Almost Chebyshev properties for \(L^ 1\)-approximation of continuous functions (Q756055)

Let K be a compact subset of \(R^ s\) (s\(\geq 1)\) satisfying \(K=int(K)\) and denote by C(K) the space of continuous, real valued functions on K. Let \(W=\{w\in L^{\infty}(K):\) \(w>0\) on \(K\}\) and, for \(w\in W_{\infty}\), let \(C_ w(K)\) denote the space C(K) endowed with the w- weighted \(L^ 1\)-norm \(\| f\|_ w=\int_{k}| f| w d\mu,\) where \(\mu\) denotes Lebesgue mesure. If U is a finite dimensional subspace of C(K) and \(f\in C(K)\), let \(P_ w(f)\) denote the set of all best \(\| \cdot \|_ w\)-approximations to f from U, and let \({\mathcal U}_ w(U)\) denote the set of all functions in C(K) that have unique best \(\| \cdot \|_ w\)-approximations from U. In this paper the denseness from \({\mathcal U}_ w(U)\) in C(K), relative to different topologies, is examined. The following results are proved: - For every finite-dimensional subspace U of C(K), \({\mathcal U}_ w(U)\) is \(\| \cdot \|_ w\) dense in C(K); - Uf \({\mathcal U}_ w(U)\) is \(\| \cdot \|_{\infty}\)-dense in C(K), then C(K)\(\setminus {\mathcal U}_ w(U)\) is of first category in C(K) relative to the \(\| \cdot \|_{\infty}\)-topology. Also, when K is locally connected and the nontrivial elements of U have sparse zero sets, a necessary and sufficient condition for \({\mathcal U}_ w(U)\) to be \(\| \cdot \|_{\infty}\)-dense is C(K) is given. Furthermore, those finite dimensional subspaces U of C(K) for which \({\mathcal U}_ w(U)\) is \(\| \cdot \|_{\infty}\)-dense in C(K) for all \(w\in W_{\infty}\) are characterized. It is not difficult to check the conditions in some cases of interest for applications.