Interpolation and Extension by Smooth Functions
The talk will discuss the following questions, and their close relatives.
Fix positive integers m,n. Let f:E→R be a function on a subset E of Rn. How can we decide whether f extends to a Cm function F on the whole Rn? If such an F exists, how small can we take its Cm norm? What can we say about the derivatives of F at a given point? Can we take F to depend linearly on f?
Suppose E is finite. How can we compute an F whose Cm norm is close to least-possible? How many computer operations does it take? What if we require merely that F agree with f on E up to a given accuracy? What if we are allowed to delete a few of the points of E? Which points should we discard? What happens in function spaces other than Cm(Rn)?