We discuss relations between the expansion coefficients of a discrete random field when analyzed with respect to different hierarchical bases. Our main focus is on the comparison of two such systems: the Walsh–Rademacher basis and the trigonometric Fourier basis. In general, spectra computed with respect to one basis will look different in the other. In this paper, we prove that, in a statistical sense, the rate of spectral decay computed in one basis can be translated to the other. We further provide explicit expressions for this translation on quadrilateral meshes. The results are illustrated with numerical examples for deterministic and random fields.

我们讨论了在不同层次基础上进行分析时离散随机场的展开系数之间的关系。我们的主要重点是比较两个这样的系统：沃尔什-拉德马赫基和三角傅立叶基。一般来说，根据一种基础计算的光谱在另一种基础上看起来会有所不同。在本文中，我们证明，从统计意义上来说，在一种基础上计算的光谱衰减率可以转换为另一种基础。我们进一步为四边形网格上的这种转换提供了显式表达式。结果通过确定性场和随机场的数值示例进行说明。