Weighted rational approximation in the complex plane (Q1305122)

Given a domain \(G\subset\mathbb{C}\), \(f\) analytic in \(G\), a weight function \(W\) also analytic in \(G\) and \(W(z)\neq 0\) in \(G\), and \(\gamma\) with \(0\leq\gamma\leq 1\), the authors study the rational approximation property of \(W\): There exist rational functions \(R_i=P_{m_i}/Q_{n_i}\) where \(P,Q\) are polynomials of degree \(\leq m_i\) and \(\leq n_i\), respectively, such that \(m_i/(m_i+n_i) \to \gamma\) \((1\to\infty)\) and \(\|f-W^{m_i+ n_i}\cdot R_i\|_E\to 0\) \((i\to \infty)\). The main result gives necessary and sufficient conditions for \(W\), expressed in terms of potential theoretic quantities, so that \(W\) has the rational approximation property. The cases \(W(z)=z^\alpha\) \((\alpha>0)\) with \(G=\{z:|z-a|<r\}\) \((a>0\) and \(r<a)\), and \(W(z)=e^{-z}\) with \(G=\{z: |z|<r\}\) \((r>0)\) are treated as special cases, as well as \(W(z)=e^{-z}\) with \(G\) a generalized Szegö domain. Earlier results by the authors [Constructive Approximation 14, 475-492 (1998; Zbl 0920.30028)] dealt with weighted polynomial approximation in \(\mathbb{C}\), while \textit{P. Borwein}, \textit{E. A. Rakhmanov} and \textit{E. B. Saff} [ibid. 12, No. 2, 223-240 (1996; Zbl 0870.41010)] considered weighted rational approximation on the real line.