Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-15T07:03:36.939Z Has data issue: false hasContentIssue false
  • English
  • Français

On strong Rieszian summability

Published online by Cambridge University Press:  18 May 2009

Martin Glatfeld
Martin Glatfeld
Affiliation:
Mathematisches Institut Deb Universität, Göttingen
Rights & Permissions [Opens in a new window]

Extract

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.

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 3 , Issue 3 , December 1957 , pp. 123 - 131
Copyright
Copyright © Glasgow Mathematical Journal Trust 1957

References

REFERENCES

1.Boyd, A. V. & Hyslop, J. M., A definition for strong Rieszian summability and its relationship to strong Cesàro summability, Proc. Glasgow Math. Assoc., 1 (1952), 94–99.CrossRefGoogle Scholar
2.Fekete, M., Vizsgálatok az absolut summabilis sorokrol, alkalmazassal a Dirichlot- és Fourier- sorokra, Math, és termész. ért, 32 (1914), 389–425.Google Scholar
3.Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
4.Hardy, G. H. & Littlewood, J. E., Some properties of fractional integrals, I, Math. Z., 27 (1928), 565–606.CrossRefGoogle Scholar
5.Hardy, G. H. & Riesz, M., The general theory of Dirichle's series (Cambridge, 1915).Google Scholar
6.Hyslop, J. M., Note on the strong summability of series, Proc. Glasgow Math. Assoc. 1 (1952), 16–20.CrossRefGoogle Scholar
7.Kogbetliantz, E., Sommation des séries et intégrales divergentes par les moyennes arithmétiques et typiques, Mémorial Sci. Math. Fascicule 51 (1931).Google Scholar
8.Kuttner, B., Note on strong summability, J. London Math. Soc, 21 (1946), 118–122.Google Scholar
9.Obreschkoff, N., Über die absolute Summierung der Dirichletschen Reihen, Math. Z., 30 (1929), 375–386.CrossRefGoogle Scholar
10.Richert, H. -E., Beitrage zur Summierbarkeit Dirichletscher Reihen mit Anwendungen auf dio Zahlentheorie, Nachr. Akad. Wiss. Göttingen, 1956, 77–125.Google Scholar
11.Titchmarsh, E. C., The theory of the Riemann zeta-function (Oxford, 1951).Google Scholar