# The linear pencil approach to rational interpolation

## TL;DR

This paper extends the theory of rational interpolation by analyzing linear pencils of matrices, providing new convergence criteria, spectral analysis, and links to biorthogonal rational functions, broadening classical results in Pade approximation.

## Contribution

It introduces a new criterion for the resolvent set of linear pencils and generalizes convergence results for rational interpolants beyond classical Pade approximants.

## Key findings

- New resolvent set criterion for linear pencils
- Generalized convergence results for rational interpolants
- Explicit spectral and numerical range calculations for specific cases

## 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, beside 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 infinity, we compute explicitly the spectrum and the numerical range of the underlying linear pencil.