Solving the inverse problem for measures using iterated function systems: a new approach

Abstract

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1] q , q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w 2 , · ·· }, subject to an ‘ ϵ -contractivity' condition, is employed. Thus, only an optimization over the associated probabilities p i is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the p i which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.