Interpolation of data

Let f be a real-valued function on a set E in R^n. How can we decide whether f extends to a C^m function F on the whole R^n? If F exists, then 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? If E is finite, is there an efficient algorithm to compute an F with close-to-minimal C^m norm? How many computer operations are needed? What if we demand merely that F and f agree approximately on E? What if we are allowed to delete a few points from E; which points should we delete? What if C^m(R^n) is replaced by some other function space?