2009/08/24 by Bernhard Beckermann, Beckermann, Bernhard, Maxim Derevyagin +3
Computer Science · Mathematics · Physics and Astronomy · #30E10 #40A15 #47A57 #47B36 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #math.CA #math.SP #msc:30E10 #msc:40A15 #msc:47A57 #msc:47B36
paper · pdf · doi:10.48550/arxiv.0908.3381
22 pages
openalex publication_date 2009/08/24 · arxiv created 2010/02/02 · arxiv updated 2010/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.