Rational approximation in the real Hardy space \(H_ 2\) and Stieltjes integrals: A uniqueness theorem (Q1205133)

Set \(V\) for the complement of the closed unit disk and \(H_{2,R}(V)\) for the real Hardy space. The paper proved that Stieltjes functions of the form \[ f(z)=\int {{d\mu(t)} \over {z-t}}, \] where \(\mu\) is a finite nonnegative Borel measure on \(\mathbb{R}\) whose support lies in \([- \lambda,\lambda]\) (\(0<\lambda<1\)) have, for any positive integer \(n\), a unique local (hence global) best approximation among rational functions of degree atmost \(n\) in \(H_{2,R}(V)\). Bounds for \(\lambda\) are explicitly estimated.