Series expansions of isotropic Gaussian random fields on $\mathbb {S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localized structure provide an alternative to the standard Karhunen–Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analysed.

中文翻译：

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球面上各向同性高斯随机场的多级表示

构造了具有独立高斯系数和局部基函数的$\mathbb {S}^2$ 上各向同性高斯随机场的级数展开。这种具有多级局部结构的表示提供了一种替代标准 Karhunen-Loève 展开的各向同性随机场的球谐函数。基函数通过将协方差算子的平方根应用于球形针获得。在随机场的角功率谱的衰减条件下显示了所得协方差相关多级基的局部化。此外，还给出了数值说明，并分析了球面上随机椭圆偏微分方程的应用。