Mellin inversion theorem

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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.

Method

If φ(s) is analytic in the strip a<(s)<b, and if it tends to zero uniformly as (s)± for any real value c between a and b, with its integral along such a line converging absolutely, then if

f(x)={1φ}=12πicic+ixsφ(s)ds

we have that

φ(s)={f}=0xs1f(x)dx.

Conversely, suppose f(x) is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral

φ(s)=0xs1f(x)dx

is absolutely convergent when a<(s)<b. Then f 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]

Boundedness condition

The boundedness condition on φ(s) can be strengthened if f(x) is continuous. If φ(s) is analytic in the strip a<(s)<b, and if |φ(s)|<K|s|2, where K is a positive constant, then f(x) as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f is φ for at least a<(s)<b.

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 φ to simply make it of polynomial growth in any closed strip contained in the open strip a<(s)<b.

We may also define a Banach space version of this theorem. If we call by Lν,p(R+) the weighted Lp space of complex valued functions f on the positive reals such that

f=(0|xνf(x)|pdxx)1/p<

where ν and p are fixed real numbers with p>1, then if f(x) is in Lν,p(R+) with 1<p2, then φ(s) belongs to Lν,q(R+) with q=p/(p1) and

f(x)=12πiνiν+ixsφ(s)ds.

Here functions, identical everywhere except on a set of measure zero, are identified.

Since the two-sided Laplace transform can be defined as

{f}(s)={f(lnx)}(s)

these theorems can be immediately applied to it also.

See also

References