Home > Abelian and tauberian theorems

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 = (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 [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 Tauber stated that if we assume also

a_n = o(1/n)

(see Big O notationIn complexity theory, computer science, and mathematics the Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. More exactly, it is used to describe an asymptotic upper bound for the magnitude of a function in) 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 theoryTraditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wide, in particular in handling Dirichlet seriesIn mathematics, a Dirichlet series one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form : The most famous of Dirichlet series is : which is the Riemann zeta function. Other Dirichlet series are: : wh.

The development of the field of tauberian theorems received a fresh turn with Norbert WienerNorbert Wiener ( November 26, 1894 March 18, 1964) was an American mathematician, known as the founder of cybernetics. He created the term in his book Cybernetics or Control and Communication in the Animal and the Machine (MIT Press, 1948). He was born in's very general result. It can now be proved by Banach algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related methods, and contains much of the previous theory in the form of corollaries.