Darboux transformations of Jacobi matrices and Pad\'e approximation

TL;DR

This paper explores the factorization of Jacobi matrices related to Padé approximants and their transformations, revealing how Christoffel transformations can lead to unbounded matrices due to pole accumulation effects.

Contribution

It introduces a novel factorization approach for Jacobi matrices associated with Padé approximants and analyzes the impact of Christoffel transformations on matrix boundedness.

Findings

Christoffel transformation can produce unbounded Jacobi matrices.

Pole accumulation at infinity affects Padé approximants.

UL-factorization of Jacobi matrices is examined.

Abstract

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_c=UL is a monic generalized Jacobi matrix associated with the function F_c(z)=zF(z)+1. It turns out that the Christoffel transformation J_c of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at infinity of the poles of the Pad\'e approximants of the function F_c although F_c is holomorphic at infinity. The case of the UL-factorization of J is considered as well.