Interpolation problems of Loewner type with infinitely many nodes (Q1611434)

Let \(L\) and \(M\) be disjoint subsets of \(\mathbb{R}\), not both empty, and \(E=L\cup M\). Let \(\varphi\) be a function from \(E\) into \(\mathbb{R}\), such that \(\varphi(x)\neq 0\) for \(x\in M\), and let \(\varphi^1\) be a function from \(L\) into \(\mathbb{R}\). The author considers the following problem \(I_\infty\). Find an analytic function \(\psi: \mathbb{C}^+\to \mathbb{C}\) satisfying (i) \(\text{Im}\psi(z)\geq 0\), \(z\in\mathbb{C}^+\); (ii) for any \(x\in L\) there exist the radial boundary value \(\lim_{y\downarrow 0}\psi(x+iy)=:\psi(x)\) and the radial derivative \(\lim_{y\downarrow 0}\psi'(x+iy)=:\psi'(x)\) and \(\psi(x)=\varphi(x)\), \(\psi'(x)=\varphi^1(x)\); and (iii) for any \(x\in M\) the radial residue \(\lim_{y\downarrow 0}iy\psi(x+iy)=:\psi^{(-1)}(x)\) equals \(\varphi(x)\). Let \(I_\infty\) denote the problem which is obtained from \(I_\infty\) by replacing the equality \(\psi'(x)=\varphi^1(x)\) in (ii) by the inequality \(\psi' (x)\leq\varphi^1(x)\), \(x\in L\), and the equality \(\psi^{(-1)}(x)=\varphi(x)\) in (iii) by the inequality \(\psi^{(-1)}(x)\leq\varphi(x)\), \(x\in M\). The problem \(I_\infty\) is a Loewner type boundary interpolation problem, whereas \(\acute{I}_\infty\) is a boundary interpolation problem of Pick-Nevanlinna type. The author proves: necessary and sufficient for the problem \(\acute{I}_\infty\) to have a solution is that the Pick kernel to the problem be nonnegative and that \[ \sum_{u\in M}\frac{-\varphi(u)}{u^2+1}<+\infty . \] A solvability condition for the problem \(I_\infty\) is established, under some additional assumptions.