Series in multiplicative systems convergent to Denjoy-integrable functions

An example of a series in the Chrestenson–Levi system with $p_j=3$, $j=0,1,\dots$, with zero-convergent coefficients is constructed such that $\lim_{n\to\infty}S_{m_n}(x)=f(x)$ everywhere on $[0,1)$ for some function $f$ that is Denjoy integrable in the extended sense, but this series is not the Denjoy–Fourier series of $f$. A series in the Price system defined by a bounded sequence $\{p_j\}_{j=0}^\infty$ that converges everywhere on $[0,1)$ (with the possible exception of some countable set) to a function Denjoy integrable in the extended sense is proved to be Denjoy–Fourier series of this function.