Mellin inversion theorem


In mathematics, the Mellin inversion formula tells us conditions under
which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.

Method

If is analytic in the strip,
and if it tends to zero uniformly as for any real value c between a and b, with its integral along such a line converging absolutely, then if
we have that
Conversely, suppose f is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral
is absolutely convergent when. Then f is recoverable via the inverse Mellin transform from its Mellin transform.

Boundedness condition

We may strengthen the boundedness condition on if
f is continuous. If is analytic in the strip, and if, where K is a positive constant, then f as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f is for at least.
On the other hand, if we are willing to accept an original f which is a
generalized function, we may relax the boundedness condition on
to
simply make it of polynomial growth in any closed strip contained in the open strip.
We may also define a Banach space version of this theorem. If we call by
the weighted Lp space of complex valued functions f on the positive reals such that
where ν and p are fixed real numbers with p>1, then if f
is in with, then
belongs to with and
Here functions, identical everywhere except on a set of measure zero, are identified.
Since the two-sided Laplace transform can be defined as
these theorems can be immediately applied to it also.