# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

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

## Abstract:

Bosanquet [2] 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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