Editing Mellin transform (section)

==Relationship to other transforms==
The [[two-sided Laplace transform]] may be defined in terms of the Mellin transform by
<math display="block">\mathcal{B}  \left\{f\right\}(s) = \mathcal{M} \left\{f(-\ln x) \right\}(s)</math>
and conversely we can get the Mellin transform from the two-sided Laplace transform by
<math display="block">\mathcal{M}  \left\{f\right\}(s) = \mathcal{B}\left\{ f(e^{-x})\right\}(s).</math>

The Mellin transform may be thought of as integrating using a kernel {{mvar|x{{sup|s}}}} with respect to the multiplicative [[Haar measure]], <math display="inline">\frac{dx}{x}</math>, which is invariant under dilation <math>x \mapsto ax</math>, so that <math display="inline">\frac{d(ax)}{ax} = \frac{dx}{x};</math> the two-sided Laplace transform integrates with respect to the additive Haar measure <math>dx</math>, which is translation invariant, so that <math>d(x+a) = dx\,.</math>

We also may define the [[Fourier transform]] in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
<math display="block">\left\{\mathcal{F} f\right\}(-s) = \left\{\mathcal{B} f\right\}(-is)
= \left\{\mathcal{M} f(-\ln x)\right\}(-is)\ .</math>
We may also reverse the process and obtain
<math display="block">\left\{\mathcal{M} f\right\}(s) = \left\{\mathcal{B} f(e^{-x}) \right\}(s) = \left\{\mathcal{F} f(e^{-x})\right\}(-is)\ .</math>

The Mellin transform also connects the [[Newton series]] or [[binomial transform]] together with the [[Poisson generating function]], by means of the [[Poisson–Mellin–Newton cycle]].

The Mellin transform may also be viewed as the [[Gelfand transform]] for the [[convolution algebra]] of the [[locally compact abelian group]] of positive real numbers with multiplication.