Generalized absolute convergence of single and double Vilenkin-Fourier series and related results

Abstract

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 $1<p\le2$. 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).

Keywords

• generalized absolute convergence
• vilenkin-fourier series
• modulus of continuity
• multiplicative system