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Proceedings of the American Mathematical Society

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

 
 

 

The Riesz summability of logarithmic type


Author: B. Kwee
Journal: Proc. Amer. Math. Soc. 45 (1974), 365-370
MSC: Primary 40D25
DOI: https://doi.org/10.1090/S0002-9939-1974-0348326-9
MathSciNet review: 0348326
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Abstract: The series $ \Sigma _{n = 1}^\infty {a_n}$ is said to be summable $ (L)$ to $ s$ if $ {(\log (1 - x))^{ - 1}}\Sigma _{n = 1}^\infty {s_n}{x^{n + 1}}/n$, where $ {s_n} = \Sigma _{v = 1}^n{a_v}$, converges for $ 0 \leq x < 1$ and tends to $ s$ when $ x \to 1 - $. The aim of this paper is to discuss the relation between summability $ (L)$ and Riesz summability $ (R,\log n,\kappa )$. It is proved that $ (R,\log n,\kappa ) \subseteq (L)$ holds for $ 0 \leq \kappa \leq 1$ and is false for $ \kappa > 1$. It is also proved that if $ \Sigma _{n = 1}^\infty {a_n} = s(L)$ and bounded $ (R,\log n,\kappa )$ for $ \kappa \geq 0$ then $ \Sigma _{n = 1}^\infty {a_n} = s(R,\log n,\kappa + \delta )$ for every $ \delta > 0$.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1974-0348326-9
Article copyright: © Copyright 1974 American Mathematical Society

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