Distributional Extension and Invertibility of the \(k\)-Plane Transform and Its Dual

We investigate the distributional extension of the k-plane transform in R d and of related operators.We parameterize the k-plane domain as the Cartesian product of the Stiefel manifold of orthonormal k-frames in R d with R d-k .This parameterization imposes an isotropy condition on the range of the k-plane transform which is analogous to the even condition on the range of the Radon transform.We use our distributional formalism to investigate the invertibility of the dual k-plane transform (the "backprojection" operator).We provide a systematic construction (via a completion process) to identify Banach spaces in which the backprojection operator is invertible and present some prototypical examples.These include the space of isotropic finite Radon measures and isotropic L p -functions for 1 < p < ∞.Finally, we apply our results to study a new form of regularization for inverse problems.