2011/10/11 by A. J. E. M. Janssen, Janssen, Augustus
Computer Science · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Physical sciences #Mathematical Physics (math-ph) #Optical measurement and interference techniques #Photorefractive and Nonlinear Optics
paper · pdf · doi:10.48550/arxiv.1110.2369
openalex publication_date 2011/10/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A generalization of the Zernike circle polynomials for expansion of functions vanishing outside the unit disk is given. These generalized Zernike functions have the form Zm,α n (ρ, ϑ) = Rm,α n (ρ) exp(imϑ), 0 ≤ ρ < 1, 0 ≤ ϑ < 2π, and vanish for ρ > 1, where n and m are integers such that n - |m| is nonnegative and even. The radial parts are O((1 - ρ2)α) as ρ \uparrow 1 in which α is a real parameter > -1. The Zm,α n are orthogonal on the unit disk with respect to the weight function (1 - ρ2)-α, 0 ≤ ρ < 1. The Fourier transform of Zm,α n can be expressed explicitly in terms of (generalized) Jinc functions Jn+α+1(2πr)/(2πr)α+1 and exhibits a decay behaviour r-α-3/2 as r → ∞. Etc.