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 $\varphi (s)$ is analytic in the strip $a<\Re (s)<b$ , 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}f(x)\,{\frac {dx}{x}}.

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}f(x)\,{\frac {dx}{x}}

is absolutely convergent when $a<\Re (s)<b$ . Then f is recoverable via the inverse Mellin transform from its Mellin transform $\varphi$ .

Boundedness condition

We may strengthen the boundedness condition on $\varphi (s)$ if f(x) is continuous. If $\varphi (s)$ is analytic in the strip $a<\Re (s)<b$ , and if $|\varphi (s)|<K|s|^{-2}$ , 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 $\varphi$ for at least $a<\Re (s)<b$ .

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

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}<\infty

where ν and p are fixed real numbers with p>1, then if f(x) is in $L_{\nu ,p}(R^{+})$ with $1<p\leq 2$ , 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

Flajolet, P.; Gourdon, X.; Dumas, P. (1995). "Mellin transforms and asymptotics: Harmonic sums" (PDF). Theoretical Computer Science. 144 (1–2): 3–58. doi:10.1016/0304-3975(95)00002-E.
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.

Mellin inversion theorem

Method

Boundedness condition

See also

References

External links