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

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$isafinitenonnegativemeasureon$[-1,1]$,$r_1$,$r_2$ are real rational functions bounded at $\mathrm{\infty }$, 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 $\text{dR}\setminus [-1,1]$ 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.