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arXiv 1702.00170math.CA
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平移不变空间的一个新的采样密度条件

A new sampling density condition for shift-invariant spaces

A. Antony Selvan

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20 pages
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摘要：

令$X=\{x_i:i\in\mathbb{Z}\}$、$\dots<x_{i-1}<x_i<x_{i+1}<\dots$是由常数$γ>0$分隔的采样集。在一定条件下，证明了若存在正整数$ν$，使得$$δ_ν:=\sup\limits_{i\in\mathbb{Z}}(x_{i+ν}-x_i)<\dfracν{2π}\left(\dfrac{c_{k}^2}{M_{2k}}\right)^{\frac{1}{4k}},$$，则属于移不变空间$V(ϕ)$的所有函数都能由其非均匀样本值$\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$稳定重构，其中$c_k$是Wirtinger-Sobolev常数，$M_{2k}$是$V(ϕ)$的Bernstein型不等式中的常数.此外，当$k=1$时，最大间隙$δ_ν<ν$对于某些移位不变空间是尖锐的。

英文摘要：

Let $X=\{x_i:i\in\mathbb{Z}\}$, $\dots<x_{i-1}<x_i<x_{i+1}<\dots$, be a sampling set which is separated by a constant $γ>0$. Under certain conditions on $ϕ$, it is proved that if there exists a positive integer $ν$ such that $$δ_ν:=\sup\limits_{i\in\mathbb{Z}}(x_{i+ν}-x_i)<\dfracν{2π}\left(\dfrac{c_{k}^2}{M_{2k}}\right)^{\frac{1}{4k}},$$ then every function belonging to a shift-invariant space $V(ϕ)$ can be reconstructed stably from its nonuniform sample values $\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$, where $c_k$ is a Wirtinger-Sobolev constant and $M_{2k}$ is a constant in Bernstein-type inequality of $V(ϕ)$. Further, when $k=1$, the maximum gap $δ_ν<ν$ is sharp for certain shift-invariant spaces.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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