Nonparametric estimation of a function on a circular domain

We consider the problem of estimating a function defined on the unit disk given a discrete and noisy data observed on a regular square lattice. An estimate of the function based on a class of orthogonal and complete polynomials over the unit disk often referred to as the Zernike polynomials is proposed. This class of polynomials has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a class of invariant estimates. We give the statistics accuracy analysis of the Zernike polynomial based estimate. It is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain. This is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle.