Tauberian Theorem
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Abelian and Tauberian theorems are
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s giving conditions for two methods of summing
divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
to give the same result, named after
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and
Alfred Tauber Alfred Tauber (5 November 1866 – 26 July 1942) was a Hungarian-born Austrian mathematician, known for his contribution to mathematical analysis and to the theory of functions of a complex variable: he is the eponym of an important class of t ...
. The original examples are
Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a powe ...
showing that if a series converges to some limit then its Abel sum 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
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
s, 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 theorems

For any summation method ''L'', its Abelian theorem is the result that if ''c'' = (''c''''n'') is a
convergent sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
, with
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
''C'', then ''L''(''c'') = ''C''. An example is given by the Cesàro method, in which ''L'' is defined as the limit of the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
s of the first ''N'' terms of ''c'', as ''N'' tends to infinity. One can prove that if ''c'' does converge to ''C'', then so does the sequence (''d''''N'') where : d_N = \frac N. To see that, subtract ''C'' everywhere to reduce to the case ''C'' = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take ''N'' large enough to make the initial segment of terms up to ''c''''N'' average to at most ''ε''/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded. The name derives from
Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a powe ...
on
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. In that case ''L'' is the ''radial limit'' (thought of within the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...
), where we let ''r'' tend to the limit 1 from below along the real axis in the power series with term : ''a''''n''''z''''n'' and set ''z'' = ''r'' ·''e''''iθ''. That theorem has its main interest in the case that the power series has
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
exactly 1: if the radius of convergence is greater than one, the convergence of the power series is
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
for ''r'' in ,1so that the sum is automatically
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
and it follows directly that the limit as ''r'' tends up to 1 is simply the sum of the ''a''''n''. When the radius is 1 the power series will have some singularity on , ''z'', = 1; the assertion is that, nonetheless, if the sum of the ''a''''n'' exists, it is equal to the limit over ''r''. This therefore fits exactly into the abstract picture.


Tauberian theorems

Partial
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to Abelian theorems are called Tauberian theorems. The original result of stated that if we assume also :''a''''n'' = o(1/''n'') (see
Little o notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Land ...
) and the radial limit exists, then the series obtained by setting ''z'' = 1 is actually convergent. This was strengthened by
John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
: we need only assume O(1/''n''). A sweeping generalization is the
Hardy–Littlewood Tauberian theorem In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if, as ''y'' ↓ 0 ...
. In the abstract setting, therefore, an ''Abelian'' theorem states that the domain of ''L'' contains the convergent sequences, and its values there are equal to those of the ''Lim'' functional. A ''Tauberian'' theorem states, under some growth condition, that the domain of ''L'' is exactly the convergent sequences and no more. If one thinks of ''L'' as some generalised type of ''weighted average'', taken to the limit, a Tauberian theorem allows one to discard the weighting, under the correct hypotheses. There are many applications of this kind of result in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, in particular in handling
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...
. The development of the field of Tauberian theorems received a fresh turn with
Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...
's very general results, namely
Wiener's Tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations ...
and its large collection of
corollaries In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
. The central theorem can now be proved by
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
methods, and contains much, though not all, of the previous theory.


See also

*
Wiener's Tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations ...
*
Hardy–Littlewood Tauberian theorem In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if, as ''y'' ↓ 0 ...
*
Haar's Tauberian theorem In mathematical analysis, Haar's Tauberian theorem named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood Taube ...


References


External links

* * * {{cite book , first1=Hugh L., last1= Montgomery , authorlink=Hugh Montgomery (mathematician) , author2-link=Robert Charles Vaughan (mathematician), first2=Robert C., last2= Vaughan , title=Multiplicative number theory I. Classical theory , series=Cambridge Studies in Advanced Mathematics , volume=97 , year=2007 , isbn=978-0-521-84903-6 , pages=147–167 , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, place=Cambridge , mr=2378655, zbl=1142.11001 Tauberian theorems Mathematical series Summability methods Summability theory