title: Haar's Tauberian theorem
text: # 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
  
      $\lim _{t\to \infty }t^{n}e^{-at}{\Bigf(t)-\sum _{j=1}^{N}{\frac {c_{j}}{\Gamma (\rho _{j})}}e^{s_{j}t}t^{\rho _{j}-1}{\Big ]}=0.$
  