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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

REFERENCES

(1)Hardy, G. H.Divergent series (Clarendon Press, London, 1949).Google Scholar
(2)Kurtz, J. C.Multipliers on some sequence spaces. Proc. Cambridge Philos. Soc. 72 (1972), 393.CrossRefGoogle Scholar
(3)Kurtz, J. C. and Sledd, W. T.Uniform summability of power series. Proc. Cambridge Philos. Soc. 71 (1972), 335–342.CrossRefGoogle Scholar
(4)Lorentz, G. G. and Zeller, K.Abschnittslimitierbarkeit und der Sats von Hardy-Bohr. Arch. Math. (Basel) 15 (1964), 208–213.CrossRefGoogle Scholar
(5)Meyer-König, W. and Zeller, K.Lückenumkehrsātze und Lückenperfektheit. Math. Z. 66 (1956), 203–224.CrossRefGoogle Scholar
(6) Peyerimhoff, A.Lectures on summability (Springer-Verlag; Berlin, 1969).CrossRefGoogle Scholar
(7) Titchmarsh, E. C.The theory of functions (Oxford University Press, 1939).Google Scholar
(8)Zeller, K.Approximation in Wirkfeldern von Surnmierungsverfahren. Arch. Math. 4 (1953), 425–431.CrossRefGoogle Scholar