Fitting a Sobolev function to data

Abstract

We exhibit an algorithm to solve the following extension problem: Given a finite set $E \subset \mathbb{R}^n$ and a function $f: E \rightarrow \mathbb{R}$, compute an extension $F$ in the Sobolev space $L^{m,p}(\mathbb{R}^n)$, $p>n$, with norm having the smallest possible order of magnitude, and secondly, compute the order of magnitude of the norm of $F$. Here, $L^{m,p}(\mathbb{R}^n)$ denotes the Sobolev space consisting of functions on $\mathbb{R}^n$ whose $m$th order partial derivatives belong to $L^p(\mathbb{R}^n)$. The running time of our algorithm is at most $C N \log N$, where $N$ denotes the cardinality of $E$, and $C$ is a constant depending only on $m$,$n$, and $p$.