Monatshefte für Mathematik

, Volume 181, Issue 3, pp 643–655 | Cite as

A note on measure-geometric Laplacians

Article

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 (Potential Anal. 16(1):265–277, 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 inhomogeneous self-similar Cantor measures and Salem measures.

Keywords

Measure-geometric Laplacians Spectral asymptotics  Singular measures Chebyshev polynomials Salem measures 

Mathematics Subject Classification

35P20 42B35 47G30 45D05 

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.FB3, MathematikUniversität BremenBremenGermany
  2. 2.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA

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