Abelian and Tauberian theorems

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Abelian and Tauberian theorems are <a href="/facts/Theorem/f2Pe1FMD">theorems</a> giving conditions for two methods of summing <a href="/facts/Divergent_series/c8Vjjjdx">divergent series</a> to give the same result, named after <a href="/facts/Niels_Henrik_Abel/6kMfdRkT">Niels Henrik Abel</a> and <a href="/facts/Alfred_Tauber/qfhBeUCQ">Alfred Tauber</a>. The original examples are <a href="/facts/Abel%2527s_theorem/edt2dPaB">Abel's theorem</a> showing that if a series <a href="/facts/Convergent_series/DTok170z">converges</a> to some limit then its <a href="/facts/Abel_sum/c8Vjjjdx">Abel sum</a> is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods.
There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense.
In the theory of <a href="/facts/Integral_transform/AB2yjHJJ">integral transforms</a>, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.

Abelian and Tauberian theorems open-in-new

Abelian and Tauberian theorems