Fourier sine basis
= Fourier sine basis
{c}
{title2=$e_n(x)=\sqrt{2/\pi}\sin(nx)$}
The functions $\sqrt{2/\pi}\sin(nx)$, $n\geq1$, form a complete <orthonormal basis> of $L^2(0,\pi)$. An odd extension to $(-\pi,\pi)$ reduces completeness to that of the <Fourier series>. Equality of all sine coefficients therefore implies equality almost everywhere; if both functions are continuous, it implies equality everywhere.