# Numerical integration for fractal measures

### Jens Malmquist

University of California, Berkeley, USA### Robert S. Strichartz

Cornell University, Ithaca, USA

## Abstract

We find estimates for the error in replacing an integral $∫fdμ$ with respect to a fractal measure $μ$ with a discrete sum $∑_{x∈E}w(x)f(x)$ 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 $Δf$, 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.

## Cite this article

Jens Malmquist, Robert S. Strichartz, Numerical integration for fractal measures. J. Fractal Geom. 5 (2018), no. 2, pp. 165–226

DOI 10.4171/JFG/60