Approximation and interpolation by entire functions with restriction of the values of the derivatives

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• 作者：Maxim R. Burke ^{burke@upei.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 30E10 ; secondary ; 54E52 ; 28A35 ; 26B35
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：213
• 期：Complete
• 页码：24-49
• 全文大小：624 K

文摘

A theorem of Hoischen states that given a positive continuous function ε:Rⁿ→R$\epsilon :{\mathbb{R}}^n\to \mathbb{R}$, an unbounded sequence 0≤c₁≤c₂≤…$0\le c_1\le c_2\le \dots$ and a closed discrete set T⊆Rⁿ$T\subseteq {\mathbb{R}}^n$, any C^∞$C^{\infty }$ function g:Rⁿ→R$g:{\mathbb{R}}^n\to \mathbb{R}$ can be approximated by an entire function f   so that for k=0,1,2,…$k=0,1,2,\dots$ , for all x∈Rⁿ$x\in {\mathbb{R}}^n$ such that |x|≥c_k$|x|\ge c_k$, and for each multi-index α   such that |α|≤k$|\alpha |\le k$,

(a)

|(D^αf)(x)−(D^αg)(x)|<ε(x)$|(D^{\alpha }f)(x)-(D^{\alpha }g)(x)|<\epsilon (x)$;

(b)

(D^αf)(x)=(D^αg)(x)$(D^{\alpha }f)(x)=(D^{\alpha }g)(x)$ if x∈T$x\in T$.

In this paper, we show that if C⊆Rⁿ⁺¹$C\subseteq {\mathbb{R}}^{n+1}$ is meager, A⊆Rⁿ$A\subseteq {\mathbb{R}}^n$ is countable and disjoint from T, and for each multi-index α   and p∈A$p\in A$ we are given a countable dense set e15f9efb2d319676ac803" title="Click to view the MathML source">A_p,α⊆R$A_{p,\alpha }\subseteq \mathbb{R}$, then we can require also that

(c)

(D^αf)(p)∈A_p,α$(D^{\alpha }f)(p)\in A_{p,\alpha }$ for p∈A$p\in A$ and α any multi-index;

(d)

if x∉T$x\notin T$, q=(D^αf)(x)$q=(D^{\alpha }f)(x)$ and there are values of p∈A$p\in A$ arbitrarily close to x   for which q∈A_p,α$q\in A_{p,\alpha }$, then there are values of p∈A$p\in A$ arbitrarily close to x   for which q=(D^αf)(p)$q=(D^{\alpha }f)(p)$;

(e)

for each α  , {x∈Rⁿ:(x,(D^αf)(x))∈C}$\{x\in {\mathbb{R}}^n:(x,(D^{\alpha }f)(x))\in C\}$ is meager in Rⁿ${\mathbb{R}}^n$.

Clause (d) is a surjectivity property whose full statement in the text also allows for finding solutions in A   to equations of the form q=h^⁎(x,(D^αf)(x))$q=h^{⁎}(x,(D^{\alpha }f)(x))$ under similar assumptions, where h(x,y)=(x,h^⁎(x,y))$h(x,y)=(x,h^{⁎}(x,y))$ is one of countably many given fiber-preserving homeomorphisms of open subsets of Rⁿ⁺¹≅Rⁿ×R${\mathbb{R}}^{n+1}\cong {\mathbb{R}}^n\times \mathbb{R}$.

We also prove a weaker corresponding result with “meager” replaced by “Lebesgue null.” In this context, the approximating function is C^∞$C^{\infty }$ rather than entire, and we do not know whether it can be taken to be entire. The first result builds on earlier work of the author which deduced special cases of it from forcing theorems using absoluteness arguments. The proofs here do not use forcing.