On the relation between Fourier and Walsh–Rademacher spectra for random fields

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.