Fitting Smooth Functions to Data
"Let E be an (arbitrary) subset of R^n, and let f be a given real-valued function on E. How can we decide whether f extends to a smooth function F on R^n? (""Smooth"" means that F belongs to one of our favorite Banach spaces e.g. C^m or W^m,p) If an F exists, then how small can we take its norm in X? What can we say about the behavior of F near E? Can we take F to depend linearly on f?
Suppose E is finite. Can we compute an F with nearly least possible norm? How many computer operations does it take? What if we require that F agree only approximately with f? What if we are allowed to delete a few points from E as ""outliers""; which points should we delete?
Joint work with Arie Israel, Bo'az Klartag, Janos Kollar, Garving (Kevin) Luli, and Pavel Shvartsman. "