Abelian and tauberian theorems

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 = (c_{n}) is a convergent sequence, with limit C, then L(c) = C. An example is given by the Cesàro 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 (d_{N}) where
 d_{N} = (c_{1} + c_{2} + ... + c_{N})/N.
To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take M large enough to make the initial segment of terms up to c_{N} average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also.
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
 a_{n}z^{n}
and set z = r.e^{iθ}. That theorem has its main interest in the case that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum of the a_{n}. When the radius is 1 the power series will have some singularity on z = 1; the assertion is that, nonetheless, if the sum of the a_{n} exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
Partial converses to abelian theorems are called Tauberian theorems. The original result of Alfred Tauber 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).
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.