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
![{\displaystyle F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfeb9b2856c960065347368fc62b7d0bbd4bba5f)
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
![{\displaystyle F(s)={\frac {C}{s}}+\psi (s),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b755fd66e235249a36637ae408ef29af16c0908)
where
has finite derivatives
and
is bounded in every finite interval;
2. The integral
![{\displaystyle \int _{0}^{\infty }e^{ity}F(x+iy)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/496817d1c18e7e208901bd0b6c75da18cacef627)
converges uniformly with respect to
for fixed
and
;
3.
as
, uniformly with respect to
;
4.
tend to zero as
;
5. The integrals
and ![{\displaystyle \int _{y_{2}}^{\infty }e^{ity}F^{(r)}(iy)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c33584a96846163f996357c779c0478117681d)
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
![{\displaystyle F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfeb9b2856c960065347368fc62b7d0bbd4bba5f)
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
![{\displaystyle {\Big |}\,\int _{\alpha }^{\beta }e^{iyt}F(x+iy)\,dy\;{\Big |}<\delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f45941af36dd108693638cfc4e8e04b3ec476388)
whenever
or
.
3. The function
has a boundary value for
of the form
![{\displaystyle F(s)=\sum _{j=1}^{N}{\frac {c_{j}}{(s-s_{j})^{\rho _{j}}}}+\psi (s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b3dd0b3f09dfa02d918df0647360fca3161c4b)
where
and
is an
times differentiable function of
and such that the derivative
![{\displaystyle \left|{\frac {d^{n}\psi (a+iy)}{dy^{n}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84130d035010d92f338666e9b118daf73481c4fe)
is bounded on any finite interval (for the variable
)
4. The derivatives
![{\displaystyle {\frac {d^{k}F(a+iy)}{dy^{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b060e7fee6d979f327d40291309a0bf073d099e3)
for
have zero limit for
and for
has the Fourier property as defined above.
5. For sufficiently large
the following hold
![{\displaystyle \lim _{y\to \pm \infty }\int _{a+iy}^{x+iy}e^{st}F(s)\,ds=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4490d6ba31268f8cb0449d329595cca8fbffd49)
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)