On the smoothness of best L 2 L_{2} approximants from nonlinear spline manifolds

Let S n k S_n^k be the nonlinear spline manifold of order k and with n - k interior variable knots. We prove that all best L 2 [ 0 , 1 ] {L_2}[0,1] approximants from S n k S_n^k to a continuous function on [0, 1] are also continuous there. We also prove that there exists a C [ 0 , 1 ] {C^infty }[0,1] function with no C 2 [ 0 , 1 ] {C^2}[0,1] best L 2 [ 0 , 1 ] {L_2}[0,1] approximants from S n k S_n^k .