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A Cauchy criterion for the summability of an infinite integral
Published online by Cambridge University Press: 20 January 2009
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If a ∈ L(1, T) for every finite T>1, then we say that the infinite integral 
a(u)du is convergent with sum s if 
a(u)du = s. It is well known that a necessary and sufficient condition for a(u)du to be convergent (with some finite sum s) is that Cauchy's criterion,
holds. The object of this note is to obtain a similar result for summability (C, α) of a(u) du which reduces to Cauchy's criterion in the case of convergence. The corresti ponding problem for summable series has been treated by A. F. Andersen in (1).
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- Copyright © Edinburgh Mathematical Society 1980
References
REFERENCES
(1)Andersen, A. F., A condition for C-summability of negative order, Math. Scand. 7 (1959), 337–346.CrossRefGoogle Scholar
(2)Bosanquet, L. S., Some extensions of M. Riesz's mean value theorem, Indian J. Math. 9 (1967), 65–90.Google Scholar
(3)Hardy, G. H., Theorems connected with Maclaurin's test for the convergence of series, Proc. London Math. Soc. (2) 9 (1911), 126–144.CrossRefGoogle Scholar
(4)Hardy, G. H., Notes on some points in the integral calculus XLVIII: On some properties of integrals of fractional order, Math. Messenger 47 (1918), 145–150.Google Scholar
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