WEAK CHEBYSHEV SUBSPACES AND CONTINUOUS-SELECTIONS FOR PARAMETRIC PROJECTIONS

We examine the existence of continuous selections for the parametric p rojection p: (p, x) --> P-r(p)(x) onto weak Chebyshev subspaces. In pa rticular, we show that if S-n,S-k(p(1), p(2),..., p(k)) := {s is an el ement of Cn-1 [a, b]:s\([pi.pi+1]) is an element of P-n for i = 0, 1, 2,..., k} is the class of polynomial splines of degree n with the k fi xed knots a = p(0) < p(1) < ... < p(k) < p(k+1) = b, then the parametr ic projection p: (p, x) --> P-sn.k(p)(x) admits a continuous selection if and only if the number of knots does not exceed the degree of spli nes plus one.