On the Finiteness of the Inverse Diffraction Transform Kernel

The diffraction transform solves both the direct and the inverse scattering problem for the Helmholtz equation. Its kernel is itself an integral. It is the representation of the evolution operator assosiated with translations of a constrained Cartesian coordinate. The corresponding momentum operator is, consequently, non-Hermitian. This fact threatens the inverse scattering problem with a divergence if the transform kernel is understood as a Cauchy integral. The kernel is, however, everywhere convergent if its integral representation is interpreted as a Cesaro summable integral. Taking a Bjorken type limit of the Cesaro representation leads to the summability of the divergent kernel with respect to a spectral measure. This accords well with the general solution of the scattering problem in which such spectral functions are admissible. The use of variously motivated cut-offs and other ad hoc filters to enforce convergence is therefore highly contrived.