Oscillating singularities in Besov spaces

Abstract

The purpose of multifractal analysis is to evaluate the Hausdorff dimensions d( h) of the sets S h of points where the pointwise Hölder exponent of a function, a signal or an image has a given value h∈[ h 0, h 1]. Inside the realm of mathematics this makes good sense but for most signals or images such calculations are out of reach. That is why Uriel Frisch and Giorgio Parisi proposed an algorithm which relates these dimensions d( h) to some averaged increments. Averaged increments are named structure functions in fluid dynamics and can be easily computed. The Frisch and Parisi algorithm is called multifractal formalism. Unfortunately multifractal formalism is not valid in full generality and one should know when it holds. A general answer is supplied by “Baire-type” results. These results show that in many function spaces, quasi-all functions (in the sense of Baire's categories) do not obey the multifractal formalism if the Hölder exponent is large. Our purpose is to understand this phenomenon. We will prove that a cause of the failure of the multifractal formalism is the presence of oscillating singularities, which was guessed by A. Arnéodo and his collaborators.