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and thus dN (Cp )1/p N 1/p Since we can take p arbitrarily close to 0, we observe that the rearrangement in a decreasing order of the sequence |df (k, j)| has fast decay, which is, once again, a way to express the lacunarity (the converse is immediate: if the sequence dn has fast decay, it belongs to all lp and thus this is also the case for the sequence df (k, j)) The Besov space for p < 1 is not locally convex, which partly explains the dif culties in their utilization Before the introduction of wavelets, these spaces were characterized either by order of approximation of f with rational fractions for which the numerator and the denominator have a xed degree, or by an order of approximation with the splines with free nodes (which means that we are free to choose the points where the polynomials in parts are connected) (see [DEV 98, JAF 01a]) However, these characterizations are dif cult to handle and hence do not have any real numerical applications Characterization (323) shows that the knowledge of the Besov spaces to which f belongs is clearly linked to the asymptotic behavior (when j + ) of the moments of distribution of the wavelet coef cients of f ; see (326) Generally, more information is available; indeed, these moments are deduced from the histogram of the coef cients at each scale j This is why it is normal to wonder which information regarding the pointwise regularity of f can be deduced from the knowledge of these histograms We present a study of this problem below We observe that the cascade type models for the evolution of the repartition function of wavelet coef cients through the scales have been proposed to model the speed of turbulent ows [ARN 98] To start with, let us point out a limitation of the multifractal analysis: functions having the same histograms of wavelet coef cients at each scale can have singularity spectra that are completely different [JAF 97a] In the multifractal analysis, it is not only the histogram of the coef cients which is important, but also their positions This is why no formula deducing the singularity spectrum from the knowledge of the histograms can be valid in general However, we can hope that some formulae are more valid than others Indeed, we have observed that if the coef cient values are independent random variables, there is a spectrum which is almost sure; we shall notice that the formula that yields this spectrum differs from the formulae proposed until now Another approach consists of specifying the information on the function and considering the functional spaces that take the positions of the large wavelet coef cients into consideration (see [JAF 05])
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342 Construction of formalisms The construction of a multifractal formalism can be based on two types of considerations: counting arguments: we consider the increments (or wavelet coef cients) having a certain size; we estimate their number and deduce their contribution to some calculable quantities; more mathematical arguments: we prove that a bound of the spectrum, according to the calculable quantities, is generally true and that this bound is generically an equality The term generically is to be understood in the sense of Baire classes if the information at the start is of functional type, or as almost sure if the information at the start is a probability We begin by describing the rst approach; we do not exactly recapitulate the initial argument of Frisch and Parisi, but rather its translation in wavelets, as found in [ARN 95b, JAF 97a] This approach admits two variants, according to the information on the function that we have Indeed, we can start from: the partition function (q) de ned from knowledge of the Besov spaces to which f belongs: (q) = sup{s : f B s/q,q } = 1 + lim inf
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