Multifractal Analysis and Wavelets

Overview & Synopsis

In this course, we give the basics of the part of multifractal theory that intersects wavelet theory. We start by characterizing the pointwise Hölder exponents by some decay rates of wavelet coefficients. Then, we give some examples of wavelet series having a multifractal behavior, and we explain how to build wavelet series with prescribed pointwise Hölder exponents. Next we develop the problematics of mul-tifractal formalism, going from the intuitive formula by Frisch and Parisi to explicit and exploitable formulas. We prove that " multifractals are everywhere " , in the sense that typical functions in Besov spaces or typical measures are multifractal in the sense of Baire's categories. We finish by some well-known examples of multifractal wavelet series, random and deterministic, focusing on the influence of certain adaptive threshold procedures to the multifractal properties of signals.