单位圆盘上Zernike方程组的新分离多项式解及基间展开
New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion
摘要：
Frits Zernike提出的微分方程，以获得一个基础上的多项式正交解的单位磁盘分类波前像差在圆形瞳孔，被证明有一组新的正交解的基础，涉及Legendre和Gegenbauer多项式，在非正交坐标系接近笛卡尔的。我们发现原来的Zernike基础和一个新的集合，这原来是Clebsch-Gordan系数的代表之间的重叠。
英文摘要：
The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the 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 non-orthogonal 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.
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