Talk:Mellin transform

Relationship with other transforms

The definition of the two-sided Laplace transform is

$\left\{{\mathcal {B}}f\right\}(s)=\varphi (s)=\int _{-\infty }^{\infty }e^{-sx}f(x)dx$

Under the substitution $x=-\ln \,t$, $t\in [0,\infty ]$ this transforms into

$\left\{{\mathcal {B}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }t^{s}f(-\ln \,t){dt \over t}\ .$

So the relationship between Mellin transform and two-sided Laplace transform should be

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

rather than the stated

$\left\{{\mathcal {B}}f\right\}(s)=\left\{{\mathcal {M}}f(e^{-x})\right\}(s)\ ?$

--212.18.24.11 14:58, 22 August 2005 (UTC)

Hello my question is ..let's suppose we know the Mellin inverse transform of F(s) then..how could we calculate the Mellin inverse transform of F(as) when a is a real number, i believe that if g(x) has F(s) as its Mellin transform then $g(x^{1/a})x^{1/a-1}$ is the inverse of F(as) but i'm not pretty sure --Karl-H 14:20, 1 October 2006 (UTC)

Request for Clarification

Hello, I have a question -- the example transform of the function

$f(x)={\frac {1}{1+x}}$

is apparently not convergent on the positive real line since the integral of $\lim _{R\rightarrow \infty }\int _{0}^{R}{\frac {1}{1+x}}dx=\lim _{R\rightarrow \infty }\ln(R)$ diverges - but you state that the function must be integrable on the positive real line to have a Mellin transform? Could you correct this apparent inconsistency or explain the condition for having a Mellin transform a bit more clearly? --81.146.66.114 14:38, 24 March 2007 (UTC)

It's supposed to be locally integrable. Corrected. -Zahlentheorie 23:26, 25 March 2007 (UTC)

Ideas for improvement

I think this article could use some restructuring.

-I don't think that the fundamental strip is nearly as important as the relationship between the Mellin Transform and Laplace Transform.

-Some talk about applications to signal/image processing of the Mellin Transform would be nice.

-Discussion of the intuitive meaning of the transform, as a hyperbolic expansion compared to the Fourier Transform's sinusoidal expansion...

Your thoughts?

--Joe056 02:19, 15 March 2007 (UTC)

The importance of the fundamental strip is most evident in the use of the Mellin-Perron formula and general Mellin inversion, which is used a lot in number theory and the analysis of algorithms (harmonic sums). Unfortunately the article on Mellin-Perron is a "stub" as of today. It would need additional detailed material to back up this claim (the importance of the fundamental strip). If you don't compute the fundamental strip, you won't know which residues will contribute to the inversion integral. -Zahlentheorie 10:51, 15 March 2007 (UTC)

Fair enough. I'll incorporate some of the suggestions I have into an "Applications" section. Joe056 14:00, 22 March 2007 (UTC)

Hello there, I do not have the time at the moment for a more detailed comment, but in harmonic sums and hence in CPSC, the key property is not only scale invariance, but the fact that when you scale the argument of the source function by, say, $\mu$ (the frequency), you get a factor of $1/\mu ^{s}$ in the transform, which produces Dirichlet series when you sum over integer frequencies. -Zahlentheorie 17:34, 22 March 2007 (UTC)

Scale Invariance

I think there's a mistake here. The article says

The Mellin Transform is widely used in computer science because of its scale invariance property. The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function. This scale invariance property is analogous to the Fourier Transform's shift invariance property. The magnitude of a Fourier transform of a time-shifted function is identical to the original function.

First of all, I presume that what is intended is The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the Mellin Transform of the original function. But I don't think even this is correct. Just as the shift invariance of the magnitude of the Fourier transform does not hold for the Laplace transform, the scale invariance property as stated here does not hold for the Mellin transform (though I suspect it holds for purely imaginary z). Consider for examle, the Mellin transform of exp(-at), as given here [1]. 72.75.103.224 03:24, 2 July 2007 (UTC)

Relationship to other transforms

The article indicates the following:

The two-sided Laplace transform may be defined in terms of the Mellin transform by

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

and conversely we can get the Mellin transform from the two-sided Laplace transform by

$\left\{{\mathcal {M}}f\right\}(s)=\left\{{\mathcal {B}}f(e^{-x})\right\}(s).$

This seems inconsistent with the following link which indicates the subsequent text: Laplace Transform

Mellin transform

The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables.

If in the Mellin transform

$G(s)={\mathcal {M}}\{g(\theta )\}=\int _{0}^{\infty }\theta ^{s}g(\theta )\,{\frac {d\theta }{\theta }}$

we set θ = e^−t we get a two-sided Laplace transform.

StvC (talk) 21:12, 27 November 2017 (UTC)