Whitney extensions and orthonormal expansions

The Whitney near extension problem for finite sets in ${\mathbb{R}}^d,d\ge 2$ 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 $\mathrm{\Phi }:{\mathbb{R}}^d\to {\mathbb{R}}^d$ 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 ${\mathbb{R}}^n,n\ge 1$ 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 $\mathrm{\Phi }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ 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 $\mathrm{\Phi }:{\mathbb{R}}^n\to \mathbb{R}$ which agrees with $\phi$ 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 $L_p(\mathbb{R}),1<p\le \mathrm{\infty }$ 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.