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1998 Cauchy transforms of self-similar measures
John-Peter Lund, Robert S. Strichartz, Jade P. Vinson
Experiment. Math. 7(3): 177-190 (1998).

Abstract

The Cauchy transform of a measure in the plane, $$ F(z) = \frac{1}{2\pi i}\int_{\C} \frac{1}{z-w} \,d\mu(w)\hbox{,} $$ is a useful tool for numerical studies of the measure, since the measure of any reasonable set may be obtained as the line integral of $F$ around the boundary. We give an effective algorithm for computing $F$ when $\mu$ is a self-similar measure, based on a Laurent expansion of $F$ for large $z$ and a transformation law (Theorem 2.2) for $F$ that encodes the self-similarity of $\mu$. Using this algorithm we compute $F$ for the normalized Hausdorff measure on the Sierpiński gasket. Based on this experimental evidence, we formulate three conjectures concerning the mapping properties of $F$, which is a continuous function holomorphic on each component of the complement of the gasket.

Citation

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John-Peter Lund. Robert S. Strichartz. Jade P. Vinson. "Cauchy transforms of self-similar measures." Experiment. Math. 7 (3) 177 - 190, 1998.

Information

Published: 1998
First available in Project Euclid: 14 March 2003

zbMATH: 0959.28006
MathSciNet: MR1676691

Subjects:
Primary: 28A80
Secondary: 30E20

Rights: Copyright © 1998 A K Peters, Ltd.

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Vol.7 • No. 3 • 1998
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