Convergence of Ces\'{a}ro means with varying parameters of Walsh-Fourier series

Abstract

In 2007 Akhobadze [1] (see also [2]) introduced the notion of Cesàro means of Fourier series with variable parameters. In the present paper we prove the almost everywhere convergence of the the Cesàro .C;˛n/ means of integrable functions ˛n n f ! f , where N˛; K 3 n ! 1 for f 2 L 1 .I /, where I is the Walsh group for every sequence˛D .˛n/, 0 <˛n < 1. This theorem for constant sequences˛that is,˛Á˛1 was proved by Fine [3].

Matrix Summability of Walsh–Fourier Series

“…The theorems can be used for various methods of summability. At the end of the article, the theorems are used in the case of Cesàro means with varying parameters; this new result improves the theorem of Gát and Abu Joudeh [6].…”

“…The almost everywhere convergence of the subsequence of Cesàro means with variable parameters has been studied by the following authors: Abu Joudeh and Gát [6], Gát and Goginava [13,14], Weisz [15].…”

Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh–Fourier series

“…The function P (n, α) was introduced by Abu Joudeh and Gát in [10]. Also, set for sequence α := {α n : n ∈ N} and positive reals K the subset of natural numbers…”

“…Under some conditions on {α n : n ∈ N} , Abu Joudeh and Gàt in [10] proved the almost everywhere convergence of the Cesàro (C, α n ) means of integrable functions. In particular, the following is proved.…”

Cesàro means with varying parameters of Walsh–Fourier series

“…divergence of Cesàro means with varying parameters of Walsh-Fourier series was investigated by Tetunashvili [25]. Abu Joudeh and Gát in [1] proved the almost everywhere convergence (with some restrictions) of the Cesàro (C, α n ) means of integrable functions.…”