= Fourier convolution of two periodic L2 functions
{c}
{title2=$\int_{-\pi}^{\pi}f(s)g(t-s)ds=\sum_n\langle f,e_n\rangle\langle g,e_n\rangle e^{int}$}
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 $\|f\|_2\|g\|_2$, so the resulting series converges absolutely and uniformly. This proves <continuity> of the convolution even though neither original <Fourier series> need converge <pointwise>.
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