The Radon transform on Abelian Groups

The Radon transform on a group A is a linear operator on the space of functions /: A-+ C. It is shown that if A = Z;: then the Radon transform with respect to a subset B c .4 is not invertible if and only if B has the same number of elements in every coset of some maximal subgroup of A. The same does not hold in general for arbitrary finite abelian groups. ' IW7 ACxhlK I%\\. 1°C Let A be a finite group and B c A, a subset. For every function j'~ A-+ @ one defines the function F,: A + Cc, the Radon transform qf,f with respect to B by F,(a)= c .f(ab). (1) h t H The principal problem we address here is: for which subsets B is the Radon transform invertible, i.e., knowledge of the function F, determines ,f uniquely. Such sets are called unique imersion sets. Unique inversion sets were investigated in Diaconis and Graham [ 11, where particular attention is given to the case A = 22;. The main result of this note gives a combinatorial description of unique inversion sets in .Z; when p is a prime. Let us say that B ts uniformly distributed modulo the subgroup A, < A if I B n aA,I is the same for all aE A. Note that this implies IA : A,\ divides 14.