Imagine a 3D vector field $\vec{E}$ in a specific volume. There exists a 3D fourier transform with wave numbers $\vec{k} = (k_x, k_y, k_z)$. Now I want to find a decomposition of the vector field into fourier components satisfying $\vec{E}\cdot\vec{k} = 0$.
Is it possible to find a decomposition with this constraint or does uniqueness of the fourier transform forbid this?