Best harmonic and superharmonic \(L^1\)-approximants in strips (Q1125463)

Let \(D\) be a domain in \(\mathbb{R}^n\), and let \(f\in C(\overline D)\). A function \(h^*\) is called a best harmonic \(L^1\)-approximant to \(f\) on \(\overline D\) if \(\|f-h^* \|_1\leq\|f-h\|_1\) for every \(h\in C( \overline D)\) that is harmonic on \(D\). In a recent paper [J. Reine Angew. Math. 478, 1-15 (1996; Zbl 0853.31002)], \textit{D. H. Armitage}, \textit{S. J. Gardiner}, \textit{W. Haussmann} and \textit{L. Rogge} gave a complete characterization of the best harmonic \(L^1\)-approximant to a subharmonic function on the closed unit ball. In the paper under review, the authors prove the analogous result for a closed infinite strip. Their approach is partly based on ideas in the earlier paper, but the unboundedness of the strip presents significant difficulties that require new techniques.