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3 Fourier analysis results

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

The results presented in this section, taken from Forte and Vrscay (1998) Sec. 6, are rather straight forward to prove but it is essential to recall them since we will use these in density estimation later on.

Given a measure

the Fourier transform

where

is the complex

space, is defined by the relation

Table 1: Approximation results for the di erent N-maps IFS (**w**, **p**) for the targe distribution function F(x) = x^{2}(3−2x) as in Forte and Vrscay (1995). N = number of maps used, AMSE = average MSE, max **p** is the contractivity constant of *T _{N}* in

with the well known properties

We denote by *FT*(X) the set of all *FT*'s associated to the measures in *M*(X). Given two elements

and

*FT*(X) the following metric can be defined

and the above integral is always finite (see cited paper). With this metric (*FT*(X), d_{FT} ) is a complete metric space. Given an *N*-maps affine IFS(**w**, **p**) and its Markov operator *M* it is possibile to define a new linear operator B : *FT*(X) → *FT*(X) as follows

where

is th*e F*T of a measure µ and is the *FT* of ν =* M*µ.

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