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On a scale of Abel-type summability methods

Published online by Cambridge University Press:  24 October 2008

D. Borwein
D. Borwein
Affiliation:
St Salvator's College University of St Andrews
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Extract

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.

Let

and let {sn} be any sequence of numbers. If

is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 318 - 322
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Amir, (Jakimovski) A.Some relations between the methods of summability of Abel, Borel, Casàro, Holder and Hausdorff. J. Anal. Math. 3 (1954), 346–81.Google Scholar
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(3)Hardy, G. H.Divergent Series (Oxford, 1949).Google Scholar
(4)Szász, O.On the product of two summability methods. Ann. Polon. Math. 25 (1952), 75–84.Google Scholar