Interpolation of scattered data in $\mathbb{R}^3$ using minimum $L_p$-norm networks, $1<p<\infty$
This work provides a complete theoretical solution for a class of interpolation problems in geometric modeling, but the results are incremental as they generalize known methods for p=2 to a broader range of p.
The paper characterizes the solution to the extremal problem of interpolating scattered data in R^3 using smooth curve networks with minimal L_p-norm of the second derivative for 1<p<∞, extending previous results for p=2. Numerical experiments support the theoretical findings.
We consider the extremal problem of interpolation of scattered data in $\mathbb{R}^3$ by smooth curve networks with minimal $L_p$-norm of the second derivative for $1<p<\infty$. The problem for $p=2$ was set and solved by Nielson (1983). Andersson et al. (1995) gave a new proof of Nielson's result by using a different approach. Partial results for the problem for $1<p<\infty$ were announced without proof in (Vlachkova (1992)). Here we present a complete characterization of the solution for $1<p<\infty$. Numerical experiments are visualized and presented to illustrate and support our results.