Fourier convolution of two periodic L2 functions (${\int }_{-\pi }^{\pi }f(s)g(t-s)ds={\sum }_n⟨f,e_n⟩⟨g,e_n⟩e^{int}$)

With $e_n(s)=(2\pi {)}^{-1/2}{e}^{ins}$, periodic $L^2$ functions have the displayed convolution identity for every real $t$. Apply the polarized Parseval identity to $f$ and $\overline{g(t-\cdot )}$; a change of variable gives the latter function's pairing with $e_n$. The Cauchy-Schwarz inequality bounds the sum of absolute coefficient products by $\Vert f{\Vert }_2\Vert g{\Vert }_2$, so the resulting series converges absolutely and uniformly. This proves continuity of the convolution even though neither original Fourier series need converge pointwise.