Abstract.
In this paper we investigate from both 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 R 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 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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November 30, 1995. Date revised: September 30, 1996.
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Daoudi, K., Lévy Véhel, J. & Meyer, Y. Construction of Continuous Functions with Prescribed Local Regularity. Constr. Approx. 14, 349–385 (1998). https://doi.org/10.1007/s003659900078
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DOI: https://doi.org/10.1007/s003659900078