On interpolation in the classes \(E^p\)
Consider a Jordan domain \(G\) in the (extended) complex plane and a sequence \(\lambda= (z_j)\) in \(G\). For \(1<p< \infty\) the author studies interpolation problems of the form (1) \(\rho_j^{1/p} f(z_j)=a_j\) \((u\in \mathbb{N})\) where \((a_j)\) is an arbitrary sequence of complex numbers with \(|a_j|\leq 1\) and \(\rho_j= \text{dist}(z_j, \partial G)\), and where the interpolating functions belong to the Smirnov class \(E^p\) in \(G\). For a certain class of Jordan domains (so-called Ahlfors class) and for a certain class of sequences \(\lambda\) he shows that (1) is solvable for all \((a_j)\) as above with a function \(f\in E^p\) in such a way that the \(E^p\) norm of \(f\) is bounded by some constant depending on \(G,\lambda,p\) and some sequence of weights.
scientific article; zbMATH DE number 1467839
scientific article