Interpolation and Extension of Functions

Interpolation and Extension of Functions

Charles Fefferman, Princeton University

What's Happening in Fine Hall

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Let X be your favorite Banach space of continuous functions on R^n. Given a function f:E->R defined on a (possibly awful) subset E in R^n, how can we tell whether f extends to a function F:R^n->R belonging to X? If such an F exists, then how small can we take its norm? What can we say about the derivatives  of F at or near points of E? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F as above? How many computer operations does it take? What if F is allowed to agree only approximately with f ? What if we are allowed to discard a few points of E as "outliers"? Which  points should we discard?