Numerical integration for fractal measures

Abstract

We find estimates for the error in replacing an integral $\int fd\mu$ 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 discrepancy that depends only on the geometry of the sample set and weights, and 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 energy and 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.