Hölder Continuity and Wavelets

There exist a lot of continuous nowhere differentiable functions, but these functions do not have the same irregularity. Hölder continuity, and more precisely Hölder exponent, allow to quantify this irregularity. If the Hölder exponent of a function takes several values, the function is said multifractal. In the first part of this thesis, we study in details the regularity and the multifractality of some functions: the Darboux function, the Cantor bijection and a generalization of the Riemann function.

The theory of wavelets notably provides a tool to investigate the Hölder continuity of a function. Wavelets also take part in other contexts. In the second part of this thesis, we consider a nonstationary version of the classical theory of wavelets. More precisely, we study the nonstationary orthonormal bases of wavelets and their construction from a nonstationary multiresolution analysis. We also present the nonstationary continuous wavelet transform.

For some irregular functions, it is difficult to determine its Hölder exponent at each point. In order to get some information about this one, new function spaces based on wavelet leaders have been introduced. In the third and last part of this thesis, we present these new spaces and their first properties. We also define a natural topology on them and we study some properties.