Minimal norm interpolation with nonnegative real part on multiply connected planar domains
type: Item
instance of: scholarly article
title: Minimal norm interpolation with nonnegative real part on multiply connected planar domains
zbMATH Open document ID: 0881.30032
published in: Michigan Mathematical Journal
publication date: 1997-08-10T00:00:00Z
review text: 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 \[ \lambda(1+ m(\text{Re F}<\lambda\})= \int_{\text{Re }F<\lambda} \text{Re }F \quad\text{and}\quad G(z_j)= w_j, \qquad j=1,\dots, n. \]
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 1046975
zbMATH Keywords: interpolation
zbMATH Keywords: minimal norm
zbMATH Keywords: nonnegative real part
zbMATH Keywords: planar domain
zbMATH Keywords: projection
author: Stephen D. Fisher
author: Charles A. Micchelli
reviewed by: Takahiko Nakazi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005628
OpenAlex ID: W2074226166
DOI: 10.1307/MMJ/1029005628
description: scientific article; zbMATH DE number 1046975