Tauberian theorems

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 [1], 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.).

References

 [1] A. Tauber, "Ein Satz aus der Theorie der unendlichen Reihen" Monatsch. für Math. , 8 (1897) pp. 273–277 [2] G.H. Hardy, "Divergent series" , Clarendon Press (1949) [3] D.V. Widder, "The Laplace transform" , Princeton Univ. Press (1941) [4] H. Pitt, "Tauberian theorems" , Oxford Univ. Press (1958) [5] A. Peyerimhoff, "Lectures on summability" , Springer (1969) (Translated from German) [6] K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1976) [7] G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70

There are many, rather unrelated seeming, results that are called Tauberian and there are a number of different but related statements about what the phrase "Tauberian theorem" means or should mean. In [a1], p. 195, is written: " a theorem is Abelian if it says something about an average of a sequence from a hypothesis about its ordinary limit; it is Tauberian if conversely the implication goes from average to limit."

Let

 (a1)

be a sequence of numbers and

 (a2)

its generating power series. In [2] the following terminology is introduced. "We call a theorem Abelian if properties of the generating function are found from the properties of its coefficients . We call a theorem Tauberian if properties of the coefficients are found from properties of the generating function."

Replacing in (a2) by and generalizing sums to integrals, a Laplace–Stieltjes transform

 (a3)

appears, and Abelian theorems derive properties of the image (under the transform) from properties of the original and Tauberian theorems do the reverse. This represents, generalizing once more, the current point of view that Tauberian theorems link the (asymptotic) behaviour of a (generalized) function in a neighbourhood of zero with that of its Fourier transform, Laplace transform, or some other integral transform at infinity. Abelian theorems do the reverse; cf. [a5], p. xiii.

The strengthened form of theorem 1) above with is Littlewood's Tauberian theorem.

The Hardy–Littlewood Tauberian theorem is the following. Let the power series (a2) converge for , let there be a such that

 (a4)

and suppose that the coefficients are non-negative. Then

 (a5)

as , where is the gamma-function.

The following result says something on the remainder term in the Hardy–Littlewood Tauberian theorem. Let, again, and consider instead of (a2) the Laplace–Stieltjes transform

 (a6)

Suppose that as ,

 (a7)

for a given , where for positive numbers , , . Then

 (a8)

Such Tauberian theorems are called Tauberian theorems with remainder or quantitative Tauberian theorems.

Wiener's generalized Tauberian theorem states: Let and let its Fourier transform have no real zeros; let be another element of and let be bounded on . Let

 (a9)

Then also

 (a10)

(and if the Fourier transform of does have a real zero, then there are and so that

holds but

does not). There are more theorems of this type. From Wiener's generalized Tauberian theorem, Tauberian theorems such as e.g. the Littlewood Tauberian theorem can be deduced, mainly by a judicious choice of and , cf. e.g. [a4], §16.

References

 [a1] D.V. Widder, "An introduction to transform theory" , Acad. Press (1971) [a2] A.G. Postnikov, "Tauberian theory and its application" Proc. Steklov Inst. Math. , 144 (1980) Trudy Mat. Inst. Steklov. , 144 (1980) [a3] N. Wiener, "Tauberian theorems" Ann. of Math. , 33 (1932) pp. 1–100 ((Reprinted in: N. Wiener, Generalized analysis and Tauberian theorems, MIT Press, 1965)) [a4] N. Wiener, "The Fourier integral and certain of its applications" , Cambridge Univ. Press (1933) [a5] V.S. Vladimirov, Yu.N. Drozzinov, B.I. Zavialov, "Tauberian theorems for generalized functions" , Kluwer (1988) (Translated from Russian) [a6] N.G. de Bruijn, "Asymptotic methods in analysis" , North-Holland & Noordhoff & Interscience (1981) [a7] T.H. Ganelius, "Tauberian remainder theorems" , Springer (1971) [a8] M.H. Subhankulov, "Tauberian theorems with remainder" , Moscow (1976) (In Russian)
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