Not the whole story indeed, but you have to dive into representation theory somewhat to get more: the Fourier transform is more or less the representation theor

Not the whole story indeed, but you have to dive into representation theory somewhat to get more: the Fourier transform is more or less the representation theory of the (abelian) group of the translations of your space, thus the homogeneity requirement. The finite-lattice version[1] (a discretized torus, basically) may serve to hint what’s in stock here. [1] https://www-users.cse.umn.edu/~garrett/m/repns/notes_2014-15/01_finite_abelian.pdf https://www-users.cse.umn.edu/~garrett/m/repns/notes_2014-15... (linear algebra required at least to the degree that one is comfortable with the difference between a matrix and an operator and knows what a direct sum is)