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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 IR such that the regularity of f at x, measured in terms of Hölder exponent, is exactly s(x), for all x [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, as for instance conditions on the box or Tricot dimension or the multifractal spectrum of these functions. Finally we present a result on the "size" of the set of functions with prescribed local regularity.
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https://hal.inria.fr/inria-00593268
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Khalid Daoudi, Jacques Lévy Véhel, yves Meyer. Construction of continuous functions with prescribed local regularity. Constructive Approximation, Springer Verlag, 1998, 14 (3), pp.349-385. ⟨10.1007/s003659900078⟩. ⟨inria-00593268⟩

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