Numerical integration for fractal measures (Q1645017)

Summary: We find estimates for the error in replacing an integral \(\int f d\mu\) with respect to a fractal measure \(\mu\) with a discrete sum \(\sum_{x \in E} w(x) f(x)\) over a given sample set \(E\) with weights \(w\). Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a \textit{discrepancy} that depends only on the geometry of the sample set and weights, and \textit{variance} that depends only on the smoothness of \(f\). We deal with p.c.f. self-similar fractals, on which Kigami has constructed notions of \textit{energy} and \textit{Laplacian}. We develop generic results where we take the variance to be either the energy of \(f\) or the \(L^1\) norm of \(\Delta f\), and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpinski gasket, both for the standard self-similar measure and energy measures, and for other fractals.