On the spherical slice transform
Boris Rubin · Analysis and Applications · 2021
We study the spherical slice transform [Formula: see text] which assigns to a function f on the unit sphere Sn in [Formula: see text] the integrals of f over cross-sections of Sn by k-dimensional affine planes passing through the north pole (0,…, 0, 1). These transforms are known when k = n. We consider all 2 ≤ k ≤ n and obtain an explicit formula connecting [Formula: see text] with the classical (k − 1)-plane Radon–John transform [Formula: see text] on [Formula: see text]. Using this connection, known facts for Rk−1, like inversion formulas, support theorems, representation on zonal functions, and some others, are reformulated for [Formula: see text].