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On dual summability methods

Published online by Cambridge University Press:  24 October 2008

B. Kuttner
B. Kuttner
Affiliation:
University of Birmingham
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Extract

1. Let A be a summability method given by the sequence-to-sequence transformation

We suppose throughout that, for each n

converges; this is a much weaker assumption than the regularity of A. Then we define

We also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say that

Let A' denote the summability method given by the series-to-sequence transformation

Following Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 67 - 73
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Cooke, R. G.Infinite matrices and sequence spaces (London, 1950).Google Scholar
(2)Darevsky, V.On intrinsically perfect methods of summation. Izv. Akad. Nauk, S.S.S.R. (Ser. Mar., N.S.), 10 (1946), 97–104 (Russian).Google Scholar
(3)Kwee, B.The relation between the sequence-to-sequence and series-to-series versions of quasi-Hausdorff summability methods. Proc. Amer. Math. Soc. 19 (1968), 45–49.CrossRefGoogle Scholar
(4)Lobentz, G. G.Über Limitierungsverfahren die von einem Stieltjes-Integral abhängen. Acta Math. 79 (1947), 255–272.CrossRefGoogle Scholar
(5)Lorentz, G. G. & Zeller, K.Summation of sequences and summation of series. Proc. Amer. Math. Soc. 15 (1964), 743–746.CrossRefGoogle Scholar
(6)Zeller, K.Merkwürdigkeiten bei Matrixverfahren: Einfolgenverfahren. Arch. Math. (Basel) 4 (1953), 1–5.CrossRefGoogle Scholar