Convergence of the minimum L_p-norm networks as p → ∞ Available to Purchase

We consider the extremal problem of interpolation of scattered data in ℝ³ by smooth curve networks with minimal L_p-norm of the second derivative for 1 < p ≤ ∞. The problem for p = 2 was set and solved by Nielson [1]. Andersson et al. [2] gave a new proof of Nielson’s result by using a different approach. Vlachkova [3] extended the results in [2] and solved the problem for 1 < p < ∞. The minimum L_p-norm network for 1 < p < ∞ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1 < p < ∞ is unique. We denote the corresponding minimum L_p-norm network by F_p. In the case where p = ∞ we establish the existence of a solution of the same type as in the case where 1 < p < ∞. This solution on each edge of the underlying triangulation is a quadratic spline function with at most one knot. We denote this solution by F_∞ and prove that the minimum L_p-norm networks F_p converge to the minimum L_∞-norm network F_∞ as p → ∞.