The definition of homomorphism is https://en.wikipedia.org/wiki/Homomorphism and definition of diagonalization needs to be understood in the context described in Fourier transform as diagonalization of convolution
The point is, one can think of Fourier transform as diagonalization of convolution. Further, Fourier transform $\mathcal{F}: L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an algebra homomorphism from $L^1$ into a sub-algebra $\mathcal(L^1(\mathbb{R}))$ of the space $C_0$ (continuous functions that vanish at $\pm\infty$).
So the question: is there a common theme between diagonalization and homomorphism or Fourier transform is the only transform which diagonalize (convolution operator) and also has a homomorphism space. Is there any other such tranformation? Can there be homomorphism space for every diagonalisable operator can it go to very diagonalisable matrix?
Please forgive if this question is non mathematical as I am trying to compare roses to apple.
