Sampling inhomogeneous turbulent fields

thus they are statistically orthogonal. The KL functions provide the "best" orthonormal set of functions in the sense that the expansion converges faster than expansions in terms of any other orthonormal set, as measured in the Lz norm. These eigenfunctions reduce to the trigonometric functions if U(y) is statistically homogeneous, so the primary significance of the KL expansion lies in its ability to efficiently represent statistically inhomogeneous processes.