Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-22T10:04:14.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Tauberian- and Convexity Theorems for Certain (N,p,q)-Means

Published online by Cambridge University Press:  20 November 2018

Rüdiger Kiesel  and
Ulrich Stadtmüller
Rüdiger Kiesel
Affiliation:
Universität Ulm, Abteilung Stochastik D-89069 Ulm, Germany, e-mail: kiesel@mathematik.uni-ulm.de, stamue@mathematik.uni-ulm.de
Ulrich Stadtmüller
Affiliation:
Universität Ulm, Abteilung Stochastik D-89069 Ulm, Germany, e-mail: kiesel@mathematik.uni-ulm.de, stamue@mathematik.uni-ulm.de
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.

The summability fields of generalized Nörlund means (N,p*α,p), α ∈ Ν, are increasing with a and are contained in that of the corresponding power series method (P,p). Particular cases are the Cesàro- and Euler-means with corresponding power series methods of Abel and Borel. In this paper we generalize a convexity theorem, which is well-known for the Cesàro means and which was recently shown for the Euler means to a large class of generalized Nörlund means.

Keywords

40E05Power series methodsgeneralized Nörlund meansconvexity theoremsTauberian theorems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 982 - 994
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Andersen, A. F., Studier over Cesàro's summabilitetsmetode, Dissertation, København, 1921.Google Scholar
2. Bingham, N. H. and Goldie, C. M., On one-sided Tauberian conditions, Analysis 3(1983), 159–188.Google Scholar
3. Bingham, N. H., Goldie, C. M. and Teugels, J. L., Regular Variation, Cambridge University Press, 1987. Google Scholar
4. Boos, J. and Tietz, H., Convexity theorems for the circle methods ofsummability, J. Comput. Appl. Math. 40(1992), 151–155.Google Scholar
5. Borwein, D., On summability methods based on Power Series, Proc. Royal Soc. Edinburgh Sect. A 64(1957), 342–349.Google Scholar
6. Borwein, D. and Kratz, W., An O-Tauberian theorem and a High Indices theorem for power series methods of summability, Math. Proc. Cambridge Philos. Soc. 115(1994), 365–375.Google Scholar
7. Das, G, On some methods of summability, Quart. J. Math. Oxford Ser. (2) 17(1966), 244–256.Google Scholar
8. Das, G, On some methods of summability II, Quart. J. Math. Oxford Ser. (2) 19(1968), 417–431.Google Scholar
9. Hardy, G. H., Divergent Series, Oxford Press, 1949.Google Scholar
10. Kiesel, R., General Nörlund transforms and power series methods, Math. Z. 214(1993), 273–286.Google Scholar
11. Kiesel, R. and Stadtmüller, U., Tauberian theorems for general power series methods, Math. Proc. Cambridge. Phil. Soc. 110(1991), 483–590.Google Scholar
12. Kratz, W. and Stadtmüller, U., O-Tauberian theorems for (Jp)-methods with rapidly increasing weights, J. London Math. Soc. (2) 41(1990), 489–502.Google Scholar
13. Kratz, W. and Stadtmüller, U., Tauberian theorems for Borel-type methods of summability, Arch. Math. 55(1990), 465–474.Google Scholar
14. Littlewood, J. E., The converse of Abel's theorem on power series, Proc. London Math. Soc. (2) 9(1911), 434–448.Google Scholar
15. Moh, T. T., On a General Tauberian theorem, Proc. Amer. Math. Soc. 36(1972), 167–172.Google Scholar
16. Motzer, W., Taubersätze zwischen Potenzreihenverfahren und speziellen Matrixverfahren, Dissertation, Universität Ulm, Ulm, 1993.Google Scholar
17. Peyerimhoff, A., Lectures on Summability, Lecture Notes in Math. 107, Springer-Verlag, 1969.Google Scholar
18. Tietz, H. and Trautner, R., Taubersätze für Potenzxeihen, Arch. Math. 50(1988), 164–174.Google Scholar
19. Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren, Springer-Verlag, 1970.Google Scholar
20. Zygmund, A., Trigonometric series, Vol. 1, Cambridge University Press, 1968.Google Scholar