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The implication of 1) 2) is relatively easy and can be found in [DAO 98] Hereafter, we present a constructive proof of the converse implication To do so, let H denote the set of functions of [0; 1] in [0; 1] which are inferior limits of continuous functions We need the following lemma LEMMA 91 ([DAO 98]) Let s H There exists a sequence {Rn }n 1 of piecewise polynomials such that: s(t) = lim inf n + Rn (t) t [0; 1] + Rn n, Rn n n 1 (93) 1 R n log n where Rn and Rn are the right and left derivatives of Rn , respectively
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Let {Rn }n 1 be the sequence given by (93) and M be the set of m-adic points of [0; 1] Now let us consider the sequence {rk }k 1 of functions on M in de ned, for k0 any t M such that t = p=1 ip m p , by: r1 (t) = R1 (i1 m 1 )
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Thanks to the continuity conditions, nding a GIFS whose attractor satis es 1) amounts to determining the double sequence (ck )i,k The latter is given by the i following result PROPOSITION 95 ([DAO 98]) Let s H and {rk }k 1 be the previously de ned sequence Then, the attractor of the GIFS whose contraction factors are given by: ck = m rk (im i
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is the graph of the continuous function f satisfying: f (t) = s(t) t [0; 1]
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This result provides an explicit method, fast and easy to execute, that allows the construction of interpolating continuous functions whose H lder function is prescribed in the class of inferior limits of continuous functions (situated between the rst and second Baire classes) Let us underline that there are two other constructive approaches to prescribe H lder functions One of them is based on a generalization of Weierstrass function [DAO 98] and the other is based on the wavelet decomposition [JAF 95] This section is concluded with some numerical simulations Figures 91 and 92 show the attractors of GIFS with prescribed H lder functions Figure 91 (respectively 92) shows the graph obtained when s(t) = t (respectively s(t) = | sin(5 t)|) In both cases, the set of interpolation points is: (0, 0); 1 ,1 ; 5 2 ,1 ; 5 3 ,1 ; 5 4 , 1 ; (1, 0) 5
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95 Estimation of pointwise H lder exponent by GIFS In this section, we address the problem of the estimation of the H lder exponent for a given discrete time signal Our approach, based on GIFS, is to be compared
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Figure 91 Attractor of a GIFS whose H lder function is s(t) = t
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Figure 92 Attractor of a GIFS whose H lder function is s(t) = | sin(5 t)|
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with two other methods which make it possible to obtain satisfactory estimations The rst method is based on the wavelet transform and is called the wavelet transform maxima modules (WTMM) (cf 3 for a detailed description of this method) The second method [GON 92b] uses Wigner-Ville distributions 951 Principles of the method For the sake of simplicity, our study is limited to continuous functions on [0; 1] The calculation algorithm of the H lder exponent is based on Proposition 94 To apply this proposition to the calculation of the H lder exponents for a real continuous signal f , we have to begin by calculating the coef cients cj of a GIFS k whose attractor is f This amounts to solving the inverse problem for GIFS, which is a generalization of the ordinary inverse problem for IFS The latter problem was studied by many authors, either from a theoretical point of view [ABE 92, BAR 88, BAR 85a, BAR 85b, BAR 86, CEN 93, DUV 92, FOR 94, FOR 95, VRS 90] or in
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