Convergence of Diagonal Padé Approximants for a Class of Definitizable Functions

Convergence of diagonal Padé approximants is studied for a class of functions which admit the integral representation F(λ) = r1(λ) R 1 −1 tdσ(t) t−λ + r2(λ), where σ is a finite nonnegative measure on [−1, 1], r1, r2 are real rational functions bounded at ∞, and r1 is nonnegative for real λ. Sufficient conditions for the convergence of a subsequence of diagonal Padé approximants of F on R \ [−1, 1] are found. Moreover, in the case when r1 ≡ 1, r2 ≡ 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 F in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of F at infinity.