Minimal norm interpolation with nonnegative real part on multiply connected planar domains (Q1363629)

Let \(\Omega\) be a domain in the plane whose boundary is composed of a finite number of disjoint smooth simple closed curves. \(H^2(\Omega)\) denotes the usual Hardy space on \(\Omega\) and \(K(\Omega)\) is the convex cone of those elements in \(H^2(\Omega)\) whose real part is nonnegative on \(\Omega\). The authors describe the projection of \(H^2(\Omega)\) onto \(K(\Omega)\) and also describe the unique element of \(K(\Omega)\) of minimal norm satisfying a finite number of interpolation conditions: \(\min\{|f|_{H^2(\Omega)}\); \(f\in K(\Omega)\) and \(f(z_j)=w_j\), \(j=1,2,\dots,n\}\), assuming that there is at least one element of \(K(\Omega)\) satisfying these conditions. When \(\Omega\) is the open unit disc and \(\text{Im } w_1=\dots= \text{Im } w_n=w_1=\dots= w_n\), they show the following: The minimization problem has a unique solution of the form \[ G(z)=i\text{ Im } F(0)= \frac{1}{2\pi} \int_0^{2\pi} \frac{e^{i\theta}+z} {e^{i\theta}-z} (\text{Re } F(e^{i\theta})- \lambda)_+ d\theta \] where \[ \lambda(1+ m(\text{Re F}<\lambda\})= \int_{\text{Re }F<\lambda} \text{Re }F \quad\text{and}\quad G(z_j)= w_j, \qquad j=1,\dots, n. \]