Generalized Tauberian theorems for Abel summability with complex coefficients
Résumé
We prove generalizations of Tauberian results of Sz\'asz (1928) or Landau (1913) and Sz\'asz (1951), respectively for the converse of Frobenius' (1880) theorem or the inversion of Abel summability on power series. We show that the present Tauberian condition, {\em i.e.} Weakly-Vanishing Mean Oscillation (\textit{W-VMO}), is not only weaker but it remains necessary and sufficient in the case of complex coefficients.
In particular, this means that the usual boundedness assumption can be dropped from the Tauberian condition.
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