Fourier inversion on a finite group ($f(x)={\sum }_{\rho }{d}_{\rho }\mathrm{tr}(\hat{f}(\rho )\rho (x{)}^{\ast })$)

The Fourier transform on a finite group convention $\hat{f}(\rho )={\mathbb{E}}_{x}f(x)\rho (x)$ has the displayed inversion formula. The sum is over one representative from each equivalence class of unitary irreducible representations. It follows from the Schur orthogonality relations and the regular representation decomposition, and holds at every group element without a limiting argument.