Editing Abelian and Tauberian theorems (section)

==Tauberian theorems==

Partial [[converse (logic)|converses]] to Abelian theorems are called '''Tauberian theorems'''. The original result of {{harvs|txt|authorlink=Alfred Tauber|last=Tauber|first=Alfred|year=1897}}<ref>{{cite journal | last=Tauber | first=Alfred | author-link=Alfred Tauber | title=Ein Satz aus der Theorie der unendlichen Reihen|trans-title=A theorem about infinite series | language=German | doi=10.1007/BF01696278 | year=1897 | journal=[[Monatshefte für Mathematik und Physik]]| volume=8 | pages=273–277 |url=http://www.literature.at/viewer.alo?viewmode=overview&olfullscreen=true&objid=12409&page=280 | jfm=28.0221.02| s2cid=120692627 }}</ref> stated that if we assume also

:''a''<sub>''n''</sub> = o(1/''n'')

(see [[Big O notation#Little-o notation|Little o notation]]) and the radial limit exists, then the series obtained by setting ''z'' = 1 is actually convergent. This was strengthened by [[John Edensor Littlewood]]: we need only assume O(1/''n'').  A sweeping generalization is the [[Hardy–Littlewood Tauberian theorem]].

In the abstract setting, therefore, an ''Abelian'' theorem states that the domain of ''L'' contains the convergent sequences, and its values there are equal to those of the ''Lim'' functional. 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 results, namely [[Wiener's Tauberian theorem]] and its large collection of [[corollaries]].<ref>{{cite journal | first=Norbert |last=Wiener | authorlink=Norbert Wiener | title=Tauberian theorems |  year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 | jstor=1968102 | issue=1 | journal=[[Annals of Mathematics]] | jfm=58.0226.02 | mr=1503035| zbl=0004.05905}}
</ref> The central theorem can now be proved by [[Banach algebra]] methods, and contains much, though not all, of the previous theory.