Convergence of diagonal Padé approximants for a class of definitizable functions

Abstract

Convergence of diagonal Padé 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 Padé 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 Padé approximants of $\mathfrak{F}$ in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of $\mathfrak{F}$ at infinity.

arXiv commentDedicated to the memory of Peter Jonas. 24 pages