On the $L^1$-convergence of series in multiplicative systems

In the paper two analogs of Garrett–Stanojević trigonometric results are established for multiplicative systems $\{\chi_n\}_{n=0}^\infty$ of bounded type. First, the modified partial sums of a series $\sum\limits^\infty_{k=0}a_k\chi_k$ with coefficients of bounded variation converge in $L^1[0,1)$ to its sum if and only if for all $\varepsilon>0$ there exists $\delta>0$ such that $\int^\delta_0\left|\sum\limits^\infty_{k=n}(a_k-a_{k+1})D_{k+1}(x)\right|\,dx<\varepsilon, \quad n\in\mathbb Z_+,$ where $D_{k+1}(x)=\sum\limits^k_{i=0}\chi_i(x)$. Secondly, if $\lim\limits_{n\to\infty}a_n\ln(n+1)=0$ and $\sum\limits^\infty_{k=n}|a_k-a_{k+1}|\leq Ca_n$, $n\in\mathbb Z_+$, then the series $\sum\limits^\infty_{n=0}a_n\chi_n(x)$ converges to its sum $f(x)$ in $L^1[0,1)$ if and only if $f\in L^1[0,1)$.