Lacunary wavelet series on Cantor sets

TL;DR

This paper analyzes lacunary wavelet series on Cantor sets, revealing their multifractal properties and how certain constructions can cause deviations from standard multifractal formalisms, with implications for numerical detection.

Contribution

It introduces a novel multifractal analysis method for lacunary wavelet series on Cantor sets and demonstrates how desynchronization affects multifractal formalisms.

Findings

Lacunary wavelet series can violate classical multifractal formalisms.

Large deviation spectra can detect failures in multifractal formalism.

Numerical methods can identify deviations caused by desynchronization.

Abstract

We provide a multifractal analysis of lacunary wavelet series on Cantor sets. Byintroducing a desynchronization between the scales of the wavelets and the scales of the steps of the construction of the Cantor set, we obtain random processes that donot satisfy multifractal formalisms based on the Legendre transform and on the large deviation of wavelet leaders. Subsequently, we show how the computation of the leader large deviation spectra of "lacunarized" versions of such wavelet series can detect the failure in the multifractal formalism using only numerical quantities.