The talk will present results with Bo'az Klartag, Arie Israel, Garving (Kevin) Luli and Pavel Shvartsman on the following questions, which have fascinated me for the last 15 years.
Let X be our favorite Banach space of continuous functions on ℝn, and let f be a real-valued function on an (arbitrary) given subset E of ℝn. How can we tell whether f extends to a function F in X? If such an F exists, how small can we take its norm? Can we take F to depend linearly on f? What can we say about the derivatives of F at a given point?
Suppose E is finite. Can we compute such an F with close-to-minimal norm in X? How many computer operations does it take? What if F is required to agree only approximately with f on E? What if we are allowed to discard a few "outliers" from E to reduce the norm of F; which points should we discard? What changes if f and F are vector-valued?