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Absolute summability of infinite series on a scale of Abel type summability methods

Published online by Cambridge University Press:  24 October 2008

Babban Prasad Mishra
Babban Prasad Mishra
Affiliation:
Department of Mathematics, The University, Gorakhpur, India
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Extract

Suppose that λ > − 1 and that

It is easy to show that

With Borwein(1), we say that the sequence {sn} is summable Aλ to s, and write sn → s(Aλ), if the series

is convergent for all x in the open interval (0, 1)and tends to a finite limit s as x → 1 in (0, 1). The A0 method is the ordinary Abel method.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 2 , April 1968 , pp. 377 - 387
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Borwein, D.On a scale of Abel type summability methods. Proc. Cambridge Philos. Soc. 53 (1957), 318–322CrossRefGoogle Scholar
(2)Hardy, G. H. and Littlewood, J. E.Some properties of fractional integrals I. Math. Z. 27 (1928), 565–606.CrossRefGoogle Scholar
(3)Flett, T. M.On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. London Math. Soc. 7 (1957), 113–141.CrossRefGoogle Scholar
(4)Flett, T. M.Some theorems on power series. Proc. London Math. Soc. 7 (1957), 211–218.CrossRefGoogle Scholar
(5)Kogbetliantz, E.Sur les séries absolument sommables par la méthode des moyennes arithmétiques. Bull. Sci. Math. 49 (1925), 234–256.Google Scholar
(6)Mishra, B. P.Strong summability of infinite series on a scale of Abel type summability methods. Proc. Cambridge Philos. Soc. 63 (1967), 119–127.CrossRefGoogle Scholar
(7)Mulholland, H. P.Some theorems on Dirichlet series with positive coefficients and related integrals. Proc. London Math. Soc. 29 (1929), 281–292.CrossRefGoogle Scholar