On a convergence test of Hardy-Littlewood's type for fourier series

Abstract

Let ф(t) be an even function integrable in the Lebesgue sense and periodic with period 2п. Let Write By an indirect method based on the method of Riesz summability for the Fourier series, the author has established the following convergence test for the oscifiating series Σan. The theorem is as follows:THEOREM A [1]. If, for some Δ > 0 as t → +0 andthen фan converges to the sums = 0. Here K is an absolute constant independent of n.