Generalized Absolute Convergence of Single and Double Series in Multiplicative Systems

Series of one- and two-dimensional Fourier coefficients in multiplicative systems $\chi$ (with bounded generating sequence ${\mathbf P}=\{p_i\}^\infty_{i=1}$) 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 ${\mathbf P}$-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 $(\Lambda,\Psi)$-fluctuation are also established.