Nonuniqueness and selections in spline approximation

This paper studies problems of nonuniqueness for the metric projection ofC(T),T a compact Hausdorff space, onto a finite-dimensional subspaceG, and discusses the results for polynomial spline approximation. Among others, we prove that the metric projection ofCa, b] ontoS _k,n , the space of polynomial splines of degree less than or equal ton withk simple knots in (a, b), is lower semicontinuous on an open, dense subset ofCa, b] and, consequently, any standard selection of the projection is continuous on this subset. We further show that continuous selections are not so easy to construct.Communicated by Ronald A. DeVore.