Ordinary smoothing splines have an integrated mean squared error which is dominated by bias contributions at the boundaries. When the estimated function has additional derivatives, the boundary contribution to the bias affects the asymptotic rate of convergence unless the derivatives of the estimated function meet the natural boundary conditions. This paper introduces relaxed boundary smoothing splines and shows that they obtain the optimal asymptotic rate of convergence without conditions on the boundary derivatives of the estimated function.
"Relaxed Boundary Smoothing Splines." Ann. Statist. 20 (1) 146 - 160, March, 1992. https://doi.org/10.1214/aos/1176348516