10.1 Introduction

We presented the conjecture of Frisch and Parisi concerning the multifractal nature of the velocity of a turbulent fluid in the previous chapter on wavelets and turbulence. They introduced the hypothesis that there is a set of points with Hausdorff dimension D( h) where the velocity increments satisfy

and from this they argued that

as ? x ? 0, where

We are interested in D( h) because it tells us about the fractal or multifractal nature of fully-developed turbulence, but ?( p) is the quantity we can compute numerically. Fortunately, under the assumption that D( h) is concave, it can be recovered from ?( p) by a classical Legendre inversion formula

Since H lder exponents and Hausdorff dimensions cannot be reasonably computed numerically, (10.3) is the only way to obtain the spectrum of singularities of a signal. Unfortunately, our understanding of this formula is quite poor; there are examples and counterexamples of its validity. (See [154] for a discussion.) The good news is that we can test (10.3) on several mathematically defined functions for which both sides of the equality can be computed independently, and this provides some intuition about the range of validity and the limitations of (10.3). We present two examples of functions that are fractal or multifractal, and we show how wavelet methods can be used to compute their H lder exponents and their spectrums of singularities. The two functions we study are the...