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Keywords:
generalized absolute convergence; Vilenkin-Fourier series; modulus of continuity; multiplicative system
generalized absolute convergence; Vilenkin-Fourier series; modulus of continuity; multiplicative system
Summary:
We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\hat {f}(n)$, $n\in \mathbb {N}$, of functions $f\in L^p(G)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series $\sum _{n=1}^{\infty }a_n|\hat {f}(n)|^r$, where $\{a_n\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative analogue of some results due to Móricz (2010), Móricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020).
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