Random Wavelet Series: Theory and Applications

TL;DR

This paper introduces Random Wavelet Series, a class of multifractal processes, and demonstrates their applications in spectrum synthesis, large deviations analysis, and turbulence modeling.

Contribution

It provides new methods for synthesizing functions with specified spectra, analyzing multifractal spectra, and connecting these processes to turbulence theory.

Findings

Able to synthesize functions with arbitrary spectra

Multifractal spectrum matches large deviations spectrum

Processes exhibit generalized selfsimilarity in turbulence

Abstract

Random Wavelet Series form a class of random processes with multifractal properties. We give three applications of this construction. First, we synthesize a random function having any given spectrum of singularities satisfying some conditions (but including non-concave spectra). Second, these processes provide examples where the multifractal spectrum coincides with the spectrum of large deviations, and we show how to recover it numerically. Finally, particular cases of these processes satisfy a generalized selfsimilarity relation proposed in the theory of fully developed turbulence.