In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.
φ ( s )
is analytic in the strip
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
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
is recoverable via the inverse Mellin transform from its Mellin transform
φ
These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem.
[1] The boundedness condition on
is analytic in the strip
, where K is a positive constant, then
as defined by the inversion integral exists and is continuous; moreover the Mellin transform of
On the other hand, if we are willing to accept an original
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.
ν , p
the weighted Lp space of complex valued functions
on the positive reals such that where ν and p are fixed real numbers with
ν , p
ν , q
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.