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Theorem 2 - Collage theorem

Let (Y, dY ) be a complete metric space. Given an

suppose that there exists a contractive map f on Y with contractivity factor 0 ≤ c < 1 such that

If is the fixed point of

then

So if one wishes to approximate a function y with the fixed point of an unknown contractive map f, it is only needed to solve the inverse problem of finding f which minimizes the collage distance dY (y,f(y)). The main result in Forte and Vrscay that we will use to build one of the IFSs estimators is that the inverse problem can be reduced to minimize a suitable quadratic form in terms of the pi given a set of affine maps wi and the sequence of moments gk of the target measure. Let

be the simplex of probabilities. Let w = (w1,w2, . . . ,wn), N = 1, 2, . . . be subsets of W = {w1,w2, . . .} the infinite set of affine contractive maps on X = [0, 1] and let g the set of the moments of any order of

Denote by M the Markov operator of the N-maps IFS (w, p) and by

, with associated moment vector of any order hN. The collage distance between the moment vector of

is a continuous function and attains an absolute minimum value Δ min on ΠN

Stefano M. Iacus, Davide La Torre

Summary: Index