International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 731-738

Some Tauberian theorems for Euler and Borel summability

J. A. Fridy1 and K. L. Roberts2

1Department of Mathematics, Kent State University, Kent 44252, Ohio, USA
2Department of Mathematics, The University of Western Ontario, Ontario, London, Canada

Received 6 August 1979; Revised 7 December 1979

Copyright © 1980 J. A. Fridy and K. L. Roberts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The well-known summability methods of Euler and Borel are studied as mappings from 1 into 1. In this setting, the following Tauberian results are proved: if x is a sequence that is mapped into 1 by the Euler-Knopp method Er with r>0 (or the Borel matrix method) and x satisfies n=0|xnxn+1|n<, then x itself is in 1.