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Analyse fractale d'une famille de fonctions aléatoires: les fonctions de bosses

Abstract : We study random series defined on R^D as
F(t) = Σ n^(-α/D)G(n^(1/D)(t − Xn)) , with α > 0, G an elementary bump or pulse and (Xn) a sequence of independent random variables. First we discuss existence of more general series, named sums of pulses, emphasizing the meaning of each parameter. Sufficient conditions of existence, continuity and integrability are stated here. We proceed with the study of path regularity and we determine a uniform Holder exponent for F. We are naturally interested in fractal dimensions of the graph of F . To this end we develop some general analysis tools suitable for Holder continuous functions. We obtain results allowing us to estimate box-dimension, regularization dimension, and a large class of dimensional indices. Some of them are linked to multifractal analysis. We also calculate the Hausdorff dimension of the graph of F . We devote the second part of our work to applications of sums of pulses, especially to model rough surfaces. We show off new theoretic properties using structure functions. They give log-log plots that also take experimental constraints into consideration. They are used to identify a sum of pulses and estimate his parameters. We propose many methods building suitable estimators. Then it is possible to model any given signal using sums of pulses. Structure curves are used again as adequation criterion. We give several examples of theoretic and experimental data.
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  • HAL Id : tel-00250060, version 1

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Yann Demichel. Analyse fractale d'une famille de fonctions aléatoires: les fonctions de bosses. Mathématiques [math]. Université Blaise Pascal - Clermont-Ferrand II, 2006. Français. ⟨NNT : 2006CLF21696⟩. ⟨tel-00250060⟩

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