Spectral decomposition of random fields defined over the generalized Cantor set

Solutions of a set of homogeneous Fredholm integral equations that appear in field analyses on approximations of fractal sets are presented. If the support is a stochastic fractal, then the field will inherit this property. A simple representation of such functions in terms of a basis is desirable. A solution to this problem is provided by the Karhunen–Loeve decomposition of stochastic functions with known covariance. In the problem at hand, the covariance is defined by the fractal nature of the support. Here we provide a rule by which the spectral decomposition of the covariance is evaluated for any iteration/scale of a generalized Cantor set. This problem is identical to the solution of a Fredholm equation of second kind.