Rational approximation
Researchers
and formerly ...Description
ORF on the circle, real part
The group investigates recursive algorithms and their applications in many problems in mathematics like continued fractions, orthogonal polynomials, rational approximation and interpolation (especially Padé approximation) etc.
A general module theoretic framework has been designed to solve very general matrix or vector interpolation problems which are generalizations of classical Newton-Padé or Padé-Hermite problems. This framework gives all the solutions of the problem and it is possible to select the one with minimal complexity. Moreover elegant algorithms can be provided to compute the successive solutions recursively. The numerical stability, more precisely the incorporation of a look-ahead strategy in these algorithms is under investigation.
The recurrence relation for polynomials orthogonal on the unit circle has been generalized to the situation of orthogonal rational functions on the unit circle. This research is done in collaboration with the NTH in Trondheim (Norway) the University of Amsterdam (The Netherlands), and the University of La Laguna (Tenerife). The most recent investigations concentrated on convergence properties when poles are on the circle and applications to numerical quadrature. The application to the case of the real line is elaborated as well. This can be used to generate quadrature formulas of Gaussian or interpolatory type, on the real line, the positive real line or the unit circle. Convergence and estimates for the rate of convergence of these quadrature formulas and of multipoint or two-point Padé or Padé type approximants were obtained.
The applications of these algorithms in signal processing, linear systems (partial realization), identification, etc. are also investigated.
