Best simultaneous approximation in Orlicz spaces
Let \(G\) be a subset of a Banach space \((X, \| \cdot \| )\). Let \(n\) be a norm on \(\mathbb R^2\). For \((f_1,f_2) \in X^2\) and \(g\in G\) put \(\text{dist}(f_1,f_2, g) = n(\| f_1-g\| , \| f_2-g\| )\), and \(\text{dist}(f_{1},f_{2}, G) = \inf \{\text{dist}(f_1,f_2, g) : g\in G\}\). A vector \(g_0 \in G\) is a best simultaneous approximation to \((f_1,f_2)\) if \(\text{dist}(f_1,f_2, g_0)= \text{dist}(f_1,f_2, G)\). Previous papers investigated basic best-approximation properties (e.g., existence, uniqueness, characterization) for classical Banach spaces \(X\) (e.g., continuous or integrable functions on a real interval). Here \(X\) is a space of Orlicz Banach space valued integrable functions on the unit interval equipped with the Luxemburg norm. The main result is a formula for dist\((f_1,f_2, G)\).
scientific article; zbMATH DE number 5282490