Construction of Continuous Functions with Prescribed Local Regularity

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.