A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory
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, \vartheta) = Rm,α n (ρ) exp(im\vartheta), 0 \leq ρ < 1, 0 \leq \vartheta < 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 \leq 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 \rightarrow \infty. Etc.
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