论文标题
双曲线和Perron-frobenius操作员的傅立叶非唯一性集
Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators
论文作者
论文摘要
令$γ$为平滑的曲线或平滑曲线的有限脱节结合，而$λ$为飞机的任何子集。令$ \ MATHCAL X（γ）$为飞机中所有有限复合物价值的Borel措施的空间，在$γ$上得到支撑，并且相对于$γ的弧长度，绝对是连续的。 \hatμ|_λ= 0 \}，$，然后我们证明以下结果：\ begin {emumerate} [（a）] \ item \ item \ item $λ_β$的合理扰动，即$λ_β^θ= \ left（（\ sathbb） z+\ {θ\}）\ times \ {0 \} \ right）\ cup \ left（\ {0 \} \timesβ\timesβ\ mathbb z \ right），$θ= 1/{p}每当$β> p。$β>p。$ \ smallskip时 \ \ $λ_γ$的有理扰动的项目，即$λ_γ^θ= \ left（（2 \ Mathbb z+\ {2θ\}）\ times \ times \ times \ {0 \} \ right）\ cup $θ=1/q,~\text{for some}~q\in\mathbb N,$ and $γ$ is a positive real, $\mathcal{AC}\left(Γ_+,Λ_γ^θ\right)$ is infinite-dimensional whenever $γ>q.$ \end{enumerate}
Let $Γ$ be a smooth curve or finite disjoint union of smooth curves in the plane and $Λ$ be any subset of the plane. Let $\mathcal X(Γ)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $Γ$ and are absolutely continuous with respect to the arc length measure on $Γ.$ Let $\mathcal{AC}(Γ,Λ)=\{μ\in \mathcal{X}(Γ) : \hatμ|_Λ=0\},$ then we prove the following results: \begin{enumerate}[(a)] \item For a rational perturbation of $Λ_β$ namely, $Λ_β^θ=\left((\mathbb Z+\{θ\})\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right),$ where $θ=1/{p},~\text{for some}~{p}\in\mathbb N,$ and $β$ is a positive real, $\mathcal{AC}\left(Γ,Λ_β^θ\right)$ is infinite-dimensional whenever $β>p.$ \smallskip \item For a rational perturbation of $Λ_γ$ namely, $Λ_γ^θ=\left((2\mathbb Z+\{2θ\})\times\{0\}\right)\cup\left(\{0\} \times2γ\mathbb Z\right),$ where $θ=1/q,~\text{for some}~q\in\mathbb N,$ and $γ$ is a positive real, $\mathcal{AC}\left(Γ_+,Λ_γ^θ\right)$ is infinite-dimensional whenever $γ>q.$ \end{enumerate}
