An interpolation algorithm for orthogonal rational functions

  • Authors:

  • J. Van Deun;A. Bultheel

  • Affiliations:

  • Department of Computer Science, Katholieke Universiteit Leuven, Belgium;Department of Computer Science, Katholieke Universiteit Leuven, Belgium

  • Venue:
  • Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
  • Year:
  • 2004

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Abstract

Let A={α1, α2...} be a sequence of numbers on the extended real line R^=R∪{∞} and µ a positive bounded Borel measure with support in (a subset of) R^. We introduce rational functions φn with poles {α1,...αn} that are orthogonal with respect to µ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system {φn} is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials.To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz-Herglotz-Nevanlinna transform Ωµ of the measure µ. Error bounds are derived and some examples serve as illustration.