This is a reference request: I was curious if there was any pre-existing work on characterizing all of the involutions from one functional space to another.
For example the Fourier Transform counts as at least on involution on $L^2(R)$, another is $I(f(x)) = -f(x)$. How many more are there? (In this case I am using this definition of Fourier Transform: Involutive fourier transform)
There are at least as many involutions on $C^1(R)$ as there are continuous involutions on $R$. (ex $I(f(x)) = f(x)^{-1}$ is an involution since $1/x$ is an involution on $R$), but the Legendre transform is also an involution on $C^1(R)$
I am interested in this because I have always found the Fourier inversion theorem remarkable, and thought this lens might help shed some light.
Please excuse the lack of precise definitions here, but since I am looking for references I am casting a wide net. I understand if this is deemed to broad for SE.