Abelian and tauberian theorems
Encyclopedia
In mathematics
, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel
and Alfred Tauber
. The original examples are Abel's theorem
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 no clear distinction between abelian and Tauberian theorems, or even a 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 to be summable in the usual sense.
sequence, with limit
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
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 (dN) where
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 M large enough to make the initial segment of terms up to cN average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also.
The name derives from Abel's theorem
on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term
and set z = r·e iθ. That theorem has its main interest in the case that the power series has radius of convergence
exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous
and it follows directly that the limit as r tends up to 1 is simply the sum of the an. 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 an exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
(see Big O notation
) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by J.E. Littlewood: we need only assume O(1/n). A sweeping generalization is the Hardy–Littlewood tauberian theorem
.
In the abstract setting, therefore, an abelian theorem states that the domain of L contains 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
, in particular in handling Dirichlet series.
The development of the field of tauberian theorems received a fresh turn with Norbert Wiener
's very general results, namely Wiener's tauberian theorem
and its large collection of corollaries. The central theorem can now be proved by Banach algebra
methods, and contains much, though not all, of the previous theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
and Alfred Tauber
Alfred Tauber
Alfred Tauber was a mathematician who was born in Bratislava, and died in the Theresienstadt concentration camp. In 1897 he proved a corrected converse of Abel's theorem. G.H. Hardy and J.E. Littlewood coined the term Tauberian to describe converse theorems like that proved by Tauber.- External...
. The original examples are Abel's theorem
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:...
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 no clear distinction between abelian and Tauberian theorems, or even a 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 to be summable in the usual sense.
Abelian theorems
For any summation method L, its abelian theorem is the result that if c = (cn) is a convergentLimit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
sequence, with limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
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
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
s of the first N terms of c, as N tends to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
. One can prove that if c does converge to C, then so does the sequence (dN) where
- dN = (c1 + c2 + ... + cN)/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 M large enough to make the initial segment of terms up to cN average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also.
The name derives from Abel's theorem
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:...
on power series. In that case L is the radial limit (thought of within the complex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with term
- anzn
and set z = r·e iθ. That theorem has its main interest in the case that the power series has radius of convergence
Radius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
and it follows directly that the limit as r tends up to 1 is simply the sum of the an. 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 an exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
Tauberian theorems
Partial converses to abelian theorems are called tauberian theorems. The original result of stated that if we assume also- an = o(1/n)
(see Big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by J.E. Littlewood: we need only assume O(1/n). A sweeping generalization is the Hardy–Littlewood tauberian theorem
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 the abstract setting, therefore, an abelian theorem states that the domain of L contains 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
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, in particular in handling Dirichlet series.
The development of the field of tauberian theorems received a fresh turn with Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
's very general results, namely Wiener's tauberian theorem
Wiener's tauberian theorem
In mathematical analysis, Wiener's tauberian theorem refers to one 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 of translations of a given...
and its large collection of corollaries. The central theorem can now be proved by Banach algebra
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 which at the same time is also a Banach space...
methods, and contains much, though not all, of the previous theory.