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,{ \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 and even. The radial parts are O((1 - { \rho}2){ \alpha}) as { \rho} \uparrow 1 in which { \alpha} is a real parameter > -1. The Zm,{ \alpha} n are orthogonal on the 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 expressed explicitly in terms of (generalized) Jinc functions Jn+{ \alpha}+1(2{ \pi}r)/(2{ \pi}r){ \alpha}+1 and exhibits a decay behaviour r-{ \alpha}-3/2 as r \rightarrow \infty. Etc.
Categories: math-ph math.MP