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This article provides a novel and simple range description for the sphericalmean transform of functions supported in the unit ball of an odd dimensionalEuclidean space. The new description comprises a set of symmetry relationsbetween the values of certain differential operators acting on the coefficientsof the spherical harmonics expansion of the function in the range of thetransform. As one application of this range characterization, we construct anexplicit counterexample proving that unique continuation type results cannothold for the spherical mean transform in odd dimensional spaces. Finally, as anauxiliary result of one of our proofs, we derive a remarkable cross productidentity for the spherical Bessel functions of the first and second kind, whichmay be of independent interest in the theory of special functions.

A simple range characterization for spherical mean transform in odd dimensions and its applications