The behavior of best \(L_{p}\)-approximations as \(p\rightarrow 1\). A counterexample of convergence (Q868827)

Let \({(\Omega, {\mathcal A},\mu )}\) be a measure space. By \(L_p\), \({p\geq 1}\), as usual we denote the class of \({\mathcal A}\)-measurable real functions \(u\) on \(\Omega\) with \({\| u\| ^p_p=\int_{\Omega} | u| ^p d\mu <+\infty. }\) Let \(C\) be a closed, convex subset of \(L_1\) and \(f\) be a function in \({L_1\setminus C}\). If \({(\Omega, {\mathcal A},\mu )}\) is a finite measure space, \({f\in \bigcup_{p>1}L_p}\), and some other conditions hold, then for \(p>1\) and \(p\) is close enough to \(1\), there exists unique best \(L_p\)-approximation \(h_p\) of \(f\) from \(C\). The well-known theorem of \textit{D. Landers} and \textit{L. Rogge} [J. Approximation Theory 33, 268--280 (1981; Zbl 0483.41034)] states that \({h_p\to g_0}\) in the \(L_1\)-norm as \({p\to 1}\). Here \(g_0\) is a best \(L_1\)-approximation of \(f\) from \(C\) called the natural best \(L_1\)-approximation. In the paper under review, it is shown that the Landers and Rogge result is not valid in the general case if the measure space \({(\Omega, {\mathcal A},\mu )}\) is infinite. Marano and Quesada consider the space \({({\mathbb N},{\mathcal A}, \mu)}\), where \({\mathbb N}\) is the set of positive integers, \(\mathcal A\) is the class of all subsets of \({\mathbb N}\), and \(\mu\) is the counting measure. They construct an example of one-dimensional affine subspace \(C\) such that \(h_p\) (the best \(L_p\) approximation of \(0\) from \(C\)) oscillates as \({p\to 1}\).