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

摘要

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. Quasi- and 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 Gauss–Radau 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 preassigned point. The space where the quadratures are exact has 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.