- Abelian and tauberian theorems
In

mathematics ,**abelian and tauberian theorems**relate to the meaningful assignment of a value as the "sum" of a class ofdivergent series . A large number of methods have been proposed for the summation of such series, generally taking the form of somelinear functional "L" with domain contained in some space "S" of numericalsequence s. 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 aconvergent 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 thearithmetic mean s of the first "N" terms of "c", as "N" tends toinfinity . 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 onpower series . In that case "L" is the "radial limit" (thought of within the complexunit 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"

^{"i"θ}. That theorem has its main interest in the case that the power series hasradius 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 ofAlfred 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 byJ.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" functionals. 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 handlingDirichlet series .The development of the field of tauberian theorems received a fresh turn with

Norbert Wiener 's very general results, namelyWiener's tauberian theorem and its large collection of corollaries. The central theorem can now be proved byBanach algebra methods, and contains much, though not all, of the previous theory.**References***cite journal | author=N. Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. of Math. | year=1932 | volume=33 | pages=1–100

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