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Uniform summability and Töplitz bases

Published online by Cambridge University Press:  24 October 2008

J. C. Kurtz
J. C. Kurtz
Affiliation:
Michigan State University
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Extract

If the series Σαn is convergent, it follows that Σαnzn is uniformly convergent in a Stolz angle at z = 1 ((7), p. 229). It was shown in (3) that for the Ces methods (C, α) with α > – 1, and for a certain class of general Nörlund means (containing (C, α) for 0 < α ≤ 1), summability of Σαn implies uniform summability in a Stolz angle at z = 1. In section 2 we prove this theorem for a class of methods which satisfy a mean value property ((6), p. 31), and for a wider class of Nörlund methods which includes (C, α) for all α > 0. An analog is also proved for absolute summability.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 73 - 81
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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