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Proceedings of the American Mathematical Society

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



Tauberian conclusions

Authors: K. A. Jukes and I. J. Maddox
Journal: Proc. Amer. Math. Soc. 53 (1975), 407-411
MSC: Primary 40E05
MathSciNet review: 0404919
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Abstract: Littlewood's celebrated Tauberian theorem states that $ \sum {a_n} = s$ (Abel) and $ n{a_n} = O(1)$ imply $ {s_n} = \sum _{k = 1}^n{a_k}$ converges to $ s$, the Tauberian condition $ n{a_n} = O(1)$ being best possible. We investigate 'best possibility' of the conclusion $ {s_n} - s = o(1)$, replacing the usual Tauberian condition by $ ({q_n}{a_n})\epsilon E$ where $ ({q_n})$ is a positive sequence and $ E$ a given sequence space.

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

  • [1] G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc (2) 8 (1910), 301-320.
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  • [4] I. J. Maddox, Elements of functional analysis, Cambridge University Press, London-New York, 1970. MR 0390692

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Keywords: Tauberian theorems, best possible conclusions, Abel, Ingham methods
Article copyright: © Copyright 1975 American Mathematical Society