Fourier and Zak transforms of multiplicative characters

04/13/2021

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by   Andrzej K. Brodzik, et al.

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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.