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.
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• HAL Id : inria-00073928, version 1

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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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