论文标题
框架频谱对和指数碱基
Frame spectral pairs and exponential bases
论文作者
论文摘要
给定带有正面和有限的lebesgue度量的域$ω\ subset \ bbb r^d $，以及一个离散集$λ\ subset \ subset \ bbb r^d $，我们说$（ω，λ）$是{\ it框架频谱对} x}：λ\inλ\} $是$ l^2（ω）$的框架。框架的特殊情况包括Riesz基地和正交基础。在有限设置$ \ bbb z_n^d $，$ d，n \ geq 1 $中，可以同样定义帧频谱对。 ％（在这里，$ \ bbb z_n $是循环的Abelian订单组。）我们通过在$ \ bbb r^{d} $和$ \ bbb z_n^d $中“添加”框架频谱对来展示如何在$ \ bbb r^d $中构建和获取新的框架光谱对。我们的构造统一了众所周知的指数框架的示例，该指数框架的结合量相等。我们还评论了域的光谱特性与采样理论之间的联系。
Given a domain $Ω\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $Λ\subset \Bbb R^d$, we say that $(Ω, Λ)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal E(Λ):=\{e^{2πi λ\cdot x}: λ\in Λ\}$ is a frame for $L^2(Ω)$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $\Bbb Z_N^d$, $d, N\geq 1$, a frame spectral pair can be similarly defined. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs in $\Bbb R^d$ by "adding" frame spectral pairs in $\Bbb R^{d}$ and $\Bbb Z_N^d$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.
