arXiv subjects

Hendrik Weyer

Publications and source records attributed to Hendrik Weyer.

4 recordsLinked to original sources

arXiv2023

Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

We consider the generalised Krein-Feller operator $Δ_{ν, μ} $ with respect to compactly supported Borel probability measures $μ$ and $ν$ with the natural restrictions that $μ$ is atomless, the supp$(ν)\subseteq$supp$(μ)$ and the atoms of $ν$ are embedded in the supp$(μ)$. We show that the solutions of the eigenvalue problem for $Δ_{ν, μ} $ can be transferred to the corresponding problem for the classical Krein-Feller operator $Δ_{ν\circ F_μ^{-1}, Λ}$ with respect to the Lebesgue measure $Λ$ via an isometric isomorphism determined by the distribution function $F_μ$ of $μ$. In this way, we obtain a new characterisation of the upper spectral dimension and consolidate many known results on the spectral asymptotics of Krein-Feller operators. We also recover known properties of and connections to generalised gap diffusions associated to these operators.

math.FA↗

arXiv2019

Measure-geometric Laplacians on the real line

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\uın, given an atomless Borel probability measure $η$ supported on a compact subset of $\mathbb{R}$, Freiberg and Zähle introduced a measure-geometric approach to define a first order differential operator $\nabla_η$ and a second order differential operator $Δ_η$, with respect to $η$. We generalise this approach to measures of the form $η= ν+ δ$, where $ν$ is continuous and $δ$ is finitely supported. We determine analytic properties of $\nabla_η$ and $Δ_η$ and show that $Δ_η$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $Δ_η$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

math.DS↗

arXiv2018

Measure-geometric Laplacians for discrete distributions

In 2002 Freiberg and Zähle introduced and developed a harmonic calculus for measure-geometric Laplacians associated to continuous distributions. We show their theory can be extended to encompass distributions with finite support and give a matrix representation for the resulting operators. In the case of a uniform discrete distribution we make use of this matrix representation to explicitly determine the eigenvalues and the eigenfunctions of the associated Laplacian.

math.DS↗

arXiv2014

A note on measure-geometric Laplacians

We consider the measure-geometric Laplacians $Δ^μ$ with respect to atomless compactly supported Borel probability measures $μ$ as introduced by Freiberg and Zähle in 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 Salem and inhomogeneous self-similar Cantor measures.

math.FA↗