2D-Zernike polynomials and coherent state quantization of the unit disc

TL;DR

This paper employs 2D-Zernike polynomials to develop a coherent state quantization of the complex unit disc, deriving associated symbols and transforms, and establishing new summation formulas for these polynomials.

Contribution

It introduces a novel quantization method of the unit disc using 2D-Zernike polynomials and constructs related coherent states and transforms.

Findings

Derived reproducing kernels and Hilbert spaces for 2D-Zernike polynomials

Constructed coherent states for the unit disc

Established summation formulas for 2D-Zernike polynomials

Abstract

Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.