CONSTRUCTION OF CONTINUOUS-FUNCTIONS WITH PRESCRIBED LOCAL REGULARITY

In this paper we investigate from both a theoretical and a practical p oint 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 te rms of Holder exponent, is exactly s(x), for all x is an element of [0 ; 1]? We obtain a necessary and sufficient condition on s and give thr ee constructions of the associated function f. We also examine some ex tensions regarding, for instance, the box or Tricot dimension or the m ultifractal spectrum. Finally, we present a result on the ''size'' of the set of functions with prescribed local regularity.