Mellin inversion theorem

Method

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

:f(x)= { \mathcal{M}^{-1} \varphi } = \frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s), ds

we have that

:\varphi(s)= { \mathcal{M} f } = \int_0^{\infty} x^{s-1} f(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

:\varphi(s)=\int_0^{\infty} x^{s-1} f(x),dx

is absolutely convergent when a . Then f is recoverable via the inverse Mellin transform from its Mellin transform \varphi. 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.

Boundedness condition

The boundedness condition on \varphi(s) can be strengthened if f(x) is continuous. If \varphi(s) is analytic in the strip a , and if |\varphi(s)| , 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 \varphi for at least a .

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 \varphi to simply make it of polynomial growth in any closed strip contained in the open strip a .

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

:|f| = \left(\int_0^\infty |x^\nu f(x)|^p, \frac{dx}{x}\right)^{1/p}

where ν and p are fixed real numbers with p1, then if f(x) is in L_{\nu, p}(R^{+}) with 1 , then \varphi(s) belongs to L_{\nu, q}(R^{+}) with q = p/(p-1) and

:f(x)=\frac{1}{2 \pi i} \int_{\nu-i \infty}^{\nu+i \infty} x^{-s} \varphi(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

: \left{\mathcal{B} f\right}(s) = \left{\mathcal{M} f(- \ln x) \right}(s)

these theorems can be immediately applied to it also.

References

General references

References

1. Debnath, Lokenath. (2015). "Integral transforms and their applications". CRC Press.