Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T11:48:52.515Z Has data issue: false hasContentIssue false
  • English
  • Français

A counter-example in summability theory

Published online by Cambridge University Press:  24 October 2008

B. Kuttner
B. Kuttner
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Extract

Let Σ denote the set of all series

of complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 3 , May 1978 , pp. 353 - 355
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Darevsky, V.On intrinsically perfect methods of summation. Izv. Akad. Nauk S.S.S.R. (Ser. Mat. N.S.) 10 (1946), 97–104 (Russian).Google Scholar
(2)Kwee, B.On Perron's method of summation. Proc. Cambridge Philos. Soc. 63 (1967), 1033–1040.CrossRefGoogle Scholar
(3)Maddox, I. J.Some general Tauberian theorems. J. London Math. Soc. (2) 7 (1974), 645–650.CrossRefGoogle Scholar
(4)Maddox, I. J.Tauberian estimates. J. London Math. Soc. (2) 15 (1977), 143–146.CrossRefGoogle Scholar
(5)Meyer-König, W. and Tietz, H.Über die Limitierungsumkehrsätze vom Typ o. Studia Math. 31 (1968), 205–216.CrossRefGoogle Scholar
(6)Meyer-König, W. and Tietz, H.Über Umkehrbedingungen in der Limitierungstheorie. Arch. Math. (Brno) 5 (1969), 177–186.Google Scholar
(7)Stieglitz, M.Über ausgezeichnete Tauber-Matrizen. Arch. Math. (Brno) 5 (1969), 227–233.Google Scholar
(8)Tietz, H.Über absolute Tauber-Bedlingungen. Math. Z. 113 (1970), 136–144.CrossRefGoogle Scholar
(9)Zeller, K.Merkwürdigkeites bei Matrixverfahren; Einfolgenverfahren. Arch. Math. (Basel) 4 (1953), 1–5.CrossRefGoogle Scholar