This paper continues the authors previous work that deals with the problem of reconstructing some function defined on a finite group G from averages of the function over various neighborhoods of points. It generalizes work done earlier by Diaconis and Graham, and Fill, among others.

Diaconis and Graham discuss the cases where G = Z^{k}_{2}, the group of binary k-tuples, and where G = S_{n}, the symmetric group on n letters, in order to provide an exposition on discrete Radon transforms which appear in applied statistics. Fill examines the case G = Z^{k}_{2}, the group of integers modulo n. This case arises in directional data analysis and circular time series. Other finite analogues of the Radon transform occur, but we shall restrict ourselves to the case G = Z^{k}_{n}, the group of k-tuples of the integers modulo n, as these results are of use in k-dimensional toroidal time series.

*with* Elinor Velasquez. SIAM J. Discrete Math. 18 (2004/05), no. 3, 472–478. **MR2134409 44A12 (43A75)
**DOI: 10.1137/S0895480103430764