Here are two generalizations of the notion of a Fourier transform. I am also aware of the Pontryagin Duality generalization for locally compact abelian groups, though I am personally more concerned with discrete cases.
Given a finite group $G$, we decompose the natural representation of $G$ in the space of functions $G \to \mathbb{C}$ into irreducibles. The Fourier coefficient at an irreducible $\rho$ is then the weighted sum $\sum_{g \in G} f(g) \rho(g)$. We recover the classical discrete transform when $G = C_n$.
Given a graph $\Gamma$, we can write any function on its vertices in the basis of eigenfunctions $\{\phi_i\}$ of its Laplacian $\mathcal{L}(\Gamma)$, with the coefficients in this basis being the Fourier coefficients. As far as I can tell, this definition is called a Fourier transform because the standard Fourier series rewrites a function in the basis of sine waves, the eigenfunctions of the continuous Laplacian $\frac{\partial^2}{\partial x^2}$.
Is there any real sense in which these two generalizations describe the same phenomenon? Can the former definition be rephrased in terms of eigenfunctions of some Laplacian? For the latter definition, maybe the natural convolution operation $f \ast g = \sum_{i}\hat{f}(i)\hat{g}(i)\phi_i$ has some nice algebraic meaning? While a truly formal common generalization would be nice, I am also okay with a conceptual one. I have seen the metaphor of "Fourier transforms break signals into frequencies" but it is unclear to me why these two types of generalized "frequencies" are both equally deserving of that name.
