Fourier sine transform ($S(k)={\int }_0^{\infty }\sin (kx)f(x)dx$)

The Fourier sine transform is a half-line transform adapted to a homogeneous Dirichlet boundary condition. With the displayed normalization, inversion is $f(x)=(2/\pi ){\int }_0^{\infty }\sin (kx)S(k)dk$. It is equivalent to the Fourier transform of an odd extension. If $f(0)=0$ and boundary terms vanish, two integrations by parts give the transform of $f^{\prime \prime }$ as $-k^{2}S(k)$.

For sufficiently regular integrable functions on a half-line, Fourier sine transform inversion recovers the odd extension at continuity points. A zero endpoint value is required when demanding continuity at the endpoint.