Whitney extensions and orthonormal expansions

Abstract

The Whitney near extension problem for finite sets in $\mathbb R^d,\, d\geq 2$ asks the following: Let $\phi:E\to \mathbb R^d$ be a near distortion on a finite set $E\subset \mathbb R^d$ with certain geometry. How to decide whether $\phi$ extends to a smooth, one to one and onto near distortion $\Phi:\mathbb R^d\to \mathbb R^d$ which agrees with $\phi$ 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 $\mathbb R^n,\, n\geq 1$ asks the following: Let $U\subset R^n$ be open and let $E\subset U$ be a compact set. Let $\phi:U\to \mathbb R^n$ 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 $\phi$ on $E$ and agrees with Euclidean motions on $\mathbb R^n$. The classical Whitney extension problem asks the following: Let $\phi:E\to \mathbb R$ be a map defined on an arbitrary set $E\subset \mathbb R^n$. How can one decide whether $\phi$ extends to a map $\Phi:\mathbb R^n\to \mathbb R$ which agrees with $\phi$ on $E$ and is in $C^m(\mathbb R^n),\, m\geq 1$, 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 $L_p(\mathbb R),\, 1<p\leq \infty$ 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.