Fitting Smooth Functions to Data – From a mathematician’s point of view

Let X be a function space on R^n. Given an arbitrary subset E in R^n and a real-valued function f on E: Q1. How can we decide whether there exists F in X such that F(x) = f(x) on E? (2) If there exists F in X such that F(x) = f(x) on E, what’s the simplest (or most efficient) way of computing F in terms of f? Moreover, if X is equipped with a norm ||.||, how small can we make ||F|| subject to F(x) = f(x) for x in E?
These questions go back to H. Whitney 1934. Complete answers to these problems for X = C^m(R^n) and X = C^{m,1}(R^n) (highest derivatives are Lipschitz continuous) have been obtained in the last few years. In this talk, I will explain some new results in the setting of Sobolev spaces, with emphasis on the second question for finite set E. Along the way, I will also briefly describe how the first question is related to the following problem considered recently by H. Brenner, M. Hochster, and J. Kollar-C. Fefferman: Solve the system of linear equations Ax = b for an unknown x in the function space X = C^{m,1}(R^n), where A is a matrix of given functions and b is a vector of functions. This talk draws on joint work with Charles Fefferman and Arie Israel.