H2- and H∞- weighting sequence approximations to rational and irrational functions (Trefethen 1981, Heltonet al. 1989) are pertinent to characteristic locus design. The most effective technique combines inverse discrete Fourier transforms with what amount to H2approximations. In this paper we discuss the theoretical background and develop some new H2and H∞ techniques. The resulting algorithms can be applied to general (rational/irrational) function and vector function approximation problems. They are shown to be particularly effective in the approximation of eigenvalue/vector functions of transfer function matrices. The derived techniques provide some very useful tools for the characteristic locus design method.