On the absolute convergence of a series associated with a Fourier series

Abstract

(1. 7) 4'(t) = {f(x + t) f(x t) }/2. The ordinary Cesaro summability of the series (1.3) was first studied by Hardy and Littlewood [5] who observed that the relation of (1.3) to the integral fo(4(t)/t)dt is very similar to that between the allied series and the integral fo(#(t)/t)dt. Zygmund [10] gave a necessary and sufficient condition for the convergence of the same series (1.3). The object of the present note is to study the absolute convergence and absolute Riesz summability of (1.3). We prove the following THEOREM 1. If (1) 4)1(t) log k/t is of bounded variation in (0, 7r), Received by the editors November 28, 1955.