Let X be our favorite Banach space of continuous functions on Rn (e.g. Cm, Cm,αα, Wm,p). Given a real-valued function f defined on an (arbitrary) given set E in Rn, we ask: How can we decide whether f extends to a function F in X? If such an F exists, then how small can we take its norm? What can we say about the derivatives of F? Can we take F to depend linearly on f?
What if the set E is finite? Can we compute an F whose norm in X has the smallest possible order of magnitude? How many computer operations does it take? What if we ask only that F agree approximately with f on E? What if we are allowed to discard a few points of E as "outliers"; which points should we discard?
A fundamental starting point for the above is the classical Whitney extension theorem.
The results are joint work with Arie Israel, Bo'az Klartag, Garving (Kevin) Luli, and Pavel Shvartsman.
