Generalization of the Fourier transform: Implications for inverse scattering theory

Abstract

An infinite number of ways are developed for representing a function in terms of the eigenfunctions of a three-dimensional scattering problem and simple known auxiliary functions. The utility of the new expansions, which generalize both the Fourier and Radon transforms, is shown by derivation of a new representation of the scatterer for the near- (far-) field inverse problem. Further, the scattering amplitude and potential are shown to be a generalized Fourier-transform pair.