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An O-Tauberian theorem and a high indices theorem for power series methods of summability*

Published online by Cambridge University Press:  24 October 2008

David Borwein  and
Werner Kratz
David Borwein
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario, CanadaN6A 557
Werner Kratz
Affiliation:
Abteilung Mathematik, Universität Ulm, Helmholtzstr. 18, D-7900 Ulm/Donau, Germany
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Abstract

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We improve known Tauberian results concerning the power series method of summability Jp based on the sequence {pn} by removing the condition that pn be asymptotically logarithmico-exponential. We also prove an entirely new Tauberian result for rapidly decreasing pn.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 365 - 375
Copyright
Copyright © Cambridge Philosophical Society 1994

References

REFERENCES

[1]Borwein, D. and Kratz, W.. On relations between weighted means and power series methods of summability. J. Math. Anal. Appl. 139 (1989), 178–186.CrossRefGoogle Scholar
[2]Borwein, D. and Meir, A.. A Tauberian theorem concerning weighted means and power series. Math. Proc. Cambridge Phil. Soc. 101 (1987), 283–286.CrossRefGoogle Scholar
[3]Hardy, G. H.. Divergent Series (Oxford University Press, 1949).Google Scholar
[4]Jakimovski, A. and Tietz, H.. Regular varying functions and power series methods. J. Math. Anal. Appl. 73 (1983), 65–84CrossRefGoogle Scholar
Jakimovski, A. and Tietz, H.. Errata, J. Math. Anal. Appl. 95 (1983) 597–598.CrossRefGoogle Scholar
[5]Kiesel, R. and Stadtmüller, U.. Tauberian theorems for general power series methods. Math. Proc. Cambridge Phil. Soc. 110 (1991), 483–490.CrossRefGoogle Scholar
[6]Kiesel, R.. Taubersätze und starke Gesetze für Potenzreihenverfahren. Dissertation, Universität Ulm, 1990.Google Scholar
[7]Kratz, W. and Stadtmüller, U.. O-Tauberian theorems for Jp-methods with rapidly increasing weights. J. London Math. Soc. (2) 41 (1990), 489–502.CrossRefGoogle Scholar
[8]Kratz, W. and Stadtmüller, U.. Tauberian theorems for Borel-type methods of summability. Arch. Math. 55 (1990), 465–474.CrossRefGoogle Scholar
[9]Kwee, B.. A Tauberian theorem for the logarithmic method of summation. Proc. Cambridge Phil. Soc. 63 (1967), 401–405.CrossRefGoogle Scholar
[10]Tietz, H. and Trautner, R.. Taubersätze für Potentzreihenverfahren. Arch. Math. 50 (1988), 164–174.CrossRefGoogle Scholar
[11]Titchmarsh, E. C.. The Theory of Functions (Oxford University Press, 1947).Google Scholar