# Tauberian theorems

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theorems of Tauberian type

Theorems establishing conditions which determine the set of series (or sequences) on which for two given summation methods and the inclusion holds. Most frequently in the theory of summation, the case in which method 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 (*)

is summable by Abel's method to a sum and , then the series converges to .

2) In order that summability of the series (*) by Abel's method to a sum implies convergence of this series to this sum , it is necessary and sufficient that Theorem 1) was later strengthened; namely, it was proved that the condition can be replaced by . 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 where is a constant, and also their generalizations, in which the natural parameter is replaced by a variable . 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 and , then the series converges to .

For every regular matrix summation method (cf. also Regular summation methods) there exists numbers such that and the condition is Tauberian for this method (that is, summability of the series by this method and the condition imply convergence of the series).

Tauberian conditions can be expressed by evaluation of the partial sums of the series or by evaluation of the difference with well-defined relations between and . Here are some examples of Tauberian theorems with such conditions: If the series (*) with partial sums is summable by Borel's method to a sum and if with , then the series converges to ; if the series (*) is summable by Abel's method to a sum and its partial sums satisfy the condition , then it is summable to by the Cesàro method (cf. Cesàro summation methods).

Lacunarity of a series, when (cf. Lacunary series), can serve as a Tauberian condition; in this case, the condition is expressed in terms of properties of the sequence .

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.).