Singular system of a periodic convolution operator (${\sigma }_n=|c_n|,v_n=e_n,u_n=(c_n/|c_n|)e_n$)

The displayed singular value system applies when the Fourier multipliers $c_n$ are nonzero. The phase in the left singular vector is necessary for complex kernels. If a multiplier vanishes, its mode instead belongs to the corresponding null spaces and is excluded from the positive singular system.