Convergency of parabolic interpolative splines

Parabolic interpolative splines are investigated with different kinds of boundary conditions: exact boundary derivative or it's difference approximation, periodic and natural, conditions. Asymptotic expressions for their error are found which are valid for small enough stepsize. It is proved that a) natural, periodic and improved difference conditions give the best accuracy, b) arbitrary nonequidistant grid leads to less order of accuracy, but quasi-equidistant grid doesn't. The adaptive grids are constructed which minimize the -error. All conclusions are illustrated with numerical examples.