In the article on the $\Gamma$ function in the Princeton Companion to Mathematics, the author states
The Mellin transform is a type of Fourier transform, but it is defined for functions on the group $(\mathbb{R}^{+}, \times)$ rather than $(\mathbb{R}, +)$ (which is the habitat of the most familiar type of Fourier transform.
I can see well enough that the Mellin transform is defined for functions defined on $\mathbb{R}^{+}$, but I can't see why group theory has anything to do with it. In what capacity is this statement true? (For instance, does the Mellin transform preserve some group action?)
Additionally, if I give you a continuous group, can you give me an integral transform for functions defined on that group? I'd like to know if there are either 1) more examples or 2) a general procedure for finding the transform.
References appreciated.