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

Adhemar Bultheel       Pablo González-Vera       Erik Hendriksen       Olav Njåstad


Abstract:

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 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.

Status:
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",
   }
   
File(s): preprint.pdf (272K)


Contact: Adhemar Bultheel