# Zernike functions, rigged Hilbert spaces and potential applications

## TL;DR

This paper explores the mathematical structure of Zernike functions, revealing their symmetries and basis properties within rigged Hilbert spaces, and discusses potential applications in optical image processing.

## Contribution

It provides a detailed analysis of Zernike functions' symmetries, basis structures, and their potential role in optical image processing, extending their mathematical understanding.

## Key findings

- Zernike functions exhibit su(1,1) + su(1,1) symmetry.
- Discrete and continuous bases coexist in rigged Hilbert spaces for Zernike functions.
- Operators on Zernike spaces may enhance optical image processing techniques.

## Abstract

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) + su(1,1). We show how they induce discrete as well continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some other interesting properties of Zernike polynomials and Zernike functions. One of the interests of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing.