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

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.