Generalized Absolute Convergence of Single and Double Series in Multiplicative Systems

Series of one- and two-dimensional Fourier coefficients in multiplicative systems χ (with bounded generating sequence P={pi}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{P}} = \{ {p_i}\} _{i = 1}^\infty $$\end{document}) with weights satisfying Gogoladze—Meskhia-type conditions are studied. Sufficient conditions for the convergence of such series for continuous (in a generalized sense) functions and functions from P-ary Hardy space are established. The question of whether these conditions are unimprovable is investigated. Sufficient conditions for generalized absolute convergence for functions of bounded (Λ, Ψ)-fluctuation are also established.