X-rays of currents and projections of forms

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. 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 Gelfand-Graev-Shapiro. 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 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.