Abelian and Tauberian Theorems

Abelian Theorems

For any summation method 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 = (c₁ + c₂ + ... + 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_nzn

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 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.