发表机构
Catholic University of Eichstätt–Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种变形方法，将球面上的各向同性核转换为轴对称核，推导出显式谐波耦合系数公式，证明其几何衰减及固有空间等价性，并通过数值实验验证了在轴对称协方差随机场中能提升预测精度。
AI 中文摘要
本文在球面上构造轴对称核。我们首先取一个各向同性核，并通过特定变形将其转换为轴对称核。对于一般变形，显式的球谐表示不一定可用。对于我们的变形，我们可以用原始各向同性核的展开系数获得变形核的这种表示。我们通过显式描述变形对球谐基的影响来推导该表示。由此，推导出耦合系数的显式有限和公式，并证明对于每个固定输入模式，系数以由我们的变形参数控制的速率几何衰减。我们进一步指出，如果各向同性核的固有空间在范数上等价于一个Sobolev空间，则变形核具有相同的固有空间且范数等价。在我们的数值实验中，我们表明当底层随机场具有轴对称协方差时，我们的变形可以提高预测精度。
英文摘要
In this paper, we construct axially symmetric kernels on the sphere. We start by taking an isotropic kernel and transform it into an axially symmetric kernel via a specific deformation. For a general deformation, an explicit spherical-harmonic representation is not necessarily available. For our deformation, we can obtain such a representation of the deformed kernel in terms of the expansion coefficients of the original isotropic kernel. We derive this representation by explicitly describing the effect of the deformation on the spherical harmonic basis. From this, an explicit finite sum formula for the coupling coefficients is deduced and we show that, for each fixed input mode, the coefficients decay geometrically at a rate controlled by our deformation parameter. We further note that if the native space of the isotropic kernel is norm equivalent to a Sobolev-space, then the deformed kernel has the same native space with an equivalent norm. In our numerical experiments, we show that our deformation can improve prediction when the underlying random field has an axially symmetric covariance.
