标题： 采样和插值在椭圆微分算子谱子空间中的必要密度条件 显示英文标题

标题： Necessary Density Conditions for Sampling and Interpolation in Spectral Subspaces of Elliptic Differential Operators

摘要： 我们证明了在具有缓慢振荡符号的二阶均匀椭圆微分算子在$R^d$上的谱子空间中采样的必要密度条件。对于常系数算子，这些条件恰好是带限函数的Landau必要密度条件，但对于更一般的椭圆微分算子，是否这样的临界密度存在一直未知。我们的结果证明了合适临界采样密度的存在性，并根据椭圆算子定义的几何结构计算了该密度。在维度1的情况下，谱子空间中的函数可以解释为变量带宽的函数，我们得到了一种新的变量带宽的临界密度。方法结合了谱理论和椭圆偏微分算子的正则性理论、一些极限算子的元素、$R^d $的某些紧化以及再生核希尔伯特空间理论。 显示英文摘要

摘要： We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on $R^d$ with slowly oscillating symbol. For constant coefficient operators, these are precisely Landaus necessary density conditions for bandlimited functions, but for more general elliptic differential operators it has been unknown whether such a critical density even exists. Our results prove the existence of a suitable critical sampling density and compute it in terms of the geometry defined by the elliptic operator. In dimension 1, functions in a spectral subspace can be interpreted as functions with variable bandwidth, and we obtain a new critical density for variable bandwidth. The methods are a combination of the spectral theory and the regularity theory of elliptic partial differential operators, some elements of limit operators, certain compactifications of $R^d $, and the theory of reproducing kernel Hilbert spaces.