The course will present recent results on the following problems:

Fix positive integers m,n. Let f be a real-valued function defined on an (arbitrary) given subset E of R^n. How can we decide whether f extends to a function F in C^m(R^n)?

If F exists, then how small can we take its C^m norm? What can we say about the derivatives of F up to order m at a given point? Can we take F to depend linearly on f? Suppose the above subset E is finite; say E contains N points. How can we compute an F as above, with C^m norm close to least possible? How many computer operations does it take?

Suppose we are allowed to delete a few points from the set E. Which points should we remove to minimize the C^m norm of the resulting F?

What if the space C^m(R^n) is replaced by a Sobolev space?