Approximation with rational interpolants in \(A^{-\infty}(D)\) for Dini domains (Q1687098)

Let \(D\subset\mathbb{C}\) be a Dini domain, i.e., \(D\) is the inner domain of a Jordan curve whose parametrisation has a Dini continuous and non-vanishing derivative. A holomorphic function \(f:D\to\mathbb{C}\) belongs to the space \( A^{-\infty}(D)\) if \(\|f\|_q= \sup_{z\in D}d(z)^q|f(z)|<+\infty\) for some positive constant \(q\), where \(d(z)=\mathrm{dist}(z, \partial D)\). Consider two collections of points \(A_n=(a_{ni})_{i=0}^{n}\subset D\) and \(B_n=(b_{ni})_{i=0}^{n}\subset \overline{\mathbb{C}}\setminus D\). For every \(n\in\mathbb{N}\), let \(r_{n,f}\) be the rational function which interpolates \(f\) in \(A_n\) and has poles in \(B_n\). The main result of the article establishes that under some conditions \[ r_{n,f}\to f\quad \text{in}\quad A^{-\infty}(D). \] The tools used involve the boundary Harnack principle, normalized point counting measures and their logarithmic potentials.