Fourier deconvolution (${\phi }_X={\phi }_Y/{\phi }_{ϵ}$)

Independent additive noise $Y=X+ϵ$ multiplies characteristic functions: ${\phi }_Y={\phi }_X{\phi }_{ϵ}$. When the noise characteristic function is nonzero, division recovers ${\phi }_X$. A smoothing kernel limits the frequencies used, controlling the amplification of sampling error where $|{\phi }_{ϵ}|$ is small.