Formulación del problema

In this subsection, we first recall some definitions and some basic topological results obtained for Sν spaces. Furthermore, we expose the multifractal formalism based on these spaces. We refer the reader to [13] for details about the topology and [9, 10, 11, 14] for more results. The spaces Sν _{are defined as function spaces through conditions on}

the wavelet coefficients. Let us first introduce the spaces Sν _{as sequence spaces.}

Following [13], the wavelet profile of a sequence ~c ∈ Ω is the function ν~c defined by

ν~c(α) := lim ε→0+lim sup_j→+∞ log #Ej(1, α + ε)(~c ) log 2j , α ∈ R, where Ej(C, α)(~c ) :=λ ∈ Λj : |cλ| ≥ C2−αj

for j ∈ N0, C > 0 and α ∈ R. Remark that the function ν~c is non-decreasing and right-

continuous. Moreover, it takes values in {−∞} ∪ [0, 1]. If ~c ∈ Cα0_{, then ν}

~c(α) = −∞

Following this remark, an admissible profile is a non-decreasing right-continuous func- tion of a real variable, with values in {−∞} ∪ [0, 1] such that there exists αmin∈ R for

which ν(α) = −∞ for all α < αminand ν(α) ≥ 0 for all α ≥ αmin.

Definition 4.6.1. Given an admissible profile ν, a sequence ~c belongs to Sν _if

ν~c(α) ≤ ν(α), ∀α ∈ R .

Equivalently, ~c belongs to Sν _{if and only if for every α ∈ R, ε > 0 and C > 0, there} exists J ∈ N0 such that

#Ej(C, α)(~c ) ≤ 2(ν(α)+ε)j, ∀j ≥ J,

with the convention that 2−∞:= 0 .

Heuristically, a sequence ~c belongs to Sν if at each large scale j, the number of k such that |cj,k| ≥ 2−αj is of order smaller than 2ν(α)j. This space is a vector space.

Jaffard [90] proved that if αmin> 0, the definition of the Sν spaces is robust, hence

independent of the chosen wavelet basis. We will then consider them equivalently as sequence or function spaces (as for Besov spaces): we say that a function f belongs to Sν _{if its sequence of wavelet coefficients belongs to the sequence space S}ν_.

In order to define a complete metrizable topology on Sν, auxiliary spaces were in- troduced. For any α ∈ R and any β ∈ {−∞} ∪ [0, +∞), the space A(α, β) is defined by

A(α, β) :=~c ∈ Ω : ∃C, C0 ≥ 0 such that #Ej(C, α)(~c ) ≤ C02βj, ∀j ∈ N0 .

This space is endowed with the distance

δα,β(~c, ~c0) := infC + C0 : C, C0 ≥ 0 and #Ej(C, α)(~c − ~c0) ≤ C02βj, ∀j ∈ N0 .

Remark that if β = −∞, then (A(α, −∞), δα,−∞) is the topological normed space Cα.

If β ≥ 1, then A(α, β) = Ω. Moreover, in the case β > 1, the topology defined by the distance δα,β is equivalent to the topology of pointwise convergence.

The properties of auxiliary spaces are summarized in the following proposition.

Proposition 4.6.2. Let α ∈ R and β ∈ {−∞} ∪ [0, +∞).

1. The addition is continuous on (A(α, β), δα,β) but the scalar multiplication is not

continuous.

2. The space (A(α, β), δα,β) has a stronger topology than the pointwise topology and

every Cauchy sequence in (A(α, β), δα,β) is also a pointwise Cauchy sequence.

3. If β > 1, the topology defined by the distance δα,β is equivalent to the uniform

topology.

4. (a) If B is a bounded set of (A(α, β), δα,β), then there exists r > 0 such that

B ⊆ ~c ∈ Ω : #{λ ∈ Λj: |cλ| ≥ r 2−αj} ≤ r 2βj, ∀j ∈ N0

(b) Let r, r0 ≥ 0, α0_{≥ α and β}0_{≤ β. The set}

B =n~c ∈ Ω : #{λ ∈ Λj : |cλ| > r 2−α

0_j

} ≤ r02β0j_{, ∀j ∈ N}0

o

is a bounded set of (A(α, β), δα,β). Moreover, B is closed for the pointwise

convergence.

5. The space (A(α, β), δα,β) is a complete metric space.

The following proposition gives the connection between auxiliary spaces A(α, β) and the space Sν_.

Proposition 4.6.3. For any sequence (αn)_{n∈N dense in R and any sequence (ε}m)m∈N

of (0, +∞) which converges to 0, one has Sν₌ \

m∈N

\

n∈N

A(αn, ν(αn) + εm).

