Orthogonal rational functions and diagonal-plus-semiseparable matrices

The space of all proper rational functions with prescribed real poles is considered. Given a set of points zi on the real line and the weights wi, we define the discrete inner product &lt,φ,ψ> := Σni=o wi2φ(zi)ψ(zi). In this paper we derive an efficient method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis functions with respect to the discrete inner product. We will show that these coefficients can be computed by solving an inverse eigenvalue problem for a diagonal-plus-semiseparable matrix.