X-rays of forms and projections of currents

Abstract

We study a new Radon-like transform that averages projected p-forms in R n over affine (n — k)-spaces. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in the work by Gelfand, Graev and Shapiro (1969). Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth p-form on R n as a (continuous) superposition of pullbacks from p-forms on k-dimensional subspaces. In turn, such a synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in R n (e.g., compact oriented k-dimensional subvarieties) from their oriented projections onto k-planes.