Fitting a Sobolev function to data I

Abstract

In this paper and two companion papers, we produce efficient algorithms to solve the following interpolation problem: Let m >= 1 and p > n >= 1. Given a finite set E subset of R-n and a function f : E -> R, compute an extension F of f belonging to the Sobolev space W-m,W-p (R-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 m, n, and p.