Cesaro Summability of a Glass of Functions

This note suggests a further use for a difference formula of L. S. Bosanquet which has already proved to be of service in dealing with certain problems involving Cesaro and Riesz means. More particularly, it is the object of the note to establish, by means of Bosanquet's formula, the integral analogue of a theorem on series, due to M. S. Macphail, [Theorem 2], which contains as a special case a result of H. L. Garabedian. Macphail's theorem itself can be obtained, as shown in the last section of the note, by an obvious adaptation of the argument used to prove its integral analogue.