Let us de ne the upper semi-continuous envelope of a function f by: f ( ) = lim sup{f ( )/| |

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Then, the above results imply that: fd ( )

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lim fd ( )

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(137)

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Let us also mention the following result PROPOSITION 17 Let A and B be the two interval functions and C = max{A, B} Let fg ( , A), fg ( , B) and fg ( , C) denote the corresponding spectra Then, for any : fg ( , C) max{fg ( , A), fg ( , B)} (138)

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Fractal and Multifractal Analysis in Signal Processing

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lim PROPOSITION 18 If d is stable, then inequality (138) is true for fd , fd and lim sup fd

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A signi cant result concerning the inverse problem for the spectra serves as a conclusion for this section PROPOSITION 19 Let f be a USC function ranging in [0, 1] { } Then, there lim lim exists an interval function A whose fd sup or fd spectrum is exactly f as soon as d is -stable or d = Let us note that fd , when d is -stable, is not necessarily USC This shows once more that this spectrum is richer than the other ones (see [LEV 98b] for a study of the structural properties of fd with d = h) Weak multifractal formalism Let us now consider the numerical estimation of fg As in the case of fh , two approaches exist: either we resort to a multifractal formalism with fg as the Legendre transform of a simple function, or we analyze in detail the de nition of fg and deduces estimation methods from this In the rst case, the heuristic justi cation is the same as for fh and we expect that, under certain conditions, fg = Since we avoid an inversion of limits as compared to the case of fh , this formalism (sometimes called weak multifractal formalism, as opposed to the strong formalism which ensures the equality of and fh ) will be satis ed more often However, the necessary condition that fg be concave, associated once again with the lack of stability, still limits its applicability An important difference between the strong and weak formalisms is that, in the latter case, a precise and reasonably general criterion ensuring its validity is known We are referring to a version of the Ellis theorem, one of the fundamental results in the theory of large deviations, which is recalled below in a simpli ed form THEOREM 19 If (q) = limn n (q) exists as a limit (rather than a lower limit), and if it is differentiable, then fg = fl When fg is not concave, it cannot equal fl , but the following result holds THEOREM 110 If fg is equal to outside a compact set, then fl is its concave hull, ie: fl = (fg ) This theorem makes it possible to measure precisely the information which is lost when fg is replaced by fl See [LEV 98b] for related results

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Scaling, Fractals and Wavelets

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Continuous spectra It is possible to prove that the previous relation is still valid in the more subtle case of continuous spectra [LEV 04b, TRI 99] These continuous spectra constitute a generalization of fg that allow us to avoid choosing a partition As already mentioned, this choice is not neutral and different partitions will, in general, lead to different spectra To begin with, interval families are de ned: R = {u interval of [0, 1] such that |u| = } R ( ) = {u : |u| = , A(u) = } R ( ) = {u : |u| = , |A(u) | DEFINITION 16 (continuous large deviation spectra)

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c fg ( ) = lim lim sup 0 0

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1 | R ( )|

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log

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1 | R ( )|

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c fg ( ) = lim sup

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c Note that fg ( ) is de ned similarly to fg , except that all the intervals of a given length are considered where the variation of X is of the order of , rather than only dyadic intervals Since the number of these intervals may be in nite, we replace 1 Nn ( ) with a measure of the average length, ie | R ( )| Within this continuous framework, R ( ) will, in general, be non-empty for in nitely many values of and 0 not only for at most 2n values, as is the case for Nn ( ): it is thus possible to get rid of c and de ne the new spectrum fg

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Legendre transform of continuous spectra For any family of R intervals, R denotes the union of all the intervals of R A packing of R is a sub-family composed of disjoint intervals DEFINITION 17 (Legendre continuous spectrum) For any real q, let: H q (R) = sup

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