FOURIER AND TAYLOR-SERIES ON FITNESS LANDSCAPES

Holland's "hyperplane transform" of a "fitness landscape", a random, real valued function of the vertices of a regular finite graph, is shown to be a special case of the Fourier transform of a function of a finite group. It follows that essentially all of the powerful Fourier theory, which assumes a simple form for commutative groups, can be used to characterize such landscapes. In particular, an analogue of the Karhunen-Loeve expansion can be used to prove that the Fourier coefficients of landscapes on commutative groups are uncorrelated and to infer their variance from the auto-correlation function of a random walk on the landscape. There is also a close relationship between the Fourier coefficients and Taylor coefficients, which provide information about the landscape's local properties. Special attention is paid to a particularly simple, but ubiquitous class of landscapes, so-called "AR(1) landscapes".