Almost everywhere $${(C, \alpha, \beta > 0)}$$ ( C , α , β > 0 ) -summability of the Fourier series of functions on the 2-adic additive group

Abstract

We investigate Cesaro summability of bivariate integrable functions on the 2-adic group. We prove the a.e. convergence of 2-adic Cesaro means \({\sigma^{\alpha, \beta}_{n, m} f \rightarrow f}\) as \({n, m \rightarrow \infty}\) for functions \({f \in\,L\,{\rm log}^{+}\,L(\mathbb{I}^2)}\) and \({\alpha, \beta > 0}\). Then it is shown that this convergence result can not be improved in the Pringsheim sense, that is, \({L\,{\rm log}^{+}\,L}\) is the maximal convergence space for \({\sigma^{1, 1}_{n, m}}\) when there are no conditions for the indices except that they tend to infinity. We prove that for all measurable functions \({\delta : [o, \infty) \rightarrow [o, \infty)}\) for which \({{\rm lim}_{t \rightarrow \infty} \delta(t) = 0}\) there is a function \({f \in\,L\,\log^{+}\,L\, \delta(L)}\) with lim sup \({|\sigma_{2^{n_{1}}, 2^{n_{2}}} f(x)| = +\infty}\) a.e. as \({{\rm min}\,\{n_{1}, n_{2}\} \rightarrow \infty}\).