ON G² CONTINUOUS SPLINE INTERPOLATION OF CURVES IN ℛ^d

In this paper the problem of G² continuous interpolation of curves in ℛ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ℛ^d. In this case the optimal approximation order is also determined.