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

:$f(x)= \ = \frac \int_^ x^ \varphi(s)\, ds$

we have that

:$\varphi(s)= \ = \int_0^ x^ f(x)\,dx.$

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

:$\varphi(s)=\int_0^ x^ f(x)\,dx$

is absolutely convergent when $a < \Re(s) < b$. 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 strengthen if
$f(x)$ is continuous. If $\varphi(s)$ is analytic in the strip $a < \Re(s) < b$, and if $, \varphi(s), < K , s, ^$, 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 $a < \Re(s) < b$.

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
$L_(R^)$ the weighted Lp space of complex valued functions $f$ on the positive reals such that

:$\, f\, = \left(\int_0^\infty , x^\nu f(x), ^p\, \frac\right)^ < \infty$

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

:$f(x)=\frac \int_^ x^ \varphi(s)\,ds.$

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

Since the two-sided Laplace transform can be defined as

:$\left\(s) = \left\(s)$

these theorems can be immediately applied to it also.

See also

*Mellin transform
*Nachbin's theorem

References

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*{{cite book , last=Zemanian , first=A. H. , title=Generalized Integral Transforms , publisher=John Wiley & Sons , year=1968

External links

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

Integral transforms
Theorems in complex analysis
Laplace transforms