A criterion for the (e, c)-summability of Fourier series

Abstract

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) ifexists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.