# New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.

@article{Pogosyan2017NewSP, title={New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.}, author={George S. Pogosyan and Kurt Bernardo Wolf and Alexander Yakhno}, journal={Journal of the Optical Society of America. A, Optics, image science, and vision}, year={2017}, volume={34 10}, pages={ 1844-1848 } }

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.

## 10 Citations

### Interbasis expansions in the Zernike system

- Mathematics
- 2017

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical…

### Zernike System Stems from Free Motion on the 3-Sphere

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- 2020

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include…

### From Free Motion on a 3-Sphere to the Zernike System of Wavefronts Inside a Circular Pupil

- PhysicsJournal of Physics: Conference Series
- 2020

Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into…

### Spherical geometry, Zernike’s separability, and interbasis expansion coefficients

- MathematicsJournal of Mathematical Physics
- 2019

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include…

### On the generalization of classical Zernike system

- Mathematics
- 2022

We generalize the results obtained recently (Nonlinearity 36 (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H = ~p 2 + F ( ~q · ~p ), ~q, ~p ∈ R 2 , for…

### RETRACTED ARTICLE: Design of Heterogeneous Data Transmission System for Industrial Enterprises Based on Fuzzy Algorithm

- Computer ScienceWireless Personal Communications
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The experimental results show that the performance of the proposed system is better, further play the role of information data in production; improve system efficiency and anti-fault ability.

### Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

- Mathematics, PhysicsNonlinearity
- 2023

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and N∈N . For N = 2,…

### On elliptic trigonometric form of the Zernike system and polar limits

- PhysicsPhysica Scripta
- 2019

The Zernike system provides orthogonal polynomial solution bases on the unit disk that separate in coordinates that are generically elliptic. This is a superintegrable system whose optical…

### Zernike functions, rigged Hilbert spaces, and potential applications

- Mathematics, PhysicsJournal of Mathematical Physics
- 2019

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…

### The Superintegrable Zernike System

- Mathematics
- 2017

We present a resume of this year’s work on what we call the Zernike system. It stems from a differential equation proposed by Frits Zernike in 1934 to describe wavefronts at circular optical pupils…

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