Fitting a sobolev function to data II

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 ⊂ ℝ ⁿ and a function f : E → ℝ , compute an extension F of f belonging to the Sobolev space W^m,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.