Haar's Tauberian theorem
# 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 Tauberian theorem.

## Simplified version by Feller

William Feller gives the following simplified form for this theorem:

Suppose that $f(t)$ is a non-negative and continuous function for $t\geq 0$, having finite Laplace transform

    $F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$

for $s>0$. Then $F(s)$ is well defined for any complex value of $s=x+iy$ with $x>0$. Suppose that $F$ verifies the following conditions:

1. For $y\neq 0$ the function $F(x+iy)$ (which is regular on the right half-plane $x>0$) has continuous boundary values $F(iy)$ as $x\to +0$, for $x\geq 0$ and $y\neq 0$, furthermore for $s=iy$ it may be written as

    $F(s)={\frac {C}{s}}+\psi (s),$

where $\psi (iy)$ has finite derivatives $\psi '(iy),\ldots ,\psi ^{(r)}(iy)$ and $\psi ^{(r)}(iy)$ is bounded in every finite interval;

2. The integral

    $\int _{0}^{\infty }e^{ity}F(x+iy)\,dy$

converges uniformly with respect to $t\geq T$ for fixed $x>0$ and $T>0$;

3. $F(x+iy)\to 0$ as $y\to \pm \infty$, uniformly with respect to $x\geq 0$;

4. $F'(iy),\ldots ,F^{(r)}(iy)$ tend to zero as $y\to \pm \infty$;

5. The integrals

    $\int _{-\infty }^{y_{1}}e^{ity}F^{(r)}(iy)\,dy$ and $\int _{y_{2}}^{\infty }e^{ity}F^{(r)}(iy)\,dy$

converge uniformly with respect to $t\geq T$ for fixed $y_{1}<0$, $y_{2}>0$ and $T>0$.

Under these conditions

    $\lim _{t\to \infty }t^{r}[f(t)-C]=0.$

## Complete version

A more detailed version is given in.

Suppose that $f(t)$ is a continuous function for $t\geq 0$, having Laplace transform

    $F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$

with the following properties

1. For all values $s=x+iy$ with $x>a$ the function $F(s)=F(x+iy)$ is regular;

2. For all $x>a$, the function $F(x+iy)$, considered as a function of the variable $y$, has the Fourier property ("Fourierschen Charakter besitzt") defined by Haar as for any $\delta >0$ there is a value $\omega$ such that for all $t\geq T$

    ${\Big |}\,\int _{\alpha }^{\beta }e^{iyt}F(x+iy)\,dy\;{\Big |}<\delta$

whenever $\alpha ,\beta \geq \omega$ or $\alpha ,\beta \leq -\omega$.

3. The function $F(s)$ has a boundary value for $\Re s=a$ of the form

    $F(s)=\sum _{j=1}^{N}{\frac {c_{j}}{(s-s_{j})^{\rho _{j}}}}+\psi (s)$

where $s_{j}=a+iy_{j}$ and $\psi (a+iy)$ is an $n$ times differentiable function of $y$ and such that the derivative

    $\left|{\frac {d^{n}\psi (a+iy)}{dy^{n}}}\right|$

is bounded on any finite interval (for the variable $y$)

4. The derivatives

    ${\frac {d^{k}F(a+iy)}{dy^{k}}}$

for $k=0,\ldots ,n-1$ have zero limit for $y\to \pm \infty$ and for $k=n$ has the Fourier property as defined above.

5. For sufficiently large $t$ the following hold

    $\lim _{y\to \pm \infty }\int _{a+iy}^{x+iy}e^{st}F(s)\,ds=0$

Under the above hypotheses we have the asymptotic formula

    $\lim _{t\to \infty }t^{n}e^{-at}{\Bigf(t)-\sum _{j=1}^{N}{\frac {c_{j}}{\Gamma (\rho _{j})}}e^{s_{j}t}t^{\rho _{j}-1}{\Big ]}=0.$
