A note on measure-geometric Laplacians

Marc Kesseböhmer; Tony Samuel; Hendrik Weyer

doi:10.1007/s00605-016-0906-0

A note on measure-geometric Laplacians

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We consider the measure-geometric Laplacians Δ^μ with respect to atomless compactly supported Borel probability measures μ as introduced by Freiberg and Zähle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of Δ^μ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of Δ^μ. 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.

Original language	English
Pages (from-to)	643-655
Number of pages	12
Journal	Monatshefte fur Mathematik
Volume	181
Issue number	3
Early online date	11 Apr 2016
DOIs	https://doi.org/10.1007/s00605-016-0906-0
Publication status	Published - Nov 2016

Keywords

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

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https://link.springer.com/article/10.1007%2Fs00605-016-0906-0Licence: None: All rights reserved

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title = "A note on measure-geometric Laplacians",

abstract = "We consider the measure-geometric Laplacians Δμ with respect to atomless compactly supported Borel probability measures μ as introduced by Freiberg and Z{\"a}hle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of Δμ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of Δμ. 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. ",

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T1 - A note on measure-geometric Laplacians

AU - Kesseböhmer, Marc

AU - Samuel, Tony

AU - Weyer, Hendrik

PY - 2016/11

Y1 - 2016/11

N2 - We consider the measure-geometric Laplacians Δμ with respect to atomless compactly supported Borel probability measures μ as introduced by Freiberg and Zähle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of Δμ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of Δμ. 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.

AB - We consider the measure-geometric Laplacians Δμ with respect to atomless compactly supported Borel probability measures μ as introduced by Freiberg and Zähle (Potential Anal. 16(1):265–277, 2002) and show that the harmonic calculus of Δμ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of Δμ. 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.

KW - Measure-geometric Laplacians

KW - Spectral asymptotics

KW - Singular measures

KW - Chebyshev polynomials

KW - Salem measures

UR - https://arxiv.org/abs/1411.2491

U2 - 10.1007/s00605-016-0906-0

DO - 10.1007/s00605-016-0906-0

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VL - 181

SP - 643

EP - 655

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 3

ER -

A note on measure-geometric Laplacians

Abstract

Keywords

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