When to use other transforms?

Question

maple code int(g*f, x=-infinity..infinity)

when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function

If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they use for? When to use them?

Answer 1

The first two kernels are related to Mellin transforms, while the last may be expressed in terms of a Laplace transform, assuming the reality of $x$ and $t$.