One also can think of transforms as the eigenfunctions of the continuous part of the spectrum of a differential operator. In differential equations (DE) theory,

One also can think of transforms as the eigenfunctions of the continuous part of the spectrum of a differential operator. In differential equations (DE) theory, a well posed DE has a structure (the DE itself), a domain and enough independent boundary conditions. Fourier transform will show up for an harmonic oscilator in the whole real line with incoming and outgoing wave boundary conditions, while Laplace will show up when working on semi-infinite interval whit initial conditions and proper convergence at infinity. These are the most common, but not the only transforms one can build. There are also Melin and Hankel transforms, and by playing with the operator, the domain and the boundary conditions, we can construct the adequate transform for each given problem. Spectral theory of DE’s is such a beautiful topic.