Necessary and sufficient Tauberian condition for both Cesàro and Abel summability

Philippe Angot (1, 2)

1 I2M - Institut de Mathématiques de Marseille
2 AMU SCI - Aix-Marseille Université - Faculté des Sciences

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 the convergence (C0) 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 the Abel (1826) theorem on power series for which (A0) ⇒(C0). 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 Tauberian condition.

Mots clés: Abel summability; Cesàro summability; Power series; Tauberian theory; Weakly-vanishing mean oscillation

Domaines: Mathématiques [math]

Philippe Angot. Necessary and sufficient Tauberian condition for both Cesàro and Abel summability. Jahresbericht der Deutschen Mathematiker-Vereinigung, inPress. ⟨hal-04227761v3⟩