arXiv subjects

Abraham Frei-Pearson

Publications and source records attributed to Abraham Frei-Pearson.

2 recordsLinked to original sources

arXiv2022

The norm of linear extension operators for $C^{m-1,1}(\mathbb{R}^n)$

Fix integers $m\ge 2$, $n\ge 1$. We prove the existence of a bounded linear extension operator for $C^{m-1,1}(\R^n)$ with operator norm at most $\exp(γD^k)$, where $D := \binom{m+n-1}{n}$ is the number of multiindices of length $n$ and order at most $m-1$, and $γ,k > 0$ are absolute constants (independent of $m,n,E$). Upper bounds on the norm of this operator are relevant to basic questions about fitting a smooth function to data. Our results improve on a previous construction of extension operators of norm at most $\exp(γD^k 2^D)$. Along the way, we establish a finiteness theorem for $C^{m-1,1}(\R^n)$ with improved bounds on the involved constants.

math.FA↗

arXiv2021

A coordinate-free proof of the finiteness principle for the Whitney extension problem

We present a coordinate-free version of Fefferman's solution of Whitney's extension problem in the space $C^{m-1,1}(\mathbb{R}^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry.

math.CA↗