Tauberian conclusions
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- by K. A. Jukes and I. J. Maddox PDF
- Proc. Amer. Math. Soc. 53 (1975), 407-411 Request permission
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
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G. H. Hardy, Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc (2) 8 (1910), 301-320.
- A. E. Ingham, Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc. 20 (1945), 171–180. MR 17392, DOI 10.1112/jlms/s1-20.3.171 J. E. Littlewood, The converse of Abel’s theorem on power series, Proc. London Math. Soc. (2) 10 (1910/11), 434-448.
- I. J. Maddox, Elements of functional analysis, Cambridge University Press, London-New York, 1970. MR 0390692
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 53 (1975), 407-411
- MSC: Primary 40E05
- DOI: https://doi.org/10.1090/S0002-9939-1975-0404919-2
- MathSciNet review: 0404919