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A criterion for the (e, c)-summability of Fourier series

Published online by Cambridge University Press:  24 October 2008

Jamil A. Siddiqi
Affiliation:
Laval University, Quebec

Extract

1. A series with partial sums {An} is said to be summable (e, c) (c > 0) if

exists, where it is to be understood that An+k = 0 when n+k < 0. The (e, c)-sum-mability method which is a regular method of summation was introduced by Hardy and Littlewood (3) (cf. also (5)) as an auxiliary method to prove the Tauberian theorem for Borel summability, viz, if Σan is summable (B) to A and an = 0(n−½), then it converges to A.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Gronwall, T.Über die Lebesguesehen Konstanten bei den Fourierschen Reihen. Math. Ann. 72 (1912), 244261.CrossRefGoogle Scholar
(2)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(3)Hardy, G. H. and Littlewood, J. E.Theorems concerning the summability of series by Borel's exponential method. Rend. Circ. Mat. Palermo 41 (1916), 3653.CrossRefGoogle Scholar
(4)Hardy, G. H. and Littlewood, J. E.Some new convergence criteria for Fourier Series. Annali di Pisa 3 (1934), 4362.Google Scholar
(5)Hardy, G. H. and Littlewood, J. E.On the Tauberian theorem for Borel summability. J. London Math. Soc. 18 (1943), 194200.CrossRefGoogle Scholar
(6)Zygmund, A.Trigonometric series, vol. 1 (Cambridge, 1959).Google Scholar