In mathematical analysis, Haar's Tauberian theorem[1] 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
[edit]
William Feller gives the following simplified form for this theorem:[2]
Suppose that
is a non-negative and continuous function for
, having finite Laplace transform

for
. Then
is well defined for any complex value of
with
. Suppose that
verifies the following conditions:
1. For
the function
(which is regular on the right half-plane
) has continuous boundary values
as
, for
and
, furthermore for
it may be written as

where
has finite derivatives
and
is bounded in every finite interval;
2. The integral

converges uniformly with respect to
for fixed
and
;
3.
as
, uniformly with respect to
;
4.
tend to zero as
;
5. The integrals
and 
converge uniformly with respect to
for fixed
,
and
.
Under these conditions
![{\displaystyle \lim _{t\to \infty }t^{r}[f(t)-C]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87b80bf9deeae0195ab369732dc5ba3c2c838ad0)
A more detailed version is given in.[3]
Suppose that
is a continuous function for
, having Laplace transform

with the following properties
1. For all values
with
the function
is regular;
2. For all
, the function
, considered as a function of the variable
, has the Fourier property ("Fourierschen Charakter besitzt") defined by Haar as for any
there is a value
such that for all

whenever
or
.
3. The function
has a boundary value for
of the form

where
and
is an
times differentiable function of
and such that the derivative

is bounded on any finite interval (for the variable
)
4. The derivatives

for
have zero limit for
and for
has the Fourier property as defined above.
5. For sufficiently large
the following hold

Under the above hypotheses we have the asymptotic formula
![{\displaystyle \lim _{t\to \infty }t^{n}e^{-at}{\Big [}f(t)-\sum _{j=1}^{N}{\frac {c_{j}}{\Gamma (\rho _{j})}}e^{s_{j}t}t^{\rho _{j}-1}{\Big ]}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9f619d21ca9ec3ded406e35f42f15bbd66cf06)