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

2022/05/14 by N. Nadirashvili, Nadirashvili, N., G. Tephnadze +3
Mathematics · #42C10 #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2205.07876

openalex publication_date 2022/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Fejér 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 Fejér means. Despite these different properties we give alternative methods to prove almost everywhere convergence of Fejér means with respect to the Vilenkin systems.

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