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.
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
is well defined for any complex value of
verifies the following conditions: 1.
the function
(which is regular on the right half-plane
) has continuous boundary values
has finite derivatives
is bounded in every finite interval; 2.
The integral converges uniformly with respect to
, uniformly with respect to
The integrals converge uniformly with respect to
Under these conditions A more detailed version is given in.
is a continuous function for
, having Laplace transform with the following properties 1.
the function
, considered as a function of the variable
, has the Fourier property ("Fourierschen Charakter besitzt") defined by Haar as for any
α , β ≥ ω
α , β ≤ − ω
has a boundary value for
times differentiable function of
and such that the derivative is bounded on any finite interval (for the variable
The derivatives for
has the Fourier property as defined above.
For sufficiently large
the following hold Under the above hypotheses we have the asymptotic formula