Mellin inversion theorem

Short description: Theorem in complex analysis

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 $a<\mathrm{\Re }(s)<b$, and if it tends to zero uniformly as $\mathrm{\Im }(s)\to \pm \mathrm{\infty }$ for any real value $c$ between $a$ and $b$, with its integral along such a line converging absolutely, then if

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

we have that

$\phi (s)=\{{ℳ}f\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{s-1}f(x)dx.$

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

$\phi (s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^{s-1}f(x)dx$

is absolutely convergent when $a<\mathrm{\Re }(s)<b$. Then $f$ is recoverable via the inverse Mellin transform from its Mellin transform $\phi$. 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 $a<\mathrm{\Re }(s)<b$, and if $|\phi (s)|<K|s{|}^{-2}$, 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 $\phi$ for at least $a<\mathrm{\Re }(s)<b$.

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 $\phi$ to simply make it of polynomial growth in any closed strip contained in the open strip $a<\mathrm{\Re }(s)<b$.

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

$\Vert f\Vert ={(\underset{0}{\overset{\mathrm{\infty }}{\int }}|x^{\nu }f(x){|}^p\frac{dx}{x})}^{1/p}<\mathrm{\infty }$

where $\nu$ and $p$ are fixed real numbers with $p>1$, then if $f(x)$ is in $L_{\nu ,p}(R^{+})$ with $1<p\le 2$, then $\phi (s)$ belongs to $L_{\nu ,q}(R^{+})$ with $q=p/(p-1)$ and

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

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

Since the two-sided Laplace transform can be defined as

$\left\{{ℬ}f\right\}(s)=\{{ℳ}f(-\ln x)\}(s),$

these theorems can be immediately applied to it also.

See also

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

General references

• Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums". Theoretical Computer Science 144 (1–2): 3–58. doi:10.1016/0304-3975(95)00002-E. https://hal.inria.fr/inria-00074307/file/RR-2369.pdf. 
• McLachlan, N. W. (1953). Complex Variable Theory and Transform Calculus. Cambridge University Press. 
• Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4. 
• Titchmarsh, E. C. (1948). Introduction to the Theory of Fourier Integrals (Second ed.). Oxford University Press. 
• Yakubovich, S. B. (1996). Index Transforms. World Scientific. ISBN 981-02-2216-5. 
• Zemanian, A. H. (1968). Generalized Integral Transforms. John Wiley & Sons. 

External links

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