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Uniform summability of power series

Published online by Cambridge University Press:  24 October 2008

J. C. Kurtz  and
W. T. Sledd
J. C. Kurtz
Affiliation:
Michigan State University
W. T. Sledd
Affiliation:
Michigan State University
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Abstract

It is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.

If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.

Applications are made to theorems on summability factors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 2 , March 1972 , pp. 335 - 341
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

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