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On the Dirichiet product of Cesàro-summable series

Published online by Cambridge University Press:  24 October 2008

K. A. Jukes
K. A. Jukes
Affiliation:
University College of Wales, Aberystwyth
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Extract

Segal(1) in his paper ‘Summability by Dirichlet convolutions’ makes direct use (in the proofs of Theorems 4 and 6) of a result on the Dirichlet product of two series which may be stated as follows:

is (C, k)-summable to the sum α (k≥0) and is (C, l)-summable to the sum β, (l ≥ 0), then the Dirichlet product

is (C, k + l + 1)-summable to the sum αβ.’

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 3 , November 1969 , pp. 563 - 567
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Segal, S. L.Summability by Dirichlet convolutions. Proc. Cambridge Philos. Soc. 63 (1967), 393–400.CrossRefGoogle Scholar
(2)Hardy, G. H.Divergent Series (Oxford, 1949).Google Scholar
(3)Hardy, G. H. and Riesz, M.The general theory of Dirichlet's series. Cambridge Tracts in Mathematics No. 18 (Cambridge, 1915).Google Scholar
(4)Pennington, W. B.On Ingham summability and summability by Lambert series. Proc. Cambridge Philos. Soc. 51 (1955), 65–80.CrossRefGoogle Scholar