wavelet transforms and their applications to turbulence - Wavelets ...

Annual Reviewswww.annualreviews.org/aronlineWAVELET TRANSFORMS 409if we want to study the scaling of a function we must take the waveletcoefficients in the L ~ norm, instead of L 2. The wavelet coefficients writtenin the L t norm 2 are related to the wavelet coeff~cients written in the Lnorm by the simple expression:fL’ = l-1/2yL2. (19)Iffbelongs to A~(x0), the H61der space of functions of exponent ~, i.e.f is continuous, not necessarily differentiable in x0 but such thatthenIf(x+xo)-f(x)[ = Clxol ~ with g < 1 and constant C > 0, (20a)3~(/, x0) Ce’f~*l~l 1/~ for l ~ 0, (20b)where @ is the phase of the wavelet coefficients.The transformed function f is regular even iff is not. The informationabout any possible singularities present in the signal, their position x0,their strength C, and their scaling exponent ~, is given by the asymptoticbehavior off(l, xo), written in norm L ~ (19), for l tending to zero. If theL~-norm wavelet coefficients diverge in the small scales at the point Xo,then f is singular at x0 and the slope of log If(l, x0) l versus log l will givethe exponent of the singularity (20). It is also very easy to localize thepossible singularities off by looking at the phase of its wavelet coefficients:The lines of constant phase converge on the singularities (Figure 2a). If,on the contrary, the modulus of the wavelet coefficient becomes zero inthe very small scales around xo, then the functionf is regular at x0. Thisresult is in fact the converse of (18), but its mathematical justificationrequires more global assumptions on the analyzed function, namely somedecay of its wavelet coefficients in the vicinity of x0 (Jaffard 1989a). Therefore,in practice, to locally analyze a singularity at x0 we should first verifythat at small scales the wavelet coefficients around x0 are not larger thanthose at x0. Then we should consider not only the coefficientsf(l, x0), butat least all coefficients belonging to the influence cone pointing towardsx0. Those properties are independent of the choice of the wavelet andthey are particularly useful for characterizing fractals and multifractals(Holschneider 1988b; Arn+odo et al 1989; Ghez & Vaienti 1989; Argoulet al 1988, 1990; Falconer 1991; Freysz et al 1990).REPRODUCING KERNEL Using the wavelet ~k, we can decompose any functionor distribution f into its wavelet coefficients. In the case of thecontinuous wavelet transform these coefficients form an over-completebasis. This implies a correlation between the wavelet coefficients which, inturn, corresponds to the existence of a reproducing kernel K6 associatedto the wavelet ~ defined by