# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by Lloyd A. Gavin
Proc. Amer. Math. Soc. 56 (1976), 130-134 Request permission

## Abstract:

Bosanquet  showed that a necessary and sufficient condition for $\Sigma _{k = 1}^\infty {x_k}{y_k}$ to be Cesàro summable of order $n$ ($n$ is a nonnegative integer) whenever $\sigma _k^n(y) = o(k)$ where $\sigma _k^n(y)$ is the $k$ th Cesàro mean of $y$ of order $n$ is that $\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| < \infty$ and ${\lim _{k \to 0}}k{x_k} = 0$. The main result of this paper is to show that a necessary and sufficient condition for $\Sigma _{k = 1}^\infty {x_k}{y_k}$ to be Cesàro summable of order $n$ ($n$ is a nonnegative integer) whenever $\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| < \infty$ and ${\lim _{k \to \infty }}k{x_k} = 0$ is that $\sigma _k^n(y) = o(k)$.
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