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 [1], 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.).

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

Comments

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

$$ \tag{a1 } a _ {0} , a _ {1} , a _ {2} \dots $$

be a sequence of numbers and

$$ \tag{a2 } f( z) = a _ {0} + a _ {1} z + a _ {2} z ^ {2} + \dots $$

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 $ z $ in (a2) by $ e ^ {-s} $ and generalizing sums to integrals, a Laplace–Stieltjes transform

$$ \tag{a3 } \Phi ( s) = \int\limits _ { 0 } ^ \infty e ^ {-su} \, d \alpha ( u) $$

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 $ a _ {n} = O( 1/n) $ is Littlewood's Tauberian theorem.

The Hardy–Littlewood Tauberian theorem is the following. Let the power series (a2) converge for $ | z | < 1 $, let there be a $ \gamma > 0 $ such that

$$ \tag{a4 } \lim\limits _ {x \uparrow 1 } ( 1- x) ^ \gamma \sum _ { n=0 } ^ \infty a _ {n} z ^ {n} = A $$

and suppose that the coefficients $ a _ {n} $ are non-negative. Then

$$ \tag{a5 } \sum _ {n \leq N } a _ {n} \sim \frac{A}{\Gamma ( 1+ \gamma ) } N ^ \gamma $$

as $ N \rightarrow \infty $, where $ \Gamma $ is the gamma-function.

The following result says something on the remainder term in the Hardy–Littlewood Tauberian theorem. Let, again, $ a _ {n} \geq 0 $ and consider instead of (a2) the Laplace–Stieltjes transform

$$ \tag{a6 } F ( s) = \sum _ { n=0 } ^ \infty a _ {n} e ^ {-ns} . $$

Suppose that as $ s \downarrow 0 $,

$$ \tag{a7 } F( s) = A s ^ {- \alpha } ( 1+ R( s) ) $$

for a given $ \alpha > 0 $, where $ | R( s) | \leq C s ^ \epsilon $ for positive numbers $ A $, $ C $, $ \epsilon $. Then

$$ \tag{a8 } \sum _ {n \leq x } a _ {n} = \ \frac{Ax ^ \alpha }{\Gamma ( 1+ \alpha ) } ( 1+ O ( 1 / { \mathop{\rm ln} x } ) ) . $$

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

Wiener's generalized Tauberian theorem (see Wiener Tauberian theorem) states: Let $ K _ {1} \in L _ {1} (- \infty , \infty ) $ and let its Fourier transform have no real zeros; let $ K _ {2} $ be another element of $ L _ {1} (- \infty , \infty ) $ and let $ f( x) $ be bounded on $ ( - \infty , \infty ) $. Let

$$ \tag{a9 } \lim\limits _ {x \rightarrow \infty } \int\limits _ {- \infty } ^ { {+ } \infty } K _ {1} ( x- y) f( y) dy = \ A \int\limits _ {- \infty } ^ { {+ } \infty } K _ {1} ( x) dx . $$

Then also

$$ \tag{a10 } \lim\limits _ {x \rightarrow \infty } \int\limits _ {- \infty } ^ { {+ } \infty } K _ {2} ( x- y) f( y) dy = A \int\limits _ {- \infty } ^ { {+ } \infty } K _ {2} ( x) dx $$

(and if the Fourier transform of $ K _ {1} $ does have a real zero, then there are $ f $ and $ K _ {2} $ so that $$ \text{(missing equation)} $$ holds but $$ \text{(missing equation)} $$ 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 $ K _ {1} $ and $ K _ {2} $, 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)