Orthogonal rational functions and diagonal-plus-semiseparable matrices

摘要

The space of all proper rational functions with prescribed real poles is considered. Given a set of points z_i on the real line and the weights w_i, we define the discrete inner product <Φ,ψ> := ∑ from i=0 to n of w_i~2Φ(z_i)ψ(z_i). 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.