Sampling on the Sierpinski Gasket

Abstract

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 {\small $d_m$} Dirichlet eigenfunctions of the Laplacian as defined by Kigami, and {\small $d_m$} 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.