On the recovery of a function on a circular domain

Abstract

We consider the problem of estimating a function f (x, y) on the unit disk f {(x, y): x/sup 2/+y/sup 2//spl les/1}, given a discrete and noisy data recorded on a regular square grid. An estimate of f (x, y) based on a class of orthogonal and complete functions over the unit disk is proposed. This class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate. For-radial functions, the orthogonal set has a particularly simple form being related to the classical Legendre polynomials. We give the statistical accuracy analysis of the proposed estimate of f (x, y) in the sense of the L/sub 2/ metric. 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. In fact, the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain.