# Measure-geometric Laplacians for discrete distributions

## TL;DR

This paper extends measure-geometric Laplacian theory to discrete distributions, providing matrix representations and explicit eigenvalues and eigenfunctions for uniform discrete cases.

## Contribution

It generalizes Freiberg and Zähle's harmonic calculus to finite support distributions and derives explicit spectral properties for uniform discrete distributions.

## Key findings

- Matrix representation of Laplacians for discrete distributions
- Explicit eigenvalues and eigenfunctions for uniform discrete case
- Extension of harmonic calculus to finite support distributions

## Abstract

In 2002 Freiberg and Z\"ahle 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.