Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 2 / ii / Solution

UseThe adjoint operator is consequentlyWith a unitary Fourier normalization, this is simply ${\mathcal{F}}^{\ast }={\mathcal{F}}^{-1}$.

At fixed incident direction and wavenumber, defineTaking the complex conjugate of the kernel givesThe least-squares minimizer of $\Vert TV-f_{\infty }{\Vert }^2$ satisfies the normal equation for a linear inverse problem $T^{\ast }TV=T^{\ast }{f}_{\infty }$. Fourier inversion on the measured transfer-vector set is exactly the corresponding Moore--Penrose reconstruction $T^{†}{f}_{\infty }$; hence the formal solution in part (i) is the minimum-norm least-squares solution when the data are incomplete or inconsistent.