Convergence rates of Padé and Padé-type approximants
Let \(f\) be holomorphic at \(\infty\) with \(f(\infty)= 0\), and let the polynomials \(p_n\), \(q_n\) and \(P_n\), \(Q_n\) of degree \(\leq n\) be so chosen that \[ q_n(z) f(z)- p_n(z)= {\mathcal O}(z^{- n- 1})\quad\text{and}\quad Q_n(z) f(z)- P_n(z)= {\mathcal O}(z^{- 1})\;(z\to \infty). \] In the first case, we get the Padé approximant (PA) \({p_n\over q_n}\) of order \(n\), in the second case, with \(Q_n\) a preassigned polynomial of degree \(n\), we get the Padé-type approximant (PTA) \({P_n\over Q_n}\). Actually, \(p_n\) and \(P_n\) are of degree \(\leq n- 1\). The authors show that if \(f\) is of the form \(f(z)= \int {w(t)\over z- t} d\mu(t)\), with an entire function \(w\) and a finite Borel measure on \(\mathbb{C}\) with compact support, the PTAs of order \(n\) give the same rate of convergence as the PAs of order \(n\). However, the PTAs are far easier to compute and need less information about \(f\). Theorem 1. If \(\mu\) is Lebesgue measure on \(\Delta= [- 1, 1]\), \(w\) an entire function, and \(Q_n\) are the Legendre polynomials, then \[ \limsup_{n\to \infty} \Biggl|f(z)- {P_n(z)\over Q_n(z)}\Biggr|^{1/n}\leq \exp(- 2g_\Omega(z)), \] locally uniformly on \(\Omega:= \mathbb{C}\backslash \Delta\), where \(g_\Omega\) is Green's function in \(\Omega\) with pole at \(\Omega\). Theorem 2 extends Theorem 1 to general measures, and Theorem 3 compares the errors \(R_n(z):= f(z)- {P_n(z)\over Q_n(z)}\), for a general entire \(w\), with those \(r_n(z)\) that are obtained when \(w(t)\equiv 1\). The convergence proofs depend on an integral representation of \(R_n(z)\) and a lemma for polynomial approximation of \({w(t)\over z- t}\).
scientific article; zbMATH DE number 938745