Sampling Theory with Average Values on the Sierpinski Gasket

TL;DR

This paper explores sampling methods based on average values over cells on fractals like the Sierpinski gasket, establishing spectral properties and sampling theorems for these structures.

Contribution

It introduces a novel sampling approach on fractals using cell averages, proves spectral decimation properties, and extends sampling theory to infinite blowups.

Findings

Cell graph approximations on SG have spectral decimation.

An analog of Shannon sampling theorem is established for SG.

Sampling on SG3 differs as spectral decimation is not useful for similar theorems.

Abstract

In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on $SG$ and $SG_{3}$. In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem.. We also investigate the numerical properties of these sampling functions and make conjectures which allow us to look at sampling on infinite blowups of $SG$. In the case of $SG_{3}$, we show that the cell graphs have the spectral decimation property, but show that it is not useful for proving an analogous sampling theorem.