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A diagonal operator on ${\ell }^p$ acts coordinatewise as $(x_n)\mapsto (a_n{x}_n)$. It is bounded when $(a_n)$ is bounded, while its inverse on its image is bounded only when the nonzero $a_n$ are bounded away from zero.

The nullspace in the Fourier diffraction theorem already prevents uniqueness, and the rapid decay of evanescent high-frequency information prevents stable recovery. Both make this an ill-posed inverse problem.

Let $A_{\pm }$ be the linear Born approximation operators mapping a real contrast supported in a known region $D$ to the measured fields on the two observation planes. For data $d_{\pm }$, a suitable Tikhonov reconstruction from two-plane scattering data minimizesA Sobolev norm penalty can instead express a smoothness preference. Reality, known support and $n=1-V/k>0$ can be imposed as constraints. For the unconstrained complex Hilbert-space problem with the displayed quadratic penalty, write $A=(A_{+},A_{-})$ and $d=(d_{+},d_{-})$. Its Tikhonov normal equation isTikhonov regularization selects a stable preferred estimate; it does not establish that the original two-plane data uniquely determine the true contrast.