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A generalization of the Zernike circle polynomials for expansion of functionsvanishing outside the unit disk is given. These generalized Zernike functionshave the form Zm,{\alpha} n ({\rho}, \vartheta) = Rm,{\alpha} n ({\rho})exp(im\vartheta), 0 \leq {\rho} < 1, 0 \leq \vartheta < 2{\pi}, and vanish for{\rho} > 1, where n and m are integers such that n - |m| is nonnegative andeven. The radial parts are O((1 - {\rho}2){\alpha}) as {\rho} \uparrow 1 inwhich {\alpha} is a real parameter > -1. The Zm,{\alpha} n are orthogonal onthe unit disk with respect to the weight function (1 - {\rho}2)-{\alpha}, 0\leq {\rho} < 1. The Fourier transform of Zm,{\alpha} n can be expressedexplicitly in terms of (generalized) Jinc functionsJn+{\alpha}+1(2{\pi}r)/(2{\pi}r){\alpha}+1 and exhibits a decay behaviourr-{\alpha}-3/2 as r \rightarrow \infty. Etc.

A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory