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

  • Authors:

  • Adhemar Bultheel;Pablo GonzáLez-Vera;Erik Hendriksen;Olav NjåStad

  • Affiliations:

  • Department of Computer Sc., K.U.Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium;Department of Mathematical Analysis, University of La Laguna, Tenerife, Spain;Beerstratenlaan 23, 2421GN Nieuwkoop, The Netherlands;Departmentof Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2013

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Abstract

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 {@f"k}"k"="0^~, with @f"0@?@?"0 and @f"k@?@?"k@?@?"k"-"1, k=1,2,... be the associated sequence of orthogonal rational functions. The zeros of @f"n 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 @f"n and @f"n"-"1 and depend on a real parameter @t. 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 @t 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.