Schur's theorem for modified discrete Fourier transform

Abstract. We find the eigenvalues of modified Fourier matrix S with entries Sk j = 1 √n ω k(1− j) , 0 k, j n− 1 , where ω = exp 2πi n . For this matrix S4 = ωI . The matrix has an interesting property: for n = 4m eigenvalues have equal multiplicities. We prove a theorem giving the multiplicities of eigenvalues for all n . The theorem is similar to Schur’s theorem (1921) for standard Fourier matrix. Our proofs are self-contained. In the proof we calculate modified Gauss sums by means of the classical analysis.