On the image of a generalized $d$-plane transform on $ R^n$

Abstract: The generalized $d$-plane transform of a function $f$ on ${\scriptstyle \RR}^n$ is defined on a set ${\cal E}$ of $d$-dimensional affine subspaces ("$d$-planes") of ${\scriptstyle\RR}^n$ by integration of $f$ over each subspace in ${\cal E}$. In general, it renders less information about the unknown function $f$ than in the special case of the well known Radon $d$-plane transform, where ${\cal E}$ contains every $d$-plane in ${\scriptstyle\RR}^n$. We study the case where ${\cal E}$ appears as an orbit of a matrix group and characterize the range of the spaces of Schwartz functions and of smooth ones with compact support.

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