Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T20:21:33.023Z Has data issue: false hasContentIssue false
  • English
  • Français

A General Tauberian Condition that Implies Euler Summability

Published online by Cambridge University Press:  20 November 2018

Mangalam R. Parameswaran
Mangalam R. Parameswaran*
Affiliation:
Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let V be any summability method (whether linear or conservative or not), 0 < p < 1 and s a real or complex sequence. Let Ep denote the matrix of the Euler method. A theorem is proved, giving a condition under which the V-summability of Eps will imply the Ep-summability of s. This extends, in generalized form, an earlier result of N. H. Bingham who considered the case where s is a real sequence and V = B (Borel's method). It is also proved that even for real sequences, the condition given in the theorem cannot be replaced by the condition used by Bingham.

Keywords

Primary: 40G05secondary: 40E99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994 , pp. 393 - 398
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Bingham, N. H., On Borel and Euler summability, J. London Math. Soc. (3) 29(1984), 141–146.Google Scholar
2. Gaier, D., Der allgemeine Lückenumkehrsatz für das Borel-Verfahren, Math. Z. 88(1965), 410–417.Google Scholar
3. Hardy, G. H., Divergent Series, Oxford, 1949.Google Scholar
4. Meyer-Kônig, W., Untersuchungen über einige verwandte Limitierungsverfahren, Math. Z. 52(1949), 257– 304.Google Scholar
5. Meyer-Konig, W. and Zeller, K., Lückenumkehrsätze and Lückenperfektheit, Math. Z. 66(1956), 203–224.Google Scholar
6. Parameswaran, M. R., On a generalization of a theorem of Meyer-König, Math. Z. 162(1978), 201–204.Google Scholar
7. Wilansky, A. and Zeller, K., Summation of bounded sequences, topological methods, Trans. Amer. Math. Soc. 78(1955), 501–509.Google Scholar
8. Zeller, K., Merkwürdigkeiten bei Matrixverfahren; Einfolgenverfahren, Arch. Math. 4(1953), 271–277.Google Scholar
9. Zeller, K. and Beekmann, W., Théorie der Limitierungsverfahren, Springer-Verlag, Berlin, Heidelberg, New York, 1970.Google Scholar