Generalized convolution of an integral transform

Summary

A convolution of an integral transform $T$ is a bilinear product $\circledast$ of functions that $T$ turns into the pointwise product: $T(f \circledast g) = Tf \cdot Tg$. The usual convolution on the line plays this role for the Fourier transform. The convolution on the half-line plays it for the Laplace transform, and the Mellin convolution for the Mellin transform. Finding such a product for a given transform gives it a convolution theorem, which is what makes it useful for solving integral and differential equations.