Mellin inversion theorem

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 [math]\displaystyle{ \varphi(s) }[/math] is analytic in the strip [math]\displaystyle{ a \lt \Re(s) \lt b }[/math], and if it tends to zero uniformly as [math]\displaystyle{ \Im(s) \to \pm \infty }[/math] for any real value c between a and b, with its integral along such a line converging absolutely, then if

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

we have that

[math]\displaystyle{ \varphi(s)= \{ \mathcal{M} f \} = \int_0^{\infty} x^{s-1} f(x)\,dx. }[/math]

Conversely, suppose [math]\displaystyle{ f(x) }[/math] is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral

[math]\displaystyle{ \varphi(s)=\int_0^{\infty} x^{s-1} f(x)\,dx }[/math]

is absolutely convergent when [math]\displaystyle{ a \lt \Re(s) \lt b }[/math]. Then [math]\displaystyle{ f }[/math] is recoverable via the inverse Mellin transform from its Mellin transform [math]\displaystyle{ \varphi }[/math]. 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 [math]\displaystyle{ \varphi(s) }[/math] can be strengthened if [math]\displaystyle{ f(x) }[/math] is continuous. If [math]\displaystyle{ \varphi(s) }[/math] is analytic in the strip [math]\displaystyle{ a \lt \Re(s) \lt b }[/math], and if [math]\displaystyle{ |\varphi(s)| \lt K |s|^{-2} }[/math], where K is a positive constant, then [math]\displaystyle{ f(x) }[/math] as defined by the inversion integral exists and is continuous; moreover the Mellin transform of [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \varphi }[/math] for at least [math]\displaystyle{ a \lt \Re(s) \lt b }[/math].

On the other hand, if we are willing to accept an original [math]\displaystyle{ f }[/math] which is a generalized function, we may relax the boundedness condition on [math]\displaystyle{ \varphi }[/math] to simply make it of polynomial growth in any closed strip contained in the open strip [math]\displaystyle{ a \lt \Re(s) \lt b }[/math].

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

[math]\displaystyle{ \|f\| = \left(\int_0^\infty |x^\nu f(x)|^p\, \frac{dx}{x}\right)^{1/p} \lt \infty }[/math]

where ν and p are fixed real numbers with [math]\displaystyle{ p\gt 1 }[/math], then if [math]\displaystyle{ f(x) }[/math] is in [math]\displaystyle{ L_{\nu, p}(R^{+}) }[/math] with [math]\displaystyle{ 1 \lt p \le 2 }[/math], then [math]\displaystyle{ \varphi(s) }[/math] belongs to [math]\displaystyle{ L_{\nu, q}(R^{+}) }[/math] with [math]\displaystyle{ q = p/(p-1) }[/math] and

[math]\displaystyle{ f(x)=\frac{1}{2 \pi i} \int_{\nu-i \infty}^{\nu+i \infty} x^{-s} \varphi(s)\,ds. }[/math]

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

Since the two-sided Laplace transform can be defined as

[math]\displaystyle{ \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(- \ln x) \right\}(s) }[/math]

these theorems can be immediately applied to it also.

See also

• Mellin transform
• Nachbin's theorem

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.