Necessary and sufficient Tauberian condition for both Cesàro and Abel summability
Résumé
We prove that the Weakly-Vanishing Mean Oscillation (W-VMO) property of a sequence or a series is a necessary and sufficient condition under which either the convergence (C0) or the Cesàro summability (C1) follows from the Abel summability (A0) to the same limit. Hence, this result shows the Tauberian converse, with the largest possible space of sequences, of both the Abel (1826) theorem, i.e. (C0) ⇒ (A0), and the Frobenius (1880) theorem, i.e. (C1) ⇒ (A0). The inversion of the Cesàro summability (C1) ⇒ (C0) is also addressed within the same unified setting and solved with the necessary and sufficient W-VMO condition.
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