Mellin inversion theorem

Template:Short description 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 $\phi (s)$ is analytic in the strip Template:Tmath, and if it tends to zero uniformly as $\mathrm{\Im }(s)\to \pm \mathrm{\infty }$ for any real value Template:Tmath between Template:Tmath and Template:Tmath, with its integral along such a line converging absolutely, then if

$f(x)=\{{ℳ}^{-1}\phi \}=\frac{1}{2\pi i}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}{x}^{-s}\phi (s)ds$

we have that

$\phi (s)=\{ℳf\}={\int }_0^{\mathrm{\infty }}{x}^{s-1}f(x)dx.$

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

$\phi (s)={\int }_0^{\mathrm{\infty }}{x}^{s-1}f(x)dx$

is absolutely convergent when Template:Tmath. Then $f$ is recoverable via the inverse Mellin transform from its Mellin transform Template:Tmath. 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 $\phi (s)$ can be strengthened if $f(x)$ is continuous. If $\phi (s)$ is analytic in the strip Template:Tmath, and if Template:Tmath, where Template:Tmath is a positive constant, then $f(x)$ as defined by the inversion integral exists and is continuous; moreover the Mellin transform of $f$ is $\phi$ for at least Template:Tmath.

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 $\phi$ to simply make it of polynomial growth in any closed strip contained in the open strip Template:Tmath.

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

$\Vert f\Vert ={({\int }_0^{\mathrm{\infty }}|x^{\nu }f(x){|}^p\frac{dx}{x})}^{1/p}<\mathrm{\infty }$

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

$f(x)=\frac{1}{2\pi i}{\int }_{\nu -i\mathrm{\infty }}^{\nu +i\mathrm{\infty }}{x}^{-s}\phi (s)ds.$

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

Since the two-sided Laplace transform can be defined as

$\left\{ℬf\right\}(s)=\{ℳf(-\ln x)\}(s),$

these theorems can be immediately applied to it also.

See also

• Mellin transform
• Fourier inversion theorem
• Nachbin's theorem

General references

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External links

• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.