On an integro-differential transform on the sphere

In a recent paper the authors have proved that a convex body K subset of R- d, d greater than or equal to 2, containing the origin O in its interior, i s symmetric with respect to O if and only if Vd-1(K boolean AND H ') greate r than or equal to Vd-1(K boolean AND H) for all hyperplanes H, H ' such th at H and H ' are parallel and H ' There Exists O (Vd-1 is (d-1)-measure). F or the proof the authors have employed a new type of integro-differential t ransform that lets to correspond to a sufficiently nice function f on Sd-1 the function R((1))f, where (R((1))f)(xi) = integralS(d-1 boolean AND xi pe rpendicular to) (partial derivativef/partial derivative psi )d eta) - with xi is an element of Sd-1, pole and psi as geographic latitude - and have de termined the null-space of the operator R-(1). In this paper we extend the definition to any integer m greater than or equal to1, defining (R((m))f)(x i) analogously as for m = 1, but using partial derivative (m)f/partial deri vative psi (m) rather than deltaf/delta psi. (The case m = 0 is the spheric al Radon transformation (Funk transformation).) We investigate the null-spa ce of the operator R-(m): up to a summand of finite dimension, it consists of the even (odd) functions in the domain of the operator, for rn odd (even ). For the proof we use spherical harmonics.