Absolute convergence of multiple series of Fourier coefficients with respect to multiplicative systems (Q1387396)

\textit{C. W. Onneweer} [Duke Math. J. 41, 679-688 (1974; Zbl 0294.43009)] obtained a bounded Vilenkin system analogue of a theorem of Szász, namely a sufficient condition (in terms of the modulus of continuity of a function \(f\in L^p\), \(1< p\leq 2\)) for the Vilenkin-Fourier coefficients of \(f\) to belong to \(\ell^\beta\) for small \(\beta\). In this paper, the author announces that Onneweer's result can be generalized to multiple Vilenkin systems of bounded type, but that if even one of the systems is of unbounded type, then there are functions in \(L^p\), \(1<p\leq 2\), which satisfy the condition but whose Vilenkin-Fourier coefficients do not belong to \(\ell^\beta\) for small \(\beta\).