Abstract:

Fix positive integers m,n. Let E be a subset of R^n, and let f be a given real-valued function on E. How can we tell whether f extends to a C^m function F on the whole R^n? If F exists, how small can we make its C^m norm? What can we say about the derivatives of F at a given point? Can we make F depend linearly on f? What if we demand only that F agree approximately with f on E? If E is finite, can we compute a nearly optimal F from f? How many operations does it take? What if we are allowed to discard a few points from E? Many of the results presented are joint work with Bo'az Klartag.