SOLVING THE INVERSE PROBLEM FOR MEASURES USING ITERATED FUNCTION SYSTEMS - A NEW APPROACH

摘要

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure mu on x = [0, 1](q), q greater than or equal to 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 X, W = {w(1), w(2), ...}, subject to an 'epsilon-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. [References: 28]