Weak Chebyshev subspaces and continuous selections for the metric projection

Abstract:

Let G be an n-dimensional subspace of $C[a,b]$. It is shown that there exists a continuous selection for the metric projection if for each f in $C[a,b]$ there exists exactly one alternation element ${g_f}$, i.e., a best approximation for f such that for some $a \leqslant {x_0} < \cdots < {x_n} \leqslant b$, \[ \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left \| {f - {g_f}} \right \|,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.\] Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, $g \ne 0$, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for $n = 1$ and Brown for $n = 5$.