Weak Chebyshev subspaces and continuous selections for parametric projections (Q1384658)

The existence of continuous selections for the parametric projection \({\mathcal P}: (p,x)\to P_{\Gamma (p)} (x)\) onto weak Chebyshev subspaces is examined in this paper. In particular, it is shown that if \(S_{n,k} (p_1,p_2, \dots, p_k): =\{s\in C^{n-1} [a,b]: s|_{[p_i, p_{i +1}]} \in p_n\) for \(i=0,1,2, \dots, k\}\) is the class of polynomial splines of degree \(n\) with the \(k\) fixed knots \(a=p_0 <p_1 <\cdots <p_k< p_{k+1} =b\), then the parametric projection \({\mathcal P}: (p,x)\to P_{S_{n,k} (p)} (x)\) admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one.