Weak Chebyshev Subspaces and Continuous Selections for Parametric Projections

We examine the existence of continuous selections for the parametric projection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\wp : (p,x) \rightarrow P_{_{\Gamma(p)}}(x)$ \end{document} onto weak Chebyshev subspaces. In particular, we show that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $S_{_{n,k}}(p_1,p_2,\ldots,p_k) := \{ s \in C^{n-1}[a,b] : s|_{_{[p_i,p_{i+1}]}} \in \mbox{\footnotesize{\bf P}} _n~~ \mbox{for}~~ i=0,~1,~2,\ldots,~k \}$ \end{document} is the class of polynomial splines of degree n with the k fixed knots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a=p_0 < p_1 < \cdots < p_k < p_{k+1} =b,$ \end{document} then the parametric projection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\wp}: (p,x) {\rightarrow} P_{_{S_{n,k}(p)}}(x)$ \end{document} admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one.