Uniqueness of Best Parametric Interpolation by Cubic Spline Curves

Abstract

Best parametric spline interpolation extends and refines the classical spline problem of best interpolation to \( {(*)}\quad\quad \inf_{\underline t} \inf_{\underline f}\left\{\int^1_0 \Vert \underline f^{(k)} (t) \Vert^2 \,dt \colon \ \underline f(t_i) = \underline y_i, 1 \leq i \leq n \right\}. \) Here \( \underline t \colon \ 0 = t_1 < \cdots < t_n = 1 \) denotes a sequence of nodes and \(\underline y_i \) data in \( \mbox{\footnotesize\bf R}^d \) with \( \underline y_i \not = \underline y_{i+1} \) . The \( \mbox{\footnotesize\bf R}^d \) -valued functions \( \underline f(t) \) lie componentwise in the Sobolev space \( L^k_2 (0,1) \) and || || denotes the Euclidean norm in R^d . This problem has been posed by H. J. Toepfer (1981) and independently by S. Marin (1984) who showed existence and uniqueness of the solution for d = 1 and k = 2 . Here we extend this result to the general parametric case \( d \geq 1\) . To this end, the problem is rewritten as a (nonclassical) saddle-point problem. It is concave for a part of the variables but not convex for the other one. Nevertheless, we can prove the existence of the unique solution of it and hence of (*). By an alternative approach via degree theory we prove further that there is at most one local strict minimum of (*). Finally, we discuss the relation to other problems of best interpolation where the minimization is performed with respect to the curvature or a certain class of β-splines.