Almost everywhere \((C, \alpha, \beta > 0)\)-summability of the Fourier series of functions on the \(2\)-adic additive group (Q2341942)

For the group \(\mathbb I\) of 2-adic integers the a.e.-convergence of Fourier series of functions \(f\) on \(\mathbb I^2\) with respect to Cesàro means \(C^{\alpha,\beta}_{m,n}\) is shown for functions in \[ L\log^+L(\mathbb I^2)=\{f\in L^1(\mathbb I^2):\,\int| f| \log^+| f| <\infty\}. \] This generalizes one-dimensional results obtained previously in [the first author, ibid. 116, No. 3, 209--221 (2007; Zbl 1136.42022)]. It is shown that this space is the maximal space of convergence for all \(\alpha,\beta>0\).