Approximation of fractals by discrete graphs: norm resolvent and spectral convergence

2017/03/31 by Olaf Post, Post, Olaf, Jan Simmer +1
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1704.00064

openalex publication_date 2017/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show a norm convergence result for the Laplacian on a class of post-critically finite fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional graph Laplacians with corresponding discrete probability measures. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.

Related

• Approximation of fractals by manifolds and other graph-like spaces
• A connection between discrete and regularized Laplacian determinants on fractals
• Spectral Convergence of the Discrete Laplacian on Models of a Metrized Graph
• Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs
• Fractal algebras of discretization sequences