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 the sequence $a_n\ge 0$ is such that there is an asymptotic equivalence

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{e}^{-ny}\sim \frac{1}{y}\text{as}y\downarrow 0$

then there is also an asymptotic equivalence

$\sum _{k=0}^n{a}_k\sim n$

as $n\to \mathrm{\infty }$. The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform.

The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood.Template:Rp In 1930, Jovan Karamata gave a new and much simpler proof.Template:Rp

Statement of the theorem

Series formulation

This formulation is from Titchmarsh.Template:Rp Suppose $a_n\ge 0$ for all $n\in \mathbb{N}$, and we have

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n\sim \frac{1}{1-x}\text{as}x\uparrow 1.$

Then as $n\to \mathrm{\infty }$ we have

$\sum _{k=0}^n{a}_k\sim n.$

The theorem is sometimes quoted in equivalent forms, where instead of requiring $a_n\ge 0$, we require $a_n=O(1)$, or we require $a_n\ge -K$ for some constant $K$.Template:Rp The theorem is sometimes quoted in another equivalent formulation (through the change of variable $x=1/e^y$).Template:Rp If,

$\sum _{n=0}^{\mathrm{\infty }}{a}_n{e}^{-ny}\sim \frac{1}{y}\text{as}y\downarrow 0$

then

$\sum _{k=0}^n{a}_k\sim n.$

Integral formulation

The following more general formulation is from Feller.Template:Rp Consider a real-valued function $F:[0,\mathrm{\infty })\to ℝ$ of bounded variation. The Laplace–Stieltjes transform of $F$ is defined by the Stieltjes integral

$\omega (s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}dF(t).$

The theorem relates the asymptotics of ω with those of $F$ in the following way. If $\rho$ is a non-negative real number, then the following statements are equivalent

• $\omega (s)\sim C{s}^{-\rho },\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{\to }\mathrm{0}$
• $F(t)\sim \frac{C}{\mathrm{\Gamma }(\rho +1)}{t}^{\rho },\text{as}t\to \mathrm{\infty }.$

Here $\mathrm{\Gamma }$ denotes the Gamma function. One obtains the theorem for series as a special case by taking $\rho =1$ and $F(t)$ to be a piecewise constant function with value ${\sum }_{k=0}^n{a}_k$ 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

$\frac{L(tx)}{L(x)}\to 1,x\to \mathrm{\infty }$

for every $t>0$. Let $L$ be a function slowly varying at infinity and $\rho \ge 0$. Then the following statements are equivalent

• $\omega (s)\sim s^{-\rho }L(s^{-1}),\text{as}s\to 0$
• $F(t)\sim \frac{1}{\mathrm{\Gamma }(\rho +1)}{t}^{\rho }L(t),\text{as}t\to \mathrm{\infty }.$

Karamata's proof

Template:Harvtxt found a short proof of the theorem by considering the functions $g$ such that

$\underset{x\to 1}{\lim }(1-x)\sum a_n{x}^{n}g(x^n)=\underset{0}{\overset{1}{\int }}g(t)dt$

An easy calculation shows that all monomials $g(x)=x^k$ have this property, and therefore so do all polynomials $g$. 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 $\sum a_n{x}^{n}g(x^n)$ is $a_0+\cdots +a_N$ 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

$\frac{1}{(1+x{)}^2(1-x)}=1-x+2x^2-2x^3+3x^4-3x^5+\cdots$

is asymptotic to $1/4(1-x)$ as $x\to 1$, 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

Template:Main In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If $a_n=O(1/n)$, and we have

$\sum a_n{x}^n\to s\text{as}x\uparrow 1$

then

$\sum a_n=s.$

This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it.Template:Rp

Prime number theorem

In 1915 Hardy and Littlewood developed a proof of the prime number theorem based on their Tauberian theorem; they proved

$\sum _{n=2}^{\mathrm{\infty }}\mathrm{\Lambda }(n)e^{-ny}\sim \frac{1}{y},$

where $\mathrm{\Lambda }$ is the von Mangoldt function, and then conclude

$\sum _{n\le x}\mathrm{\Lambda }(n)\sim x,$

an equivalent form of the prime number theorem.Template:RpTemplate:Rp Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.Template:Rp

Notes

• Template:Cite journal

External links

• Template:Springer
• Template:MathWorld