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Representation of functions in the space \(\phi\) (L) by Vilenkin series. (English) Zbl 0589.42018

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.

MSC:

42C15 General harmonic expansions, frames
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