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A counter-example in summability theory

Published online by Cambridge University Press:  24 October 2008

B. Kuttner
B. Kuttner
Affiliation:
University of Birmingham
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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

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References

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