A paper I'm reading now defines invariant signal "representations" as those functions $\Phi$ of signals $x$ in a Hilbert space such that $\Phi(g\cdot x) = \Phi(x)$ where $g\cdot x$ is the action of some group element.
I'm not quite sure what is meant by "representation" in this case. The paper goes on to cite the Fourier transform modulus and the function $\Phi(x) = x(u-a(x))$ for some function $a$ as examples of these representations. What is meant by this term in general? A google search hasn't produced anything concrete.
Thank you.
After some more looking, I found that a represenation is any operator $\Phi : L^2(\mathbb{R}^d) \to H$ for some Hilbert space $H$.