A note on measure-geometric Laplacians

Abstract

We consider the measure-geometric Laplacians ${\Delta }^{\mu }$ with respect to atomless compactly supported Borel probability measures $\mu$ as introduced by Freiberg and Zähle in 2002 and show that the harmonic calculus of ${\Delta }^{\mu }$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of ${\Delta }^{\mu }$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.

9 pages, 10 figures

References in corpus (1)

• Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces

Cited by in corpus (4)

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• A Laplacian on the Full Shift Space
• Measure-geometric Laplacians for discrete distributions