The beat of a fuzzy drum: fuzzy Bessel functions for the disc

TL;DR

This paper introduces fuzzy Bessel functions as eigenfunctions of a fuzzy Laplacian on the fuzzy disc, providing a matrix approximation that preserves rotational symmetry and converges to classical Bessel functions in the commutative limit.

Contribution

It develops a basis for functions on the fuzzy disc using eigenfunctions of a fuzzy Laplacian, bridging fuzzy and classical Bessel functions.

Findings

Eigenfunctions form a basis for the fuzzy disc algebra

Convergence to classical Bessel functions in the commutative limit

Preservation of rotational symmetry in the fuzzy approximation

Abstract

The fuzzy disc is a matrix approximation of the functions on a disc which preserves rotational symmetry. In this paper we introduce a basis for the algebra of functions on the fuzzy disc in terms of the eigenfunctions of a properly defined fuzzy Laplacian. In the commutative limit they tend to the eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of the first kind, thus deserving the name of fuzzy Bessel functions.