Various aspects of gravity field modeling rely upon analytical mathematical functions for calculating spherical harmonic coefficients. Such functions allow quick and efficient evaluation of cumbersome convolution integrals defined on the sphere. In this work, we present a new analytical method for determining spherical harmonic coefficients of isotropic polynomial functions. This method in computationally flexible and efficient, since it makes use of recurrence relations. Also, its use is universal and could be extended to piecewise polynomials and polynomials with compact support. Our numerical investigation of the proposed method shows that certain recurrence relations lose accuracy as the order of implemented polynomials increases because of accumulation of numerical errors. Propagation of these errors could be mitigated by hybrid methods or using extended precision arithmetic. We demonstrate the relevance of our method in gravity field modeling and discuss two areas of application. The first one is the design of B-spline windows and filter kernels for the low-pass filtering of gravity field functionals (e.g., GRACE Follow-On monthly geopotential solutions). The second one is the calculation of spherical harmonic coefficients of isotropic polynomial covariance functions.

重力场建模的各个方面依赖于计算球谐系数的分析数学函数。这些函数可以快速有效地评估球体上定义的繁琐的卷积积分。在这项工作中，我们提出了一种新的分析方法来确定各向同性多项式函数的球谐系数。这种方法计算灵活且高效，因为它利用了递归关系。此外，它的用途是通用的，可以扩展到分段多项式和具有紧凑支持的多项式。我们对所提出方法的数值研究表明，由于数值误差的累积，随着所实施多项式阶数的增加，某些递推关系会失去准确性。这些误差的传播可以通过混合方法或使用扩展精度算术来减轻。我们展示了我们的方法在重力场建模中的相关性，并讨论了两个应用领域。第一个是用于重力场泛函低通滤波的 B 样条窗口和滤波器内核的设计（例如，GRACE Follow-On 每月位势解决方案）。第二个是各向同性多项式协方差函数的球谐系数的计算。