Integrability of series with respect to multiplicative systems and generalized derivatives (Q6599709)

Let \(\left\{p_n\right\}_{n=1}^{\infty}\subset \mathbb N\), \( p_n \geq 2\), be uniformly bounded. Each \(x \in[0,1)\) has the expansion \[ x=\sum_{n=1}^{\infty} x_n / (p_1 \ldots p_n), \quad x_n \in \mathbb{Z}, \quad 0 \leq x_n<p_n . \] If additionally \(k \in \mathbb N \cup \{0\}\) is written as \[ k=k_0 + \sum_{i=2}^{\infty} k_i p_1 \ldots p_{i-1}, \quad k_i \in \mathbb{Z}, \quad 0 \leq k_i<p_i, \] then the set of functions \(\left\{\chi_k(x)\right\}_{k=0}^{\infty}\), with \[ \chi_k(x)=\exp \left(2 \pi i \sum_{j=1}^{\infty} x_j k_j / p_j\right), \] is called the multiplicative system with the generating sequence \(\left\{p_n\right\}_{n=1}^{\infty}\).\N\NAmong other results, the authors prove that under some assumptions on a sequence \(\{a_n\}\) (it belongs to the \(S_{p\alpha r}\)-class, with \(1<p \leq 2\), \(\alpha \geq 0\), \(0 \leq r \leq \alpha\)),\N \[ \lim _{n \rightarrow \infty} a_n\left\| \sum_{k=0}^{n} k^r \chi_k(x) \right\|_1=0 \] is equivalent to the convergence of \[ S_n^{[r]}(x)=\sum_{k=0}^{n-1} a_k k^r \chi_k(x) \] in \( L^1[0,1)\) to some function \( g \in L^1[0,1) \).\N\NThis extends some previous results of \textit{W. O. Bray} and \textit{C. V. Stanojevic} [Trans. Am. Math. Soc. 275, 59--69 (1983; Zbl 0513.42004)] , \textit{C. V. Stanojevic} [Proc. Am. Math. Soc. 82, 209--215 (1981; Zbl 0462.42007)], and \textit{S. Sheng} [ibid. 110, No. 4, 895--904 (1990; Zbl 0717.42004)].