Mellin convolution

Summary

The Mellin convolution of two functions $f$ and $g$ on $(0, \infty)$ is $(f \star g)(x) = \int_0^\infty f(x/y) g(y) \, \frac{dy}{y}$. It is the convolution of the multiplicative group $(0, \infty)$ with its invariant measure $dy/y$. The Mellin transform turns it into a product: $\mathcal{M}(f \star g) = \mathcal{M}f \cdot \mathcal{M}g$ where both transforms converge absolutely. It describes, for example, the law of a product of independent positive random variables and many integral operators with homogeneous kernels.