# Tauberian theorems

theorems of Tauberian type

Theorems establishing conditions which determine the set of series (or sequences) on which for two given summation methods $A$ and $B$ the inclusion $A \subset B$ holds. Most frequently in the theory of summation, the case in which method $B$ is equivalent with convergence is considered. In Tauberian theorems concerning such cases, conditions on a series (sequence) are established under which convergence follows from summability by a given method. The name of these theorems goes back to A. Tauber , who was the first to prove two theorems of this type for the Abel summation method:

1) If the series

$$\tag{* } \sum _ { n=0 } ^ \infty a _ {n}$$

is summable by Abel's method to a sum $S$ and $a _ {n} = o ( 1/n)$, then the series converges to $S$.

2) In order that summability of the series (*) by Abel's method to a sum $S$ implies convergence of this series to this sum $S$, it is necessary and sufficient that

$$\sum _ { k=0 } ^ { n } ka _ {k} = o( n).$$

Theorem 1) was later strengthened; namely, it was proved that the condition $a _ {n} = o( 1/n)$ can be replaced by $a _ {n} = O( 1/n)$. Conditions other than summability imposed on the series are called Tauberian conditions in such cases. These conditions can be expressed in various forms. For a series (*), the most widespread conditions are of the type

$$a _ {n} = o \left ( \frac{1}{n} \right ) ,\ \ a _ {n} = O \left ( \frac{1}{n} \right ) ,\ \ a _ {n} > - \frac{H}{n} ,\ \ a _ {n} < \frac{H}{n} ,$$

where $H$ is a constant,

$$\sum _ { k=0 } ^ { n } ka _ {k} = o ( n),$$

and also their generalizations, in which the natural parameter $n$ is replaced by a variable $\tau _ {n}$. In Tauberian theorems, such conditions include, apart from those adduced above, for instance, the following one: If the series (*) is summable by Borel's method (cf. Borel summation method) to a sum $S$ and $a _ {n} = O( 1 / \sqrt n )$, then the series converges to $S$.

For every regular matrix summation method (cf. also Regular summation methods) there exists numbers $\tau _ {n} \geq 0$ such that $\sum _ {n=0} ^ \infty \tau _ {n} = \infty$ and the condition $a _ {n} = o( \tau _ {n} )$ is Tauberian for this method (that is, summability of the series by this method and the condition $a _ {n} = o( \tau _ {n} )$ imply convergence of the series).

Tauberian conditions can be expressed by evaluation of the partial sums $S _ {n}$ of the series or by evaluation of the difference $S _ {n} - S _ {m}$ with well-defined relations between $n$ and $m$. Here are some examples of Tauberian theorems with such conditions: If the series (*) with partial sums $S _ {n}$ is summable by Borel's method to a sum $S$ and if

$$\varliminf\limits _ {{ {m \rightarrow \infty }}\; } ( S _ {n} - S _ {m} ) \geq 0,\ \ n > m ,$$

with $( n- m) / \sqrt m \rightarrow 0$, then the series converges to $S$; if the series (*) is summable by Abel's method to a sum $S$ and its partial sums $S _ {n}$ satisfy the condition $S _ {n} = O( 1)$, then it is summable to $S$ by the Cesàro method $( C, 1 )$( cf. Cesàro summation methods).

Lacunarity of a series, $a _ {n} = 0$ when $n = n _ {k}$( cf. Lacunary series), can serve as a Tauberian condition; in this case, the condition is expressed in terms of properties of the sequence $\{ n _ {k} \}$.

Apart from ordinary summability, in the theory of summation Tauberian theorems are considered for special types of summability (absolute, strong, summability with a weight, etc.).

How to Cite This Entry:
Tauberian theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tauberian_theorems&oldid=51714