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A note on measure-geometric Laplacians

By Marc Kesseböhmer, Tony Samuel and Hendrik Weyer

Abstract

We consider the measure-geometric Laplacians $\Delta^{\mu}$ with respect to atomless compactly supported Borel probability measures $\mu$ as introduced by Freiberg and Z\"ahle 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.Comment: 9 pages, 10 figure

Topics: Mathematics - Functional Analysis, Mathematics - Spectral Theory, 35P20, 42B35, 47G30, 45D05
Publisher: 'Springer Science and Business Media LLC'
Year: 2014
DOI identifier: 10.1007/s00605-016-0906-0
OAI identifier: oai:arXiv.org:1411.2491

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