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 ~(?0)? ~(?1)? ~(?2)? of rational functions with prescribed poles in [-∞,0]. Let ~(φk)k=0∞, with ~(φ0)∈ ~(?0) and ~(φk)∈ ~(?k)?k- _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 ~(?n)·?n- _1, a space of dimension 2n. Quasi- and pseudo-orthogonal functions are functions in ~(?n) that are orthogonal to some subspace of ?n- _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 ~(?n)·?n- _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 ?n- _1·?n- _1 in the quasi-orthogonal case and it is in ~(?n)·?n- _2 in the pseudo-orthogonal case.