Littlewood's Tauberian theorem

In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by Littlewood (1911).

Statement

Littlewood showed the following: If an = O(1/n ), and as x ↑ 1 we have

\sum a_n x^n \to s,

then

\sum a_n = s.

Hardy and Littlewood later showed that the hypothesis on an could be weakened to the "one-sided" condition an ≥ –C/n for some constant C. However in some sense the condition is optimal: Littlewood showed that if cn is any unbounded sequence then there is a series with |an| ≤ |cn|/n which is divergent but Abel summable.

History

Littlewood (1953) described his discovery of the proof of his Tauberian theorem. Tauber's original theorem was similar to Littlewoods, but with the stronger hypothesis that an=o(1/n). Hardy had proved a similar theorem for Cesàro summation with the weaker hypothesis an=O(1/n), and suggested to Littlewood that the same weaker hypothesis might also be enough for Tauber's theorem. In spite of the fact that the hypothesis in Littlewood's theorem seems only slightly weaker than the hypothesis in Tauber's theorem, Littlewood's proof was far harder than Tauber's, though Karamata later found an easier proof.

Littlewoods theorem follows from the later Hardy–Littlewood tauberian theorem, which is in turn a special case of Wiener's tauberian theorem, which is a special case of various abstract Tauberian theorems about Banach algebras.

References