欧几里得球上的局部化框架
Localized frames on Euclidean balls
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中文总结 AI 辅助
研究构造适用于欧几里得球的显式波包框架，通过该框架对时空谱极限算子进行定量特征值估计，利用框架的定量傅里叶局部化估计得出相关 plunge 区域的上界。
中文摘要 AI 辅助
我们构造了适用于欧几里得球的显式波包框架，并利用它们获得时空谱极限算子的定量特征值估计。设\(d\geq 2\)，\(B_d(R)\subset \R^d\)是半径为\(R\)的欧几里得球，\(S\subset \R^d\)是一个可测集，其边界\(\partial S\)对于\(0 < \eta \leq 1\)具有有限的\((d - \eta)\)-上闵可夫斯基内容。我们为\(L^2(B_d(R))\)构造了一个单位范数框架，其框架界仅取决于维度\(d\)，其元素适应于球的径向和角向几何形状。我们证明了该框架的定量傅里叶局部化估计：相对于\(S\)，框架分解为集中在\(S\)中的包、集中在\(\R^d\setminus S\)中的包以及一个例外族，其基数根据\(R\)和\(\partial S\)的闵可夫斯基内容明确界定。作为应用，我们推导了与集合\(B_d(R)\)和\(S\)相关的时空谱极限算子的 plunge 区域的上界。
英文摘要
We construct explicit wave packet frames adapted to Euclidean balls and use them to obtain quantitative eigenvalue estimates for spatio--spectral limiting operators. Let \(d\geq 2\), let \(B_d(R)\subset \R^d\) be the Euclidean ball of radius \(R\), and let \(S\subset \R^d\) be a measurable set such that $\partial S$ has finite $(d-η)$-upper Minkowski content for $0 < η\leq 1$. We construct a unit-norm frame for \(L^2(B_d(R))\), with frame bounds depending only on the dimension $d$, whose elements are adapted to the radial and angular geometry of the ball. We prove quantitative Fourier localization estimates for this frame: Relative to \(S\), the frame decomposes into packets concentrated in \(S\), packets concentrated in \(\R^d\setminus S\), and an exceptional family whose cardinality is bounded explicitly in terms of \(R\), and the Minkowski content of \(\partial S\). As an application, we derive an upper bound for the plunge region of the spatio-spectral limiting operator associated to the sets $B_d(R)$ and $S$.
