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Published online by Cambridge University Press: 24 October 2008
This note is a sequel to a former one, a knowledge of which will be assumed. We first prove a theorem which is analogous to O Theorem I, and is indeed a simple consequence of it. As an easy inference from this theorem we obtain a necessary condition that the series allied with a Fourier series may be summable (C, k) for all k >p, where p is any number greater than 1. We then show that the condition thus obtained is sufficient. We finally show that the same method can be applied to Fourier series. This note thus contains a solution of the Cesàro summability problem for Fourier series and for the allied series, which is the most precise one yet obtained.
* Proc. Camb. Phil. Soc. 26 (1930), 152–157. Referred to as O.CrossRefGoogle Scholar
† In this case we need only suppose that p > 0.
* For the details, see Sayers, E. I., Proc. Lond. Math. Soc. 31 (1930), 29–39, §§ 4–7.CrossRefGoogle Scholar
* The proof is very similar to an argument given by Paley, , Proc. Camb. Phil. Soc. 26 (1930), 173–203, § 3.CrossRefGoogle Scholar