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histograms of wavelet coef cients Generally, for each j, let Nj ( ) = #{k : |df (k, j)| E(Nj ( )) = 2j j ([0, a]) If: ( , ) = lim sup
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(324)
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It is important to note that the information provided by ( ) is richer than that provided by (q); indeed, (q) can be deduced from the histograms with: (q) = lim inf
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because, by de nition of Nj , we have deduce that:
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2 qj dNj ( ) It is easy to (327)
(q) = inf q ( ) + 1
On the other hand, we cannot reconstitute ( ) from (q); indeed, it is clear from (327) that the two functions ( ) and ( ) having the same concave envelope lead to the same function (q) Based on (q), we can thus only obtain the envelope of ( ), by carrying out a Legendre transformation once again We will now describe the heuristic arguments based on the construction of these multifractal formalisms We will divide them into four steps, highlighting the implicit assumptions that we make for each of them S TEP 1 The rst assumption, common to both approaches, is that the H lder exponent at each point x0 is given by the order of magnitude of the wavelet coef cients of f in C2 j With respect to (36), if they are decreasing as 2 Hj , a cone |k2 j x0 | we then have hf (x0 ) = H This assumption is veri ed if f does not have cusp type singularities ([MEY 98]), ie, the oscillation exponent is zero everywhere If we go from the data of (H), we have, as an assumption, 2 (H)j wavelet coef cients of size 2 Hj By using the supports of the corresponding wavelets to cover EH , we expect to obtain fH (H) = (H) Thus, we also obtain a rst form of multifractal formalism: the formalism said to be of large deviation , which simply af rms that: fH (H) = (H) Let us brie y justify this name as well as that of the large deviation spectrum that we sometimes give to function The theory of large deviations takes care of the calculation of the probabilities, which are so small that we can only correctly estimate them on logarithmic scales The basic example is as follows: if Xi are n 1 independent reduced centered Gaussian, and if we have Sn = n i=1 Xi , then we 1 2 obtain n log P(|Sn | ) /2 The analogy with (324) and (325) is striking since, in the common law of wavelet coef cients, the parts of very small probabilities, which we measure with the help of a logarithmic scale, are those that provide the relevant information ( ) for calculating the spectrum (see 4 for a more detailed study) The effective calculation of function ( ) is numerically delicate because its de nition leads to a double limit, which generally results into problems said to be of nite size In theory, we must go completely to the limit in j in (324) before taking the limit in in (325) Practically speaking, the two limits must effectively be taken together The problem is then to know how to take j suf ciently large according to , which creates signi cant numerical stability problems In any case, a
Scaling, Fractals and Wavelets
calculation of ( ) which is numerically reliable requires us to know the signal on a large number of scales, ie, with an excellent precision This is why we often prefer to work from averages, such as k |df (k, j)|q , ie, nally, based on the partition function for which the de nition leads to only one limit From now, this is the point of view that we shall adopt (however, let us note that the direct method introduced by Chhabra and Jensen in [CHH 89] is a method for calculating ( ) without going through a double limit, or through a Legendre transformation; we shall nd a mathematical discussion of this method adapted to the framework of the wavelets in [JAF 04a]) S TEP 2 We will estimate, for each H, the contribution of H lder singularities of exponent H at: |ef (k, j)|q
(328)
Each singularity of this type brings a contribution of C2 Hqj and there must be 2fH (H)j intervals of length 2 j to recover these singularities; the total contribution of the H lder singularities of exponent H at (328) is thus as follows: 2fH (H)j 2 Hqj = 2 (Hq fH (H))j (329)
This is a critical step of reasoning; it contains an inversion of limits that implicitly assumes that all H lder singularities have coef cients 2 Hj simultaneously from a certain scale J and that Hausdorff dimension can be estimated as if it were a box dimension It is notable that the multifractal formalism leads to the correct singularity spectrum in several situations where these two assumptions are not veri ed S TEP 3 The third step is an argument of the Laplace method type When j + , we note that, among the (329) terms, the one that brings the main contribution to (328) is that for which the exponent H carries out the minimum of Hq fH (H), from which comes the heuristic formula: (q) 1 = inf hq fH (H)
S TEP 4 If fH (H) is concave, then fH (H) and (q) + 1 are conjugate convex functions and each of them can be deduced from the other by a Legendre transformation Thus, if we de ne the Legendre spectrum with: fL (H) = inf Hq (q) + 1