Almost everywhere and norm convergence of Approximate Identity and Fejér means of trigonometric and Vilenkin systems

Abstract: In this paper, we investigate very general approximation kernels with special properties, called an approximate identity, and prove almost everywhere and norm convergence of these general methods, which consists of a class of summability methods and provide norm and a.e. convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fej\'er means with respect to the Vilenkin system also, but these methods are not useful to study a.e. convergence in this case, because of some special properties of the kernels of Fej\'er means. Despite these different properties we give alternative methods to prove almost everywhere convergence of Fej\'er means with respect to the Vilenkin systems.