In mathematics, a large number of methods have been proposed for the summation of divergent series. These generally take the form of some linear functional L with domain contained in some space S of numerical sequences. That is, firstly, a useful method for attributing a sum to a series that doesn't converge should at least be linear. Secondly, the sequence of partial sums of the series is considered, which is an equivalent way of presenting it.
For any such L, its abelian theorem is the result that if c = (cn) is a convergent sequence, with limit C, then L(c) = C. An example is given by the Cesaṛ method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity. One can prove that if c does converge to C, then so does the sequence (dN) where
- dN = (c1 + c2 + ... + cN)/N.
The name derives from Abel's theorem on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term
- anzn
Partial converses to abelian theorems are called Tauberian theorems. The original result of Tauber stated that if we assume also
- an = o(1/n)
In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, and its values there are equal to the Lim functional's. 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 result. It can now be proved by Banach algebra methods, and contains much of the previous theory in the form of corollaries.