论文标题
一些线性空间工会的光谱特性
Spectral properties of some unions of linear spaces
论文作者
论文摘要
我们考虑\ textit {addive spaces}，由两个单位长度的间隔或两个一般概率度量的间隔组成，$ {\ mathbb r}^1 $，位于$ {\ Mathbb r}^2 $中的轴上，并带有天然的附加措施$ρ$。我们研究$ l^2（ρ）$的指数框架，riesz碱基和正统基础与其组件空间之间的关系。我们发现，指数基础的存在很大程度上取决于我们如何将措施定位在$ {\ mathbb r}^1 $上。我们表明，非重叠的添加剂空间具有Riesz Bases，我们为重叠空间提供了必要的条件。我们还表明，Lebesgue类型的一些重叠的添加空间具有指数正态碱基，而有些则没有。一个特定的示例是原点的“ l”形状，它具有独特的正顺序基础，直到形式的翻译为\ [\ left \ {e^{e^{2πi（λ_1x_1 x_1 x_1 +λ_2x_2 x_2）}：（λ_1，λ_1，λ_2）\inλ_1，λ_2） \ Mid n \ in {\ Mathbb z} \}。 \]
We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $ρ$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(ρ)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the "L" shape at the origin, which has a unique orthonormal basis up to translations of the form \[ \left\{e^{2 πi (λ_1 x_1 + λ_2 x_2)} : (λ_1, λ_2) \in Λ\right\}, \] where \[ Λ= \{ (n/2, -n/2) \mid n \in {\mathbb Z} \}. \]
