# Interpolation of scattered data in $\mathbb{R}^3$ using minimum $L_p$-norm networks, $1<p<\infty$

## TL;DR

This paper fully characterizes the solution to the problem of interpolating scattered data in three-dimensional space using smooth curve networks that minimize the $L_p$-norm of the second derivative for all $p$ between 1 and infinity, extending previous partial results.

## Contribution

It provides a complete theoretical solution for the $L_p$-norm interpolation problem in $R^3$ for all $p$ in (1,∞), building on prior partial results and proofs.

## Key findings

- Complete characterization of the $L_p$-norm interpolation solution for $1<p<
finite$
- Numerical experiments illustrating the theoretical results
- Extension of previous partial results to a full solution

## Abstract

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.