International Journal of Mathematics and Mathematical Sciences 
Volume 3 (1980), Issue 4, Pages 731-738
doi:10.1155/S0161171280000531

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

Abstract

The well-known summability methods of Euler and Borel are studied as mappings from ℓ1 into ℓ1. In this ℓ−ℓ setting, the following Tauberian results are proved: if x is a sequence that is mapped into ℓ1 by the Euler-Knopp method Er with r>0 (or the Borel matrix method) and x satisfies ∑n=0∞|xn−xn+1|n<∞, then x itself is in ℓ1.