# Construction of Continuous Functions with Prescribed Local Regularity

Abstract : In this work we investigate both from a theoretical and a practical point of view the following problem¸: Let $s$ be a function from $[0¸;1]$ to $[0¸;1]$. Under which conditions does there exist a continuous function $f$ from $[0¸;1]$ to $\RR$ such that the regularity of $f$ at $x$, measured in terms of Hölder exponent, is exactly $s(x)$, for all $x \in [0¸;1]$¸? \\ We obtain a necessary and sufficient condition on $s$ and give three constructions of the associated function $f$. We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally we present a result on the «size» of the set of functions with prescribed local regularity.
keyword :
Type de document :
Rapport
[Research Report] RR-2763, INRIA. 1995
Domaine :
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https://hal.inria.fr/inria-00073928
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Soumis le : mercredi 24 mai 2006 - 14:06:27
Dernière modification le : vendredi 25 mai 2018 - 12:02:05
Document(s) archivé(s) le : lundi 5 avril 2010 - 00:01:57

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• HAL Id : inria-00073928, version 1

### Citation

Khalid Daoudi, Jacques Lévy Véhel, Yves Meyer. Construction of Continuous Functions with Prescribed Local Regularity. [Research Report] RR-2763, INRIA. 1995. 〈inria-00073928〉

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