Best near-interpolation by curves: Existence (Q2706382)

For specified subsets \(K_{ij},i=1,\dots,n, j=0,\dots,m-1\) of \(\mathbb{R}^d\), a curve \(f:[0,1]\to\mathbb{R}^d\) is called feasible if \(f^{(j)}(t_i)\in K_{ij}\) for \(j=0,\dots,m-1\) and for a sequence of points \(t_1,\dots,t_n\) in the interval \([0,1]\). The author studies the following problem: For given sets \(K_{ij}\) find a feasible curve in the appropriate Sobolev space which minimizes the \(L^2\)-norm of \(f^{(m)}\). This is a generalization of the parametric interpolation studied by \textit{K. Scherer} and \textit{P. W. Smith} [SIAM J. Math. Anal. 20, No. 2, 160-168 (1989; Zbl 0675.41009)]. The condition of ``asymptotically polynomial'' of Scherer and Smith is replaced by an analogous condition ``near \(m\)-order''. The main result of the paper states that either a polynomial solution exists, or if this is not the case, a best near-interpolant exists if and only if the given data are not near \(m\)-order.