Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.
  • [2] A. E. Ingham, Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc. 20 (1945), 171–180. MR 0017392 (8,147i)
  • [3] J. E. Littlewood, The converse of Abel's theorem on power series, Proc. London Math. Soc. (2) 10 (1910/11), 434-448.
  • [4] I. J. Maddox, Elements of functional analysis, Cambridge University Press, London, 1970. MR 0390692 (52 #11515)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40E05

Retrieve articles in all journals with MSC: 40E05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0404919-2
PII: S 0002-9939(1975)0404919-2
Keywords: Tauberian theorems, best possible conclusions, Abel, Ingham methods
Article copyright: © Copyright 1975 American Mathematical Society