Discrete Bessel and Mathieu functions

TL;DR

This paper introduces discrete Bessel and Mathieu functions derived from the separation of variables in elliptic and polar coordinates, approximating continuous functions via finite sums and discrete transforms.

Contribution

It defines new discrete analogs of classical special functions using N-point Fourier transforms, closely approximating their continuous counterparts and maintaining key properties.

Findings

Discrete functions closely match continuous function values

They preserve key special function relations

Approximate Fourier transforms over the circle with finite sums

Abstract

The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of translations and orthogonal transformations; we replace the latter by the discrete dihedral group of N discrete rotations and reflections. The separation of variables in polar and elliptic coordinates is then used to define discrete Bessel and Mathieu functions, as approximants to the well-known continuous Bessel and Mathieu functions, as N-point Fourier transforms approximate the Fourier transform over the circle, with integrals replaced by finite sums. We find that these 'discrete' functions approximate the numerical values of their continuous counterparts very closely and preserve some key special function relations.