Fourier sine basis ($e_n(x)=\sqrt{2/\pi }\sin (nx)$)

The functions $\sqrt{2/\pi }\sin (nx)$, $n\ge 1$, 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.