Whitney extensions and orthonormal expansions

Abstract

The Whitney near extension problem for finite sets in ${\mathbb{R}}^d,d\ge 2$ asks the following: Let $\varphi :E\to {\mathbb{R}}^d$ be a near distortion on a finite set $E\subset {\mathbb{R}}^d$ with certain geometry. How to decide whether $\varphi$ extends to a smooth, one to one and onto near distortion $\Phi :{\mathbb{R}}^d\to {\mathbb{R}}^d$ which agrees with $\varphi$ on $E$ and with Euclidean motions in ${\mathbb{R}}^d$. The Whitney near extension problem for compact sets $E\subset U$ in open subsets $U$ of ${\mathbb{R}}^n,n\ge 1$ asks the following: Let $U\subset R^n$ be open and let $E\subset U$ be a compact set. Let $\varphi :U\to {\mathbb{R}}^n$ be a smooth near isometry. How to decide if there exists a smooth one-to-one and onto near isometry $\Phi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ which extends $\varphi$ on $E$ and agrees with Euclidean motions on ${\mathbb{R}}^n$. The classical Whitney extension problem asks the following: Let $\varphi :E\to \mathbb{R}$ be a map defined on an arbitrary set $E\subset {\mathbb{R}}^n$. How can one decide whether $\varphi$ extends to a map $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ which agrees with $\varphi$ on $E$ and is in $C^m({\mathbb{R}}^n),m\ge 1$, the space of functions from ${\mathbb{R}}^n$ to $\mathbb{R}$ whose derivatives of order $m$ are continuous and bounded. In this paper, we survey some of our work on the near Whitney extension problem [2] in ${\mathbb{R}}^n$. Thereafter, we survey some of our work on weighted $L_p(\mathbb{R}),1<p\le \infty$ convergence of orthonormal expansions in $\mathbb{R}$ [3] and present a result of [13]. The motivation for doing this is motivated by interesting connections between Whitney extension theorems, Taylor series and Fourier expansions. Finally, we raise various open questions to study.

arXiv commentarXiv admin note: text overlap with arXiv:2103.09748