Abstract
We consider the problem of designing optimal smoothing spline curves and surfaces for a given set of discrete data. First we derive concise expressions for the optimal solutions using normalized uniform B-splines as the basis functions. Then, assuming that a set of data in a plane is obtained by sampling some curve with or without noises, we prove that, under certain condition, optimal smoothing splines converge to some limiting curve as the number of data increases. Such a limiting curve is obtained as a functional of given curve to be sampled. The case of surfaces is treated in parallel, and it is shown that the results for the case of curves can be extended to the case of surfaces in a straightforward manner.
