The linear pencil approach to rational interpolation

Abstract

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi)orthogonal rational functions and linear pencils z B − 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 Padé 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.

A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

“…Here we will consider the recurrence relations (4.5) as a particular case of the theory of linear pencils of tridiagonal matrices that was elaborated in [5], [10], and [12], which, in turn, had their origin in [31].…”

“…This is what is actually done in Section 3 and Section 4 through the findings of H. S. Wall. The section after that is where we simply adopt a more general theory developed in [5], [10], and [12] to this very particular case of Wall's continued fraction representation of Nevanlinna functions. Besides, Section 5 reveals the relation between the spectral theories of OPUC and linear pencils of Jacobi matrices.…”

The Jacobi matrices approach to Nevanlinna–Pick problems

“…for every n ∈ Z + (see [11]). Going further in this direction, we should note that, sometimes, it is very useful to have (3.1) in the following matrix form:…”

“…, j. Consequently, the classical Christoffel-Darboux formula is the limiting case of (4.5). Moreover, it is shown in [23, Theorem 2.2] (see also [11,Section 4]…”

Padé approximants to certain elliptic-type functions

“…For restricted classes of functions, such approximants were proven to converge, locally uniformly in the domain of analyticity, as n goes large. These classes include Markov functions and rational perturbation thereof [37,23,47,12], Cauchy transforms of continuous non-vanishing functions on a segment [11,42,36] (in these examples interpolation takes place at infinity), and certain entire functions such as Polya frequencies or functions with smooth and fast decaying Taylor coefficients [5,33,34] (interpolation being now at the origin). However, such favorable cases do not reflect the general situation which is that Padé approximants often fail to converge locally uniformly, due to the occurrence of "spurious" poles that may wander about the domain of analyticity.…”