vilenkin systems are Schauder bases showing that the Vilenkin systems are bases in a separable Orlicz space if and only if the space if reflexive. Cited in 2 ReviewsCited in 3 Documents MSC: 42C15 General harmonic expansions, frames Keywords:Vilenkin systems; Orlicz spaces; Schauder bases PDFBibTeX XMLCite \textit{S. Fridli} et al., Acta Sci. Math. 48, 143--154 (1985; Zbl 0589.42018)