We consider a positive measure on [0,∞) and a sequence of
nested spaces ℒ_{0} ⊂ ℒ_{1} ⊂ ℒ_{2} ...
of rational functions with prescribed poles in [-∞,0].
Let {φ_{k} : k = 0 ... ∞}, with
φ_{0} ∈ ℒ_{0} and φ_{k} ∈ ℒ_{k}∖ℒ_{k-1}, k = 1,2,...
be the associated sequence of orthogonal
rational functions. The zeros of φ_{n} can be used as the nodes of
a rational Gauss quadrature formula that is exact for all functions in
ℒ_{n} . ℒ_{n-1}, a space of dimension 2n.
Quasi- and pseudo-orthogonal functions are functions in ℒ_{n}
that are orthogonal to some subspace of ℒ_{n-1}.
Both of them are generated from φ_{n} and φ_{n-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 ℒ_{n} . ℒ_{n-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 ℒ_{n-1} . ℒ_{n-1} in the quasi-orthogonal case
and it is in ℒ_{n} . ℒ_{n-2} in the pseudo-orthogonal case.

Online since 12 July 2012

**BiBTeX entries:**

@article{ArtBGHN10b, author = "A. Bultheel and P. Gonz{\'a}lez-Vera and E. Hendriksen and O. Nj{\aa}stad", journal = "Journal of Computational and Applied Mathematics", pages = "589-602", title = "Quadratures associated with pseudo-orthogonal rational functions on the real half line with poles in $[-\infty,0]$", volume = "237", number = "1", year = "2013", url = "http://nalag.cs.kuleuven.be/papers/ade/PORF/index.html", DOI = "10.1016/j.cam.2012.06.037", LIRIAS = "351157", ZBL = "06097330", MR = "2966931", }

Contact: Adhemar Bultheel