Quadratures associated with pseudo-orthogonal rational functions on the real half line with poles in [-∞, 0]

We consider a positive measure on [0,∞) and a sequence of nested spaces L0 ⊂ L1 ⊂ L2 · · · of rational functions with prescribed poles in [−∞, 0]. Let {φk}k=0, with φ0 ∈ L0 and φk ∈ Lk \ Lk−1, k = 1, 2, . . . be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in Ln · Ln−1, a space of dimension 2n. Quasiand pseudo-orthogonal functions are functions in Ln that are orthogonal to some subspace of Ln−1. Both of them are generated from φn and φn−1 and depend on a real parameter τ . Their zeros can be used as the nodes of a rational GaussRadau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of Ln · Ln−1 where the quadrature is exact. The parameter τ is used to fix a node at a pre-assigned point. The space where the quadratures are exact have dimension 2n− 1 in both cases but it is in Ln−1 · Ln−1 in the quasi-orthogonal case and it is in Ln ·Ln−2 in the pseudo-orthogonal case.