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On Riesz and Riemann Summability

Published online by Cambridge University Press:  24 October 2008

H. Burkill
H. Burkill
Affiliation:
The UniversitySheffield
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Extract

Several relations between the Cesàro and the Riemann methods of summation are known. For instance, Verblunsky(3) has shown that a series summable (C, k − δ), where k is a positive integer, is also summable (R,k+ 1) and Kuttner (2) has proved that, for k = 1, 2, summability (R, k) implies summability (C, k + δ). In this paper we consider Riesz's typical means and a generalized Riemann summability both of which are intimately connected with almost periodic functions. The result we establish is similar to Verblunsky's, except that we start from a Riesz mean of integral order k*.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 1 , January 1961 , pp. 55 - 60
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Burkill, H., and Petersen, G. M., Proc. Amer. Math. Soc. (in Press).Google Scholar
(2)Kuttner, B., Proc. Land. Math. Soc. (2), 38 (1935), 273–83.CrossRefGoogle Scholar
(3)Verblunsky, S., Proc. Camb. Phil. Soc. 26 (1930), 34–42.CrossRefGoogle Scholar