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A converse theorem on overconvergence of sequences of partial sums

Published online by Cambridge University Press:  24 October 2008

M. E. Noble
M. E. Noble
Affiliation:
The UniversityNottingham
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Extract

A number of authors ((2), (4), (5)) have recently considered problems relating to the convergence of specified sequences of partial sums of a gap Taylor series on its circle of convergence. For convenience in stating such results we shall standardize the Taylor series as Σanzn with the unit circle as circle of convergence, and denote the partial sums and the sum function by sn(z) and f(z) respectively. Following the notation of ((4)) and (5) we shall define φ(x) for 0 ≤ x < ∞ to be the least concave majorant of log (|an| + 2). One of the recent results in this field ((5), Theorem 1, and see also (2)) may be stated as

THEOREM A. Suppose that f(z) is regular on an arc α < arg z < β. Suppose also that there are sequences of integers nk, Nk → ∞ such that Nk – nk → ∞ and am = 0 if nk < m < Nk.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 3 , July 1957 , pp. 592 - 598
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Bourion, G.Récherches sur l'ultraconvergence. Ann. sci. Ée. norm, sup., Paris, 50 (1933), 245–318.Google Scholar
(2)Erdos, P. and Piraniian, G.Overconvergence on the circle of convergence. Duke Math. J. 14 (1947), 647–58.CrossRefGoogle Scholar
(3)Landau, E.Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie (New York, 1946).Google Scholar
(4)Noble, M. E.On Taylor series with gaps. J. Lond. Math. Soc. 28 (1953), 197–203.Google Scholar
(5)Noble, M. E.A further note on Taylor series with gaps. J. Lond. Math. Soc. 30 (1955), 220–8.Google Scholar
(6)Ostrowski, A.Über Potenzreihen die überkonvergente Abschittsfolgen bezitzen. S. B. preuss. Akad. Wiss. (1923), 185–92.Google Scholar