Scaling, Fractals and Wavelets

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has a H lder exponent that is constant and equal to H; similarly, the sample paths of the Brownian motion verify with near certainty that hB (x) = 1 for all x In a more 2 general manner, a fractional Brownian motion of exponent H has at every point a H lder exponent equal to H Such functions are irregular everywhere Our objective is to study functions whose H lder exponent can jump from one point to another In such a situation, the numerical calculation of functions hf (x0 ) is completely unstable and of little signi cance We are rather trying to extract less precise information: whether or not the function hf takes a certain given value H and, if it does, what is the size of the sets of points where hf takes on this value Here we are faced with a new problem: what is the right notion of size in this context We will not be able to fully justify the answer to this question because it is a result of the study of numerous mathematical examples Let us just keep in mind that the term size does not signify Lebesgue measure because, in general, there exists a H lder exponent that is the most probable and that appears almost everywhere The other exponents thus appear on all zero sets and the Lebesgue measure does not make it possible to differentiate them Besides, the right notion of size cannot be the box dimension because these sets are usually dense In fact, we expect them to be fractal A traditional mathematical method to measure the size of such dense sets of zero measure is to calculate their Hausdorff dimension Let us recall its de nition DEFINITION 33 Let A be a subset of For > 0, let us note:

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d M = inf R i

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d i

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where R signi es a generic covering of A by intervals ]xi , xi + i [ of a length i The operator inf is thus taken on all these coverings For all d [0, 1], Hausdorff d-dimensional measure of A is: mes d (A) = lim M d

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This measure takes on a value of + or 0 except for, at the most, a value of d and the Hausdorff dimension of A is: dim(A) = sup d: lim M d = + = inf d: lim M d = 0

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0 0

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DEFINITION 34 Let f be a function and H 0 If H is a value taken by function hf (x), let us note by EH the set of points x where we have hf (x) = H Therefore, the singularity spectrum (or H lder spectrum) of the signal being studied is: fH (H) = dim(EH ) (we use the convention fH (H) = if H is not a H lder exponent of f )

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Wavelet Methods for Multifractal Analysis of Functions

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The concept of multifractal function is not precisely de ned (just like the concept of a fractal set) For us, a multifractal function is a function whose spectrum of singularities is non-trivial , ie unreduced to a point In the examples, fH (H) takes on positive values on an entire interval [Hmin , Hmax ] Its assessment thus requires a study of an in nity of fractal sets EH , hence the term multifractal 322 Wavelets and pointwise regularity For many reasons that we shall gradually discover, wavelet methods of analysis are a favorite tool for studying multifractal functions The rst reason is that we have a simple criteria that allows us to characterize the value of the H lder exponent by a decay condition of a given function s wavelet coef cients Let us begin by recapitulating certain points related to the wavelet analysis methods An orthonormal base of wavelets of L2 ( ) has a particularly simple algorithmic form: we start from a function (the mother wavelet) that is regular and well-localized; the technical assumptions are: i = 0, , N, m N, | (i) (x)| C(i, m) (1 + |x|)m

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for a relatively big N We can choose such functions as , such that, moreover, the translation-dilation of : j,k (x) = 2j/2 (2j x k), j, k Z

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form an orthonormal base of L2 ( ) (see [MEY 90]) (we will choose N to be bigger than the maximum regularity that we expect to nd in the signal analyzed; we can also take N = + and the wavelet will thus belong to the Schwartz class) We verify that the wavelet has a corresponding number of zero moments: i = 0, , N, (x)xi dx = 0

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Thus, every f L2 ( ) function can be written as: f (x) =

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