On Functions with Zero Integral Moments over Balls of Fixed Radius

We consider precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. The case where, together with zero integrals for f over congruent balls, the functions x_jf for all j have zero integrals over these balls is completely investigated. An inversion formula recovering a function in the class C^∞ from indicated integral moments is derived.