Whitney extensions and orthonormal expansions

TL;DR

This paper surveys recent work on Whitney extension problems for near isometries and smooth functions, explores connections with orthonormal expansions and Fourier analysis, and discusses open questions in these areas.

Contribution

It provides a comprehensive survey of recent advances in Whitney extension problems, especially for near isometries, and links these with orthonormal expansions and Fourier analysis.

Findings

Survey of near Whitney extension problem results

Analysis of weighted $L_p$ convergence of orthonormal expansions

Presentation of new open questions in the field

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.