Some Tauberian theorems for Euler and Borel summability

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.

Keywords

• Tauberian condition
• &#8467;&#8722;&#8467; method
• Eular-Knopp means
• Borel exponential method.