Convergence of diagonal Pad\'e approximants for a class of definitizable functions

TL;DR

This paper investigates the convergence properties of diagonal Padé approximants for a specific class of functions represented by integral formulas involving rational functions and measures, providing conditions for convergence and analyzing special cases.

Contribution

It establishes sufficient conditions for the convergence of subsequences of diagonal Padé approximants for a class of definitizable functions, including cases with spectral gaps.

Findings

Convergence of subsequences of Padé approximants is proven under certain conditions.

In the case of a spectral gap containing zero, convergence occurs within the gap.

Operator representations using generalized Jacobi matrices are used to analyze convergence.

Abstract

Convergence of diagonal Pad\'e approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(\lambda)=r_1(\lambda)\int_{-1}^1\frac{td\sigma(t)}{t-\lambda}+r_2(\lambda), $ where $\sigma$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\infty$, and $r_1$ is nonnegative for real $\lambda$. Sufficient conditions for the convergence of a subsequence of diagonal Pad\'e approximants of $ {\mathfrak F}$ on $\dR\setminus[-1,1]$ are found. Moreover, in the case when $r_1\equiv 1$, $r_2\equiv 0$ and $\sigma$ has a gap $(\alpha,\beta)$ containing 0, it turns out that this subsequence converges in the gap. The proofs are based on the operator representation of diagonal Pad\'e approximants of $ {\mathfrak F}$ in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of $ {\mathfrak F}$ at infinity.