Multifractal analysis based on p-exponents and lacunarity exponents

TL;DR

This paper introduces a new multifractal analysis method based on p-exponents, capable of analyzing signals with unbounded behavior, and demonstrates its application to lacunary wavelet series.

Contribution

It develops a multifractal analysis framework using p-exponents, extending the applicability to signals with unbounded singularities, and adapts wavelet methods for this setting.

Findings

p-exponent can take large negative values, modeling lacunarity.

Mathematical properties of p-exponents are characterized.

Application to lacunary wavelet series demonstrates effectiveness.

Abstract

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the H\"older exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of "lacunarity" of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.