Generalized absolute convergence of single and double Vilenkin-Fourier series
and related results
Nayna Govindbhai Kalsariya, Bhikha Lila Ghodadra
Received February 15, 2022. Published online March 23, 2023.
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
Affiliations: Nayna Govindbhai Kalsariya (corresponding author), Bhikha Lila Ghodadra, Department of Mathematics, Faculty of Science, The M. S. University of Baroda, Vadodara 390 002, India, e-mail: kalsariyanayna.g125@gmail.com, bhikhu_ghodadra@yahoo.com