Sampling on the Sierpinski Gasket
Sampling on the Sierpinski Gasket
复制标题
在谢尔宾斯基垫片上取样
DOI：
--
复制
发表时间：
2003
影响因子：
0.5
通讯作者：
R. Strichartz
中科院分区：
文献类型：
--
作者：
R. Oberlin;B. Street;R. Strichartz
我们研究了 Sierpinski Gasket (SG) 上定义的函数的规则和不规则采样，其中我们将“带限”解释为函数在 Kigami 定义的拉普拉斯算子的第一个 dm 狄利克雷本征函数中具有有限展开，dm 是采样集的基数。在常规情况下，我们将采样集作为近似SG的m级图的非边界顶点。我们证明定期采样总是可能的，并且基于福岛和岛马的频谱抽取方法的扩展以包括内积，我们给出了计算采样函数的算法。我们给出的实验证据表明，采样函数在远离采样点时会迅速衰减，这与经典理论中 sinc 函数表现出极其缓慢的衰减的直线形成鲜明对比。类似的行为似乎也适用于某些狄利克雷核。我们通过示例表明，采样公式提供了一种有吸引力的逼近函数的方法，这些函数不一定是带限的，因此可能对数值分析有用。我们提供的实验证据表明，其中一个常规采样集的合理扰动仍然是采样集。与单位间隔上发生的情况相反，并非所有正确基数的集合都是采样集合。
We study regular and irregular sampling for functions defined on the Sierpinski Gasket (SG), where we interpret “bandlimited” to mean the function has a finite expansion in the first dm Dirichlet eigenfunctions of the Laplacian as defined by Kigami, and dm is the cardinality of the sampling set. In the regular case, we take the sampling set to be the nonboundary vertices of the level m graph approximating SG. We prove that regular sampling is always possible, and we give an algorithm to compute the sampling functions, based on an extension of the spectral decimation method of Fukushima and Shima to include inner products. We give experimental evidence that the sampling functions decay rapidly away from the sampling point, in striking contrast to the classical theory on the line where the sinc function exhibits excruciatingly slow decay. Similar behavior appears to hold for certain Dirichlet kernels. We show by example that the sampling formula provides an appealing method of approximating functions that are not necessarily bandlimited, and so might be useful for numerical analysis. We give experimental evidence that reasonable perturbations of one of the regular sampling sets remains a sampling set. In contrast to what happens on the unit interval, it is not true that all sets of the correct cardinality are sampling sets.