Weighted averages of Radon transforms on Z(k2).

Weighted averages of Radon transforms on the group of binary k-tuples under modulo 2 addition, which arise in applied statistics, are investigated. The inversion formula is derived by means of (discrete) Fourier transforms, and also by means of an expansion in terms of Krawtchouk polynomials. Alternate explicit representations are obtained for the coefficients in the inversion formula in several particular cases. Moreover, when the Radon transform is over the subset of Hamming distance 2, the asymptotic behavior of these coefficients is investigated when k is large (and not the square of a integer).