Publications of Adhemar Bultheel | |
We consider a positive measure on [0,∞) and a sequence of nested spaces â_{0} ⊂ â_{1} ⊂ â_{2} ... of rational functions with prescribed poles in [-∞,0]. Let {φ_{k} : k = 0 ... ∞}, with φ_{0} ∈ â_{0} and φ_{k} ∈ â_{k}∖â_{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 â_{n} . â_{n-1}, a space of dimension 2n. Quasi- and pseudo-orthogonal functions are functions in â_{n} that are orthogonal to some subspace of â_{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 â_{n} . â_{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 â_{n-1} . â_{n-1} in the quasi-orthogonal case and it is in â_{n} . â_{n-2} in the pseudo-orthogonal case.
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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", }File(s): preprint.pdf (272K)