Hostname: page-component-7bb8b95d7b-495rp Total loading time: 0 Render date: 2024-09-11T07:45:03.004Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform summability and Töplitz bases

Published online by Cambridge University Press:  24 October 2008

J. C. Kurtz
J. C. Kurtz
Affiliation:
Michigan State University
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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