Mellin Transform - Relationship To Other Transforms

Relationship To Other Transforms

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

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

The Mellin transform may be thought of as integrating using a kernel xs with respect to the multiplicative Haar measure, which is invariant under dilation, so that ; the two-sided Laplace transform integrates with respect to the additive Haar measure, which is translation invariant, so that .

We also may define the Fourier transform in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then

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

We may also reverse the process and obtain

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

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.

Read more about this topic: Mellin Transform

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