On the Spherical Slice Transform

TL;DR

This paper investigates the spherical slice transform, relating it to the Radon-John transform, and derives explicit formulas and properties for a range of dimensions, extending known results to new cases.

Contribution

It establishes an explicit connection between the spherical slice transform and the Radon-John transform for all relevant dimensions, enabling the transfer of known properties.

Findings

Derived explicit formulas linking the transforms.

Reformulated inversion formulas and support theorems.

Extended known results to new dimensions.

Abstract

We study the spherical slice transform which assigns to a function on the $n$-dimensional unit sphere the integrals of that function over cross-sections of the sphere by $k$-dimensional affine planes passing through the north pole. These transforms are well known when $k=n$. We consider all $1< k < n+1$ and obtain an explicit formula connecting the spherical slice transform with the classical Radon-John transform over $(k-1)$-dimensional planes in the $n$-dimensional Euclidean space. Using this connection, known facts for the Radon-John transform, like inversion formulas, support theorems, representation on zonal functions, and others, can be reformulated for the spherical slice transform.