Tauberian theorems for general power series methods

Abstract

Let us assume throughout that (pn) denotes a sequence of reals which satisfiesFor real sequences (sn) with increments an = sn – sn−1 for n ≥ 0,(where s−1 = 0), we consider the power seriesmethod of summability (P), where we sayThe power series methods (P) containthe so-called (Jp)-methods (R = 1)and the Borel-type methods (Bp)(R = ∞). We consider only regular (P)-methods, i.e. sn → s implies sn → s(P). By theorem 5 in [5], p.49, we have regularity if and only ifHere we are interested in the converse conclusion, namely sn → s(P) implies sn → s, which can only be validiffurther conditions, so-called Tauberian conditions are satisfied by (sn). These so-called Tauberian theorems for power series methods have a long history; see e.g. the books [5, 14, 23], and they found new attentionrecently in the papers [6, 18, 19, 20] and [8, 9, 10, 11, 12]. The latter papers contain certain o- Tauberian theorems for all power series methods in question and O-Tauberian theorems, if the weight sequence (pn) can be interpolated by alogarithmico-exponential function g(·)(see e.g. [4]), i.e.