# Interpolation and Approximation

# Interpolation and Approximation

Suppose f is a real-valued function on an awful set E in R^n. How can we decide whether f extends to a smooth function F on the whole R^n? "Smooth" means that F belongs to our favorite space X of continuous functions, e.g. C^m, C^{m, alpha}, or W^{m,p}. If such an F exists, how small can we take its norm in X? What can we say about its derivatives at a given point in or near E? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F with close-to-minimal norm in X? How many computer operations does it take? What if we require merely that F agree approximately with f on E? What if we are allowed to discard a few points of E as "outliers"? Which points should we discard to make the norm of F as small as possible? The subject started with Whitney in 1934. I've been working on it for many years. The talk will be completely accessible.