Wavelets techniques for pointwise anti-Holderian irregularity

TL;DR

This paper introduces a new notion of weak pointwise anti-Holder regularity, develops a multifractal formalism for it, and demonstrates differences from traditional spectra using wavelet series and Davenport series.

Contribution

It defines weak pointwise anti-Holder regularity, constructs wavelet series satisfying the multifractal formalism, and shows the weak spectrum can differ from the strong spectrum.

Findings

Weak spectrum of singularities can be disconnected from the strong spectrum.

Constructed wavelet series satisfy the multifractal formalism.

Davenport series exhibit differences between weak and strong spectra.

Abstract

In this paper, we introduce a notion of weak pointwise Holder regularity, starting from the de nition of the pointwise anti-Holder irregularity. Using this concept, a weak spectrum of singularities can be de ned as for the usual pointwise Holder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (denoted here strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum di ers from the strong one.