In the paper two analogs of Garrett – Stanojevic´ trigonometric results are established for multiplicative systems {χn} ∞n=0 of bounded type. First, the modified partial sums of a series P∞ k=0 akχk with coefficients of bounded variation converge in L 1 [0, 1) to its sum if and only if for all ε > 0 there exists δ > 0 such that R δ 0 ¯ ¯ ¯ ¯ P∞ k=n (ak − ak+1)Dk+1(x) ¯ ¯ ¯ ¯ dx < ε, n ∈ Z+, where Dk+1(x) = Pk i=0 χi(x).
