Optimal $L^2$ and $L^\infty;$ Error Estimates for Continuous and Discrete Least Squares Methods for Boundary Value Problems

A priori error estimates in both the $L^2$ and $L^\infty$ norms are derived for continuous and discrete least squares approximates to solutions of two-point boundary value problems from approximating spaces of Hermite splines. Superconvergence at knots is established, and the estimates are shown to be optimal in the sense that no better rate of convergence is possible in the spaces employed.