8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"On the norm of linear extension operators in smooth function spaces"

Israel, Arie

We discuss the solution to Whitney’s extension problem in the space $C^m(\mathbb{R}^n)$. We establish the validity of the Brudnyi-Shvartsman-Fefferman finiteness principle with an improvement to the involved constants. As a byproduct, we construct a linear extension operator $T : C^m(E) \rightarrow C^m(\mathbb{R}^n)$ with operator norm at most $O(\exp(poly(n)))$ for fixed $m$, or $O(\exp(poly(m)))$ for fixed $n$, where $E$ is an arbitrary finite subset of $\mathbb{R}^n$. Connections will be made to the algorithmic problem of interpolation of data by $C^m$ functions. This is a joint work with Jacob Carruth and Abraham Frei-Pearson.

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