MULTIFRACTAL FORMALISM FOR THE GENERALIZED DE RHAM FUNCTION

Mourad Ben Slimane (Tunisia)

Received January 25, 2008

Abstract

The classical de Rham function is continuous but nowhere differentiable. We analyse the multifractal properties of a straightforward generalization of the de Rham function and obtain generalized de Rham functions with pointwise Hölder regularity larger than 1 (so differentiable at many points). The spectrum of singularities (which associates to each given pointwise Hölder regularity the Hausdorff dimension of the set of points for which the function has exactly that value of pointwise Hölder regularity) of the generalized de Rham function can be determined directly. That spectrum can also be deduced from “global” quantities numerically computable extracted from the function. Finally, we prove the validity of the multifractal formalism.

Keywords and phrases: Hölder regularity, Hausdorff dimension, Besov spaces, multifractal formalism, spline wavelet basis.