The weighted $L^1$-integrability of functions and the Parseval equality with respect to multiplicative systems

In this paper we prove necessary and sufficient conditions for the weighted $L^1$-integrability of functions defined on $[0,1)$ in terms of Fourier coefficients with respect to a multiplicative system of bounded type. These results are counterparts of trigonometric ones by M. and S. Izumi and M. M. Robertson.