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}(λ)=r_1(λ)\int_{-1}^1\frac{tdσ(t)}{t-λ}+r_2(λ), $ where $σ$ 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 $λ$. 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 $σ$ has a gap $(α,β)$ 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.