The spectral projections of convolution operator

Question

Given a self-adjoint operator $A$ in a Hilbert space $H$. How can one find its spectral projections $\{E_{\lambda}\}_{\lambda\in\sigma(A)}$? In particular, given a convolution operator on $L^2(G)$, where $G$ is a locally compact separable(+add any reasonable assumptions so that we get a "nice" harmonic analysis), how can one find its spectral projections?