Numerical integration for fractal measures
Jens Malmquist
University of California, Berkeley, USARobert S. Strichartz
Cornell University, Ithaca, USA
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Abstract
We find estimates for the error in replacing an integral with respect to a fractal measure with a discrete sum over a given sample set with weights . 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 . 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 or the norm of , 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