The linear pencil approach to rational interpolation

Authors:
Bernhard Beckermann;Maxim Derevyagin;Alexei Zhedanov
Affiliations:
Laboratoire Painlevé UMR 8524 (ANO-EDP), UFR Mathématiques-M3, UST Lille, F-59655 Villeneuve d'Ascq Cedex, France;Laboratoire Painlevé UMR 8524 (ANO-EDP), UFR Mathématiques-M3, UST Lille, F-59655 Villeneuve d'Ascq Cedex, France and Department of Nonlinear Analysis, Institute of Applied Mathematics a ...;Institute for Physics and Engineering, R. Luxemburg Street 72, 83114 Donetsk, Ukraine
Venue:
Journal of Approximation Theory
Year:
2010

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Abstract

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Pade approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils zB-A of two tridiagonal matrices A,B, following Spiridonov and Zhedanov. In the present paper, as well as revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Pade approximants in terms of complex Jacobi matrices to the more general case of convergence of rational interpolants in terms of the linear pencil. We also study generalizations of the Darboux transformations and the link to biorthogonal rational functions. Finally, for a Markov function and for pairwise conjugate interpolation points tending to ~, we compute the spectrum and the numerical range of the underlying linear pencil explicitly.