Whitney扩张与标准正交扩张
Whitney extensions and orthonormal expansions
摘要：
$\mathbb R^d,\, d\geq 2$中有限集的Whitney近延拓问题提出了如下问题：设$ϕ:E\to \mathbb R^d$是具有一定几何的有限集$E\subset \mathbb R^d$上的近畸变。如何判断$ϕ$是否扩展到一个光滑的，一对一的和接近失真的$Φ:\mathbb R^d\to \mathbb R^d$，这与$ϕ$在$E$和$\mathbb R^d$中的欧几里得运动一致。紧集$E\subset U$在$\mathbb R^n,\, n\geq 1$的开子集$U$中的Whitney近延拓问题要求如下：令$U\subset R^n$是开的，令$E\subset U$是紧集。设$ϕ:U\to \mathbb R^n$是光滑的近等距。如何判断是否存在一个光滑的一对一的近似等距$Φ:\mathbb R^n\to \mathbb R^n$，它在$E$上扩展了$ϕ$，并与$\mathbb R^n$上的欧几里德运动一致。经典的Whitney扩张问题要求如下：设$ϕ:E\to \mathbb R$是定义在任意集合$E\subset \mathbb R^n$上的映射。如何判断$ϕ$是否延伸到一个映射$Φ:\mathbb R^n\to \mathbb R$，它与$ϕ$在$E$上一致，并且在$C^m(\mathbb R^n),\, m\geq 1$中，即从$\mathbb R^n$到$\mathbb R$的函数空间，其阶$m$的导数是连续的和有界的。本文综述了我们在$\mathbb R^n$中对近Whitney延拓问题[2]所做的一些工作。然后，我们综述了在$\mathbb R$ [3]中关于标准正交展开式的加权$L_p(\mathbb R),\, 1<p\leq \infty$收敛性的一些工作，并给出了[13]的一个结果。这样做的动机是由惠特尼扩展定理，泰勒级数和傅立叶展开之间的有趣联系所激发的。最后，我们提出了一些有待研究的开放性问题。
英文摘要：
The Whitney near extension problem for finite sets in $\mathbb R^d,\, d\geq 2$ asks the following: Let $ϕ:E\to \mathbb R^d$ be a near distortion on a finite set $E\subset \mathbb R^d$ with certain geometry. How to decide whether $ϕ$ extends to a smooth, one to one and onto near distortion $Φ:\mathbb R^d\to \mathbb R^d$ which agrees with $ϕ$ 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 $ϕ:U\to \mathbb R^n$ be a smooth near isometry. How to decide if there exists a smooth one-to-one and onto near isometry $Φ:\mathbb R^n\to \mathbb R^n$ which extends $ϕ$ on $E$ and agrees with Euclidean motions on $\mathbb R^n$. The classical Whitney extension problem asks the following: Let $ϕ:E\to \mathbb R$ be a map defined on an arbitrary set $E\subset \mathbb R^n$. How can one decide whether $ϕ$ extends to a map $Φ:\mathbb R^n\to \mathbb R$ which agrees with $ϕ$ 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.
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