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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

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