Fitting a Sobolev function to data III

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 .