The convergence of padé approximants and the size of the power series coefficients

Abstract

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ≢ 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)