Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-27T23:43:07.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Abel Transformations Into I 1

Published online by Cambridge University Press:  20 November 2018

J. A. Fridy
J. A. Fridy*
Affiliation:
Kent State University, Kent Ohio
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 t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by a nk = t n(1-t n )k . The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l 1 into l 1 if and only if t is in l 1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ x k has bounded partial sums, then A t x is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At .

Keywords

40D0540D2540E0540G10

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 421 - 427
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. DeFranza, J., Norlund methods of summability that map the set of absolutely convergent series into itself, Ph.D. Dissertation, Kent State Univ., Kent, Ohio, 1979.Google Scholar
2. Fridy, J. A., Absolute summability matrices that are stronger than the identity mapping, Proc. Amer. Math. Soc. 47 (1975), pp. 112-118.Google Scholar
3. Fridy, J. A., Tauberian theorems via block dominated matrices, Pacific J. Math. 81 (1979), pp. 81-91.Google Scholar
4. Fridy, J. A. and Roberts, K. L., Some Tauberian theorems for Euler and Borel summability, Internat. J. Math. & Math. Sci., 3 (1980) pp. 731-738.Google Scholar
5. Powell, R. L. and Shah, S. M., Summability Theory and Its Applications. Van Nostrand Reinhold Co., London, 1972.Google Scholar
6. Knopp, K. and Lorentz, G. G., Beiträge zur absoluten Limitierung, Arch Math. 2 (1949), pp. 10-16.Google Scholar
7. Tauber, A., Ein Satz aus der Théorie der Unendlichen Reihen, Monatash. Math. Phys. 8 (1897), pp. 273-277.Google Scholar