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 , then x itself is in ℓ1.
Highlights
Hardy described a Tauberian theorem as one which asserts that a particular summability method cannot sum a divergent series that oscillates too slowly
In this paper we shall state the results in sequence-tosequence form, so a typical order-type Tauberian theorem for a method A would have
By a direct application of the Knopp-Lorentz Theorem [5], one can show that B is an matrix
Summary
H. Hardy described a Tauberian theorem as one which asserts that a particular summability method cannot sum a divergent series that oscillates too slowly. In this paper we shall state the results in sequence-tosequence form, so a typical order-type Tauberian theorem for a method A would have. AXk=X the form, "if x is a sequence such that Ax is convergent and k -Xk+I o(), x itself is convergent." Our present task is not to give more theorems in the setting of ordinary convergence, but rather, we shall develop analogous
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