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On the recovery of a function on a circular domain

机译：关于循环域上函数的恢复

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摘要

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.
机译：考虑到给定离散且嘈杂的数据，我们考虑了估计单位磁盘f {（x，y）：x / sup 2 / + y / sup 2 // spl les / 1}上的函数f（x，y）的问题记录在规则的正方形网格上。提出了基于单位磁盘上一类正交和完整函数的f（x，y）估计。这类函数具有一个独特的特性，即，轴围绕坐标原点的旋转不变，因此产生旋转不变的估计。对于径向函数，正交集具有与经典勒让德多项式有关的特别简单的形式。我们从L / sub 2 /度量的角度对f（x，y）的建议估计值进行统计准确性分析。发现由于圆域的几何性质，估计精度存在固有的局限性。这是通过将精度问题与解析数论中称为圆的晶格点的著名问题联系起来来解释的。实际上，所获得的平均积分平方误差的范围由迄今为止关于圆域内晶格点问题的最著名结果确定。

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