Let \(P=\{p_n\}\) be a sequence of natural numbers betwen 2 and a fixed \(N\) and let \(G=G(P)\) denote the direct product of \(\prod_{n=1}^\infty \mathbb{Z}_{p_n}\) of integers mod \(p_n\). The subgroups \(G_n\) are defined by those \(x\in G\) whose first \(n\) coordinates are the respective identities in \(\mathbb{Z}_{p_k}\), \(k=1,\dots, n\), which have cosets \(G_n(y)=\{x\in G: x_1=y_1,\dots, x_n=y_n\}\). Set \(r_k(x)=\exp(2\pi i x_k/p_k)\) and define mutliplicative characters \(\chi_n(x)=\prod r_k^{n_k}(x)\) where \(n=\sum_{k=1}^\infty n_k m_{k-1}\), (\(m_n=p_1\cdots p_n\)) is the \(P\)-adic representation of \(n\). The Fourier coefficients \(\hat{f}(n)\) are defined by integration against characters with respect to the Haar measure on \(G\) and analogues \(D_n=\sum_{k=0}^{n-1} \chi_k\) of the Dirichlet kernel and partial sums operators \(S_nf=\sum_{k=0}^{n-1} \hat{f}(n)\, \chi_k \) are defined as usual. Also, \(\mathcal{P}_n=\{f\in L^1(G): \hat{f}_k=0, k\geq n\}\).
Define moduli of continuity of \(f\) on \(G\) by \[ (\omega_n f)_\infty=\sup_{h\in G_n}\| f(x\oplus h)-f(x)\|_\infty,\quad n\in \mathbb{Z}_+ \] and \[ (E_nf)_\infty=\inf\{\| f-t_n\|_\infty: t_n\in \mathcal{P}_n\}\, . \]
This article establishes multiplicative analogues of the following result of V. Totik [Acta Math. Acad. Sci. Hung. 35, 151–172 (1980; Zbl 0454.42001)] for Fourier series: \[ \Bigl(\frac{1}{r}\sum_{i=1}^r|(S_{k_i}^Tf)(x)-f(x)|^p\Bigr)^{1/p}=O(E_{k_1}^Tf)_\infty\log (2n/r)),\quad 0<k_1<\dots<k_r\leq n, \] and necessary and sufficient conditions for the inequality \[ \frac{1}{n}\sum_{k=n+1}^{2n}\Phi(|(S_k^Tf)(x)-f(x)|)\leq K\Phi(E_n^Tf)_\infty),\quad n\in\mathbb{N} \] to hold for every continuous \(f\). In the Fourier case the necessary and sufficient conditions are that \(\Phi(t)\leq e^{A t}\), \(0<t<\infty\), for some \(A>0\), and \(\Phi(2t)\leq A\Phi(t)\) in \((0,1)\). The sufficiency of these conditions is proved here for the case of multiplicative Fourier series. The authors also provide a certain characterization of weak differentiability using strong de la Vallée Poussin means and strong power means.