A nonnegative version of Whitney's Extension Problem

Friday, February 5, 2021 - 3:00pm
Kevin O'Neill

Whitney's Extension Problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $f:E\to\mathbb{R}$, how can we tell if there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials $P^x$ at each $x\in E$, there is such an extension F if and only if the $P^x$'s are compatible under Taylor's theorem. However, this leaves open the question of how to tell solely from $f$. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when $f\ge0$ and we require $F\ge0$.

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