Fitting a Sobolev function to data III

Abstract

In this paper and two companion papers, we produce efficient algorithms to solve the following interpolation problem: Let $\mathfrak{m}\ge 1$ and $\mathfrak{p}>\mathfrak{n}\ge 1$. Given a finite set E $\subset {\mathbb{R}}^{\mathfrak{n}}$ and a function f: E $\to \mathbb{R}$, compute an extension F of f belonging to the Sobolev space W${}^{\mathfrak{m},\mathfrak{p}}({\mathbb{R}}^{\mathfrak{n}})$ with norm having the smallest possible order of magnitude; secondly, compute the order of magnitude of the norm of F. The combined running time of our algorithms is at most CN log N, where N denotes the cardinality of E, and C depends only on $\mathfrak{m}$, $\mathfrak{n}$, and $\mathfrak{p}$.