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Theorem 4 (Iacus and La Torre, 2001)

On fractal distribution function estimation and applications

Theoretical background for affine IFS

Theorem 3 (Forte and Vrscay, 1995)

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Theorem 8 (Forte and Vrscay, 1998)

Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 12 (Gill and Levit, 1995)

4.2 Characteristic function and Fourier density estimation

Under conditions i) to v):

**1.** T_{p} is an operator from

to itself.

**2.** Suppose that w_{i}(x) = x, P _{i} = p, and δ_{i} ≥ −p, then

**3.**

then *T _{p}* is a contraction on

with contractivity

constant c.

**4.** Let

such that T_{p}F_{1} = F_{1} and T_{p}* F_{2} = F_{2}. Then

where c is the contractivity constant of T_{p}. The theorem above assures the IFS nature of the operator T_{p} that can be denoted, as in the previous section, as a N-maps IFS(w, p) with obvious notation. The goal is again the solution of the inverse problem in terms of p. Consider the following convex set:

then we have the following results:

... in Theorem 5

Stefano M. Iacus, Davide La Torre

In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems.

Stefano M. Iacus, Davide La Torre