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 $f:E\to \mathbb{R}$, 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 $CN\log N$, where $N$ denotes the cardinality of $E$, and $C$ is a constant depending only on $m$,$n$, and $p$.