On the absolute summability and convergence of Fourier series and associated series

Abstract

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x byWe denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the typeWe writeThe object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)