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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some remarks on summability factors


Author: Lloyd A. Gavin
Journal: Proc. Amer. Math. Soc. 56 (1976), 130-134
MSC: Primary 40G05
MathSciNet review: 0412664
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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}}\vert{\Delta ^{n + 1}}{x_k}\vert < \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}}\vert{\Delta ^{n + 1}}{x_k}\vert < \infty $ and $ {\lim _{k \to \infty }}k{x_k} = 0$ is that $ \sigma _k^n(y) = o(k)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0412664-3
PII: S 0002-9939(1976)0412664-3
Keywords: Summabllity factors
Article copyright: © Copyright 1976 American Mathematical Society