Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:12:25.239Z Has data issue: false hasContentIssue false
  • English
  • Français

On Cesàro and Abel Summability Factors for Integrals

Published online by Cambridge University Press:  20 November 2018

David Borwein  and
Brian Thorpe
David Borwein
Affiliation:
The University of Western Ontario, London, Ontario
Brian Thorpe
Affiliation:
The University of Birmingham, Birmingham, England
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.

Many results have been obtained about factors transforming integrals summable by ordinary and absolute Cesàro methods of non-negative orders into integrals summable by such methods (see [4], [2], [6], [3]) and also into integrals summable by the ordinary and absolute Abel methods (see [7]). Since the Cesàro summability methods (C, α) and |C, α| for integrals are defined for α ≦ –1, it is natural to try to extend the above mentioned results for α ≦ 0 to the case –1 ≧ α < 0. In this paper we restrict attention to the simplest case α = –1, and classify the summability factors from (C, –1) and |C, –1| to (C, –1) , |C, –1|, (C, λ), |C, λ|, A and |A|, where λ ≦ 0 and A denotes Abel summability.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 2 , 01 April 1986 , pp. 453 - 477
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bosanquet, L. S. and Tatchell, J. B., A note on summability factors, Mathematika 4 (1957), 25–40.Google Scholar
2. Borwein, D., A summability factor theorem, J. London Math. Soc. 25 (1950), 302–315.Google Scholar
3. Borwein, D., Note on summability factors, J. London Math. Soc. 29 (1954), 198–206.Google Scholar
4. Cossar, J., A theorem on Cesaro summability, J. London Math. Soc. 16 (1941), 56–68.Google Scholar
5. Lorentz, G. G., Über Limitierungsverfahren, die von einem Stieltjes-Integral abhängen, Acta Math. 79 (1947), 255–272.Google Scholar
6. Sargent, W. L. C., On the summability of infinite integrals, J. London Math. Soc. 29 (1952), 401–413.Google Scholar
7. Sargent, W. L. C., Some summability factor theorems for infinite integrals, J. London Math. Soc. 32 (1957), 387–396.Google Scholar
8. Tatchell, J B., On some integral transformations, Proc. London Math. Soc. (3), 3 (1953), 257–266.Google Scholar