 Odd periodic extension

Given a function on $(0,L)$, reflect it oddly onto $(-L,0)$ and repeat with period $2L$. Nonzero endpoint limits produce jumps at the joins. The resulting Fourier series is the Fourier sine series of the original function; its value at a jump converges to the mean of the one-sided limits under the Dirichlet convergence theorem for Fourier series.