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The p-spectrum of Random Wavelet Series

Esser Céline, Lambert Thelma, Vedel Béatrice

TL;DR

This work extends multifractal analysis to p-exponents by developing a wavelet-based method that estimates the p-spectrum from the scale distribution of wavelet coefficients. It derives a rigorous upper bound for the p-spectrum in terms of the wavelet density and proves its sharpness for Random Wavelet Series, where the bound is realized almost surely on a natural interval. Furthermore, the authors establish a prevalence result in S^ν spaces, showing that a large (in the sense of prevalence) class of functions attains this bound, thereby capturing typical multifractal behaviour in constrained function spaces. The framework unifies p-exponent analysis with large-deviation techniques and mass transference principles to connect local regularity with geometric size in a robust, wavelet-based formalism.

Abstract

The goal of multifractal analysis is to characterize the variations in local regularity of functions or signals by computing the Hausdorff dimension of the sets of points that share the same regularity. While classical approaches rely on Hölder exponents and are limited to locally bounded functions, the notion of $p$-exponents extends multifractal analysis to functions locally in $L^p$, allowing a rigorous characterization of singularities in more general settings. In this work, we propose a wavelet-based methodology to estimate the $p$-spectrum from the distribution of wavelet coefficients across scales. First, we establish an upper bound for the $p$-spectrum in terms of this distribution, generalizing the classical Hölder case. The sharpness of this bound is demonstrated for \textit{Random Wavelet Series}, showing that it can be attained for a broad class of admissible distributions of wavelet coefficients. Finally, within the class of functions sharing a prescribed wavelet statistic, we prove that this upper bound is realized by a prevalent set of functions, highlighting both its theoretical optimality and its representativity of the typical multifractal behaviour in constrained function spaces.

The p-spectrum of Random Wavelet Series

TL;DR

This work extends multifractal analysis to p-exponents by developing a wavelet-based method that estimates the p-spectrum from the scale distribution of wavelet coefficients. It derives a rigorous upper bound for the p-spectrum in terms of the wavelet density and proves its sharpness for Random Wavelet Series, where the bound is realized almost surely on a natural interval. Furthermore, the authors establish a prevalence result in S^ν spaces, showing that a large (in the sense of prevalence) class of functions attains this bound, thereby capturing typical multifractal behaviour in constrained function spaces. The framework unifies p-exponent analysis with large-deviation techniques and mass transference principles to connect local regularity with geometric size in a robust, wavelet-based formalism.

Abstract

The goal of multifractal analysis is to characterize the variations in local regularity of functions or signals by computing the Hausdorff dimension of the sets of points that share the same regularity. While classical approaches rely on Hölder exponents and are limited to locally bounded functions, the notion of -exponents extends multifractal analysis to functions locally in , allowing a rigorous characterization of singularities in more general settings. In this work, we propose a wavelet-based methodology to estimate the -spectrum from the distribution of wavelet coefficients across scales. First, we establish an upper bound for the -spectrum in terms of this distribution, generalizing the classical Hölder case. The sharpness of this bound is demonstrated for \textit{Random Wavelet Series}, showing that it can be attained for a broad class of admissible distributions of wavelet coefficients. Finally, within the class of functions sharing a prescribed wavelet statistic, we prove that this upper bound is realized by a prevalent set of functions, highlighting both its theoretical optimality and its representativity of the typical multifractal behaviour in constrained function spaces.

Paper Structure

This paper contains 13 sections, 37 theorems, 221 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Wavelets and leaders
  4. Hausdorff measure and dimension
  5. Upper bound for the $p$-spectrum
  6. Large deviation estimates of $p$-leaders
  7. Proof of Theorem \ref{['mainSup']}
  8. Study of the $p$-spectrum of Random Wavelet Series
  9. Random Wavelet Series
  10. Proof of Theorem \ref{['main_RWS']}
  11. Prevalent $p$-spectrum in $S^\nu$ spaces
  12. Prevalence and $S^\nu$ spaces
  13. Proof of Theorem \ref{['mainPrev']}

Key Result

Theorem 1.4

If $f$ is a function for which $p_0(f)>0$, then for every $0<p<p_0(f)$ and every $h\geq \frac{-1}{p}$,

Figures (1)

  • Figure 1: The almost sure $p$-spectrum of a Random Wavelet Series (in red) together with the corresponding wavelet density (in blue).

Theorems & Definitions (71)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • ...and 61 more