In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, the Hardy–Littlewood Tauberian theorem is a
Tauberian theorem 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 th ...
relating the
asymptotics of the partial sums of a
series with the asymptotics of its
Abel summation
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 ...
. In this form, the theorem asserts that if, as ''y'' ↓ 0, the non-negative sequence ''a''
''n'' is such that there is an
asymptotic equivalence
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
:
then there is also an asymptotic equivalence
:
as ''n'' → ∞. The
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
formulation of the theorem relates in an analogous manner the asymptotics of the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Eve ...
of a function with the asymptotics of its
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
.
The theorem was
proved in 1914 by
G. H. Hardy and
J. E. Littlewood.
[
] In 1930,
Jovan Karamata gave a new and much simpler proof.
Statement of the theorem
Series formulation
This formulation is from Titchmarsh.
Suppose ''a''
''n'' ≥ 0 for all ''n'', and as ''x'' ↑ 1 we have
:
Then as ''n'' goes to ∞ we have
:
The theorem is sometimes quoted in equivalent forms, where instead of requiring ''a''
''n'' ≥ 0, we require ''a''
''n'' = O(1), or we require ''a''
''n'' ≥ −''K'' for some constant ''K''.
[
] The theorem is sometimes quoted in another equivalent formulation (through the change of variable ''x'' = 1/''e''
''y'' ).
If, as ''y'' ↓ 0,
:
then
:
Integral formulation
The following more general formulation is from Feller. Consider a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued function ''F'' :
,∞) → R of bounded variation.
[Bounded variation is only required locally: on every bounded subinterval of [0,∞). However, then more complicated additional assumptions on the convergence of the Laplace–Stieltjes transform are required. See ] The Laplace–Stieltjes transform of ''F'' is defined by the Stieltjes integral
:
The theorem relates the asymptotics of ω with those of ''F'' in the following way. If ρ is a non-negative real number, then the following statements are equivalent
*
*
Here Γ denotes the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
. One obtains the theorem for series as a special case by taking ρ = 1 and ''F''(''t'') to be a piecewise constant function with value
between ''t'' = ''n'' and ''t'' = ''n'' + 1.
A slight improvement is possible. According to the definition of a
slowly varying function, ''L''(''x'') is slow varying at infinity iff
:
for every positive ''t''. Let ''L'' be a function slowly varying at infinity and ρ a non-negative real number. Then the following statements are equivalent
*
*
Karamata's proof
found a short proof of the theorem by considering the functions ''g'' such that
:
An easy calculation shows that all
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s ''g''(''x'') = ''x''
''k'' have this property, and therefore so do all
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s ''g''. This can be extended to a function ''g'' with simple (step)
discontinuities by approximating it by polynomials from above and below (using the
Weierstrass approximation theorem and a little extra fudging) and using the fact that the coefficients ''a''
''n'' are positive. In particular the function given by ''g''(''t'') = 1/''t'' if 1/''e'' < ''t'' < 1 and 0 otherwise has this property. But then for ''x'' = ''e''
−1/''N'' the sum Σ''a''
''n''''x''
''n''''g''(''x''
''n'') is ''a''
0 + ... + ''a''
''N'', and the integral of ''g'' is 1, from which the Hardy–Littlewood theorem follows immediately.
Examples
Non-positive coefficients
The theorem can fail without the condition that the coefficients are non-negative. For example, the function
:
is asymptotic to 1/4(1–''x'') as ''x'' tends to 1, but the partial sums of its coefficients are 1, 0, 2, 0, 3, 0, 4, ... and are not asymptotic to any linear function.
Littlewood's extension of Tauber's theorem
In 1911
Littlewood proved an extension of
Tauber
The Tauber () is a river in Franconia (Baden-Württemberg and Bavaria), Germany. It is a left tributary of the Main and is in length. The name derives from the Celtic word for water (compare: Dover).
It flows through Rothenburg ob der Tauber ...
's
converse of
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 pow ...
. Littlewood showed the following: If ''a''
''n'' = O(1/''n''), and as ''x'' ↑ 1 we have
:
then
:
This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it.
Prime number theorem
In 1915 Hardy and Littlewood developed a proof of the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
based on their Tauberian theorem; they proved
:
where Λ is the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mango ...
, and then conclude
:
an equivalent form of the prime number theorem.
[
][
]
Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.
Notes
External links
*
*
{{DEFAULTSORT:Hardy-Littlewood tauberian theorem
Tauberian theorems