Revisiting the Problem of Recovering Functions in $\Bbb R^{n}$ by Integration on $k$ Dimensional Planes

TL;DR

This paper develops inversion methods for the Radon transform on specific subsets of $k$-dimensional planes in $R^n$, addressing the challenge of reconstructing functions from their integrals over these planes.

Contribution

It introduces new inversion techniques for the Radon transform on carefully chosen subsets of planes in $R^n$, expanding the classical theory to less studied cases.

Findings

Inversion methods are established for certain subsets of $k$-planes.

The dimension of these subsets is exactly $n$, ensuring well-posedness.

The methods extend classical Radon inversion to new geometric configurations.

Abstract

The aim of this paper is to present inversion methods for the classical Radon transform which is defined on a family of $k$ dimensional planes in $\Bbb R^{n}$ where $1\leq k\leq n - 2$. For these values of $k$ the dimension of the set $\mathcal{H}(n,k)$, of all $k$ dimensional planes in $\Bbb R^{n}$, is greater than $n$ and thus in order to obtain a well-posed problem one should choose proper subsets of $\mathcal{H}(n,k)$. We present inversion methods for some prescribed subsets of $\mathcal{H}(n,k)$ which are of dimension $n$.