  In this paper we derive formulas for the N-point discrete Fourier transform
and the R1 x R2 finite Zak transform of multiplicative characters on Z/N, where
N is an odd integer, and R1 and R2 are co-prime factors of N. In one special
case this permits computation of the discrete Fourier transform and the finite
Zak transform of the Jacobi symbol, the modified Jacobi sequence, and the
Golomb sequence. In other cases, not addressed here, this permits computation
of the discrete Fourier transform and the finite Zak transform of certain
complex-valued sequences. These results constitute, to our knowledge, the first
unified treatment of key Fourier and Zak space properties of multiplicative
characters. These results also provide a convenient framework for the design of
new character-based sequences.
