On the Lebesgue test for the convergence of Vilenkin-Fourier series (Q1803915)

In this note the authors adapt Waterman's localization of the Salem test to obtain an extension of the Onneweer-Waterman Lebesgue test for convergence of Fourier series of functions defined on a bounded Vilenkin group. The main result is Theorem 3.3: Let \(G\) be a bounded Vilenkin group, and suppose that \(f\in L^ 1(G)\). Then the Fourier series of \(f\) converges to \(f^*(x)\) at every point \(x\in G\) at which the following conditions hold: (S1) \(f^*(x)\) exists, and \[ \lim_{k\to\infty}{1\over m(G_{k+1})}\int_{G_{k+1}}\sum^{m_ k- 1}_{\alpha=1}{1\over\alpha}\Bigl|\sum^{p _{k+1}-1}_{j=0}f(x- z^{(k)}_ \alpha - jx_ k -t)\zeta^{ja_ k}_ k\Bigr| dt = 0\tag{S2} \] \[ \text{ uniformly in }a_ k\in\{1,2,\dots,p_{k+1}-1\}. \] The convergence is uniform in \(x\) on any closed set of points where \(f\) is continuous and (S2) holds uniformly.