Abelian and Tauberian Theorems - Tauberian Theorems

Partial converses to abelian theorems are called tauberian theorems. The original result of Tauber (1897) stated that if we assume also

a_n = o(1/n)

(see Big O notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by J.E. Littlewood: we need only assume O(1/n). A sweeping generalization is the Hardy–Littlewood tauberian theorem.

In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, and its values there are equal to those of the Lim functional. A tauberian theorem states, under some growth condition, that the domain of L is exactly the convergent sequences and no more.

If one thinks of L as some generalised type of weighted average, taken to the limit, a tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in number theory, in particular in handling Dirichlet series.

The development of the field of tauberian theorems received a fresh turn with Norbert Wiener's very general results, namely Wiener's tauberian theorem and its large collection of corollaries. The central theorem can now be proved by Banach algebra methods, and contains much, though not all, of the previous theory.