Distributional Extension and Invertibility of the [math]-Plane Transform and Its Dual

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4662-4686, August 2024.
Abstract. We investigate the distributional extension of the [math]-plane transform in [math] and of related operators. We parameterize the [math]-plane domain as the Cartesian product of the Stiefel manifold of orthonormal [math]-frames in [math] with [math]. This parameterization imposes an isotropy condition on the range of the [math]-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 [math]-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 [math]-functions for [math]. Finally, we apply our results to study a new form of regularization for inverse problems.