Radon-John transforms and spherical harmonics

The d-plane Radon-John transform takes functions on R-n to functions on the set of all d-dimensional planes in R-n by integration over these planes. We study the action of this transform on degenerate functions of the form f(x) = f(0)(r) Y-k(theta), where r = vertical bar x vertical bar > 0, theta = x/vertical bar x vertical bar, and Yk is a spherical harmonic of degree k. It is shown that the results for d < n - 1 are surprisingly different from those in the known case d = n - 1.