The topology of Sν is defined as the projective limit topology, i.e. the coarsest topology that makes each inclusion Sν ⊆ A(αn, ν(αn) + εm) continuous. This topology

is equivalent to the topology given by the distance δ = +∞ X m=1 +∞ X n=1 2−(m+n) δm,n 1 + δm,n , where δm,ndenotes the distance δαn,ν(αn)+εm. The topology of S

ν_{is independent of the}

sequences (αn)n∈N and (εm)m∈N chosen as above. Therefore, this distance is denoted δ,

independently of the sequences chosen. The next result concerns the compact subsets of (Sν_{, δ).}

Proposition 4.6.4. For m, n ∈ N, let C(m, n) and C0(m, n) be positive constants and let us define Km,n:= n ~c ∈ Ω : #{λ ∈ Λj : |cλ| > C(m, n) 2−αnj} ≤ C0(m, n) 2(ν(αn)+εm)j, ∀j ∈ N0 o and K := \ m∈N \ n∈N Km,n.

Every sequence of K which converges pointwise converges also in (Sν_{, δ) to an element}

of K. It follows that K is a compact of (Sν_{, δ).}

Let us now present some connections with Besov spaces. If we define the concave conjugate η of the admissible profile ν by

η(p) := inf

α≥αmin

(αp − ν(α) + 1) , p > 0, we get the following embedding of Sν _{spaces into Besov spaces.}

Proposition 4.6.5. [13] If (pn)_n∈N is a dense sequence of (0, +∞) and if (εm)m∈N is

a sequence of (0, +∞) which converges to 0, then Sν _⊆ \ p>0 \ ε>0 b η(p) p −ε p,∞ = \ n∈N \ m∈N b η(pn) pn −εm pn,∞

This result justifies the introduction of the Sν spaces: the spaces Bη do not contain more information about the multifractal spectra of their functions than their concave hull since for most of the functions in this intersection, the multifractal spectrum is given by a Legendre transform of η (see Theorem 4.5.2). By contrast, if ν is not concave, the space Sν _{gives an additional information and leads to estimation of spectra which are}

not concave, as presented below.

In order to state the multifractal formalism based on the Sν _{spaces and justify its}

validity, we assume now that αmin > 0 and we consider the space Sν as a function

space. The multifractal formalism based on the Sν _{spaces, called also the wavelet profile}

method, consists in the estimation of the spectrum of a function f by the formula df(h) = h sup h0_∈(0,h] νf(h0) h0 , ∀h ≤_α≥αinf min α νf(α) . (4.3)

Aubry and Jaffard [12] proved that for any Random Wavelet Series (see Chapter 5), formula (4.3) holds. They also established that if f is a uniformly H¨older function, then

df(h) ≤ h sup h0_∈(0,h] νf(h0) h0 , ∀h ≤_α≥αinf min α νf(α) .

Furthermore, an implementation of this formalism has been proposed by Kleyntssens et al. [99] where it is tested on several theoretical examples such as fractional Brownian motions, L´evy processes, sum of binomial cascades...

The results about the validity of this formalism are given by the following theorem.

Theorem 4.6.6. [11, 14] If ν is an admissible profile, we denote

νI(h) =        −∞ if h < αmin, h sup h0_∈(0,h] ν(h0) h0 if αmin≤ h ≤ hmax, 1 otherwise,

where hmax= infh≥αmin

h

ν(h). If αmin> 0, the set of functions f ∈ S

ν _{such that}

df(h) =



νI(h) if h ≤ hmax,

−∞ otherwise, and ν = νf is residual and prevalent in Sν.

It follows that for a generic function in Sν, formula (4.3) holds. Although it allows to estimate non-concave spectra, this formalism is still limited to the increasing part of spectra. Moreover, the conversion of ν into νI transforms the admissible profile into

another admissible profile with an additional property, called the increasing-visibility (see [106] and the definition below), and is therefore limited to spectra enjoying this property.

Definition 4.6.7. Take 0 ≤ a < b ≤ +∞. A function g : [a, b] 7→ [0, +∞) is with

increasing-visibility on [a, b] if g is continuous at a and if the function x 7→g(x)

x is increasing on (a, b].

In other words, a function g is with increasing-visibility if for all x ∈ (a, b], the segment [(0, 0); (x, g(x))] lies above the graph of g on (a, x].

0 1

0 hmax

Figure 4.4: Example of ν (---) and νI (—)