February Fourier Talks 2010

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 C^m function F on the whole Rⁿ? If such an F exists, how small can we take its C^m 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 C^m 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 C^m(Rⁿ)?