On the Fejér kernel functions with respect to the character system of the group of \(2\)-adic integers

Abstract. It was a long time open problem the almost everywhere convergence of the Cesàro \((C,1)\)-means of integrable functions \(\sigma _nf \to f\) for \(f\in L^1(I)\), where \(I\) is the group of \(2\)-adic integers. In his paper the author verified [2] this a.e. relation with the help of an inequality for the integral of the maximal function of the Fejér kernels. Later, the author also discussed [3] the almost everywhere convergence of the Cesàro \((C,\alpha)\)-means for every \(\alpha>0\). In general, in the investigation of the Fejér means with respect to convergence or divergence issue, a formula for the kernel function plays an important role. In the Walsh–Paley case see [1] or alternatively [7]. The Walsh–Kaczmarz version is due to Skvortsov [10], [9]. For the time being, there is no known formula for this \(2\)-adic integers case. This paper fills this gap.