Marianne Clausel
email: clausel@univ-paris12.fr

Samuel Nicolay
email: s.nicolay@ulg.ac.be

Wavelets techniques for pointwise anti-Hölderian irregularity

Keywords: Pointwise Hölder regularity, Wavelets, Spectrum of singularities, Multifractal formalism. Mathematics Subject Classification 26A16, 42C40

In this paper, we introduce a notion of weak pointwise Hölder regularity, starting from the definition of the pointwise anti-Hölder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise Hölder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (denoted here strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.

Introduction

The concept of Hölderian regularity has been introduced to study nowhere differentiable functions (several examples are given in [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF][START_REF] Nicolay | About the pointwise Hölder exponents of some functions[END_REF]). An archetype of such functions is maybe the Weierstraß function

W H (x) = +∞ n=0 a -nH cos(2πa n x) (0 < H < 1)
exhaustively studied by Hardy in [START_REF] Hardy | Weierstrass's non differentiable function[END_REF]. He proved that for every a > 1, this function is nowhere differentiable. More precisely, the function W H satisfies the two following conditions on [0, 1] 2 ,

|W H (y) -W H (x)| ≤ C 1 |x -y| H and sup (u,v)∈[x,y] 2 |W H (u) -W H (v)| ≥ C 2 |x -y| H
for two constants C 1 , C 2 that do not depend on x or y. The first inequality gives the regularity of W H , which is said uniformly Hölder with exponent H on [0, 1]. The second one reflects the irregularity of the function; in particular, W H is nowhere differentiable. One says, following [START_REF] Tricot | Courbes et dimension fractale[END_REF], that W H is uniformly anti-Hölder with exponent H on (0, 1).

An increasing interest has been paid to functions f that are both uniform Hölder and uniformly anti-Hölder with exponent H, since these two properties ensure that the box-counting dimension of the graph of f is equal to 2 -H (see e.g. [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF]). Canonical Weierstraß functions, i.e. functions of the form

f (x) = n b -nα g(b n x)
where g is 1-periodic, 1 < b < ∞ and 0 < α < 1, have been extensively studied from the irregularity point of view by many authors (see [START_REF] Kaplan | The Lyapounov dimension of a nowhere differentiable torus[END_REF][START_REF] Mauldin | On the Hausdorff dimensions of some graphs[END_REF][START_REF] Przytycki | On the Hausdorff dimension of some fractal sets[END_REF][START_REF] Bousch | On oscillations of Weierstrass-type functions[END_REF][START_REF] Bousch | Caloric measure on domains bounded by Weierstrass type graphs[END_REF][START_REF] Heurteaux | Weierstrass function with random phases[END_REF][START_REF] Heurteaux | Weierstrass function in Zygmund's class[END_REF][START_REF] Demichel | Analysis of fractal sum of pulses[END_REF][START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF]). Other well-known examples of such functions are provided by sample paths of Gaussian fields, generalizing the fractional Brownian motion. Irregularity properties, such as law of the iterated logarithm, are established using fine results concerning the regularity of the local time of the studied fields (see [START_REF] Berman | Gaussian sample functions: uniform dimension and Hölder conditions nowhere[END_REF][START_REF] Geman | Occupation densities[END_REF][START_REF] Adler | The Geometry of Random Field[END_REF]). In particular, in this class of examples are included the so-called index-α Gaussian fields studied in [START_REF] Adler | The Geometry of Random Field[END_REF], or more generally non locally deterministic Gaussian fields (see e.g. [START_REF] Berman | Local Nondeterminism and Local Times of Gaussian Processes[END_REF][START_REF] Pitt | Local times for Gaussian vector fields[END_REF]) and strongly non locally deterministic Gaussian fields (see [START_REF] Xiao | Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields[END_REF][START_REF] Xiao | Properties of Local Nondeterminism of Gaussian and Stable Random Fields and Their Applications[END_REF][START_REF] Xiao | Sample Path Properties of Anisotropic Gaussian Random Fields[END_REF]).

In this paper, we focus on the pointwise anti-Hölderian irregularity, which is the pointwise counterpart of the concept of uniform anti-Hölderian irregularity. Our main goal is to answer quite natural questions: Can we overstep the usual framework of functions both uniform Hölder and uniformly anti-Hölder? More precisely, can we give some explicit examples of functions for which the pointwise anti-Hölderian behavior is different from point to point? What are the main characteristics of such a behavior?

In the case of the usual pointwise regularity, multifractal functions provide examples of functions for which the Hölder exponent vary from point to point. So, we naturally tend to be interested in defining some multifractal functions for this notion of pointwise anti-Hölderian irregularity. We also need suitable tools to describe the multifractal behavior of such functions. It raises the problem of the related multifractal formalism. Indeed, let us recall that, in general settings, it is not possible to estimate the regularity index (which will be defined hereafter) of a function at a given point. The relevant information is then the "size" of the sets of points where the regularity is the same. This "size" is mathematically formalized as the Hausdorff dimension. The function that associates the dimension of the set of points sharing the same regularity index with this index is referred to as the spectrum of singularities. The goal of any multifractal formalism is to provide a method which allows to estimate this spectrum of singularities from numerically computable quantities derived from the signal. The same problem arises when dealing with pointwise anti-Hölderian irregularity.

Section 2 is devoted to the definitions related to the Hölder regularity. In Section 3 we investigate the structure of the irregularity exponent and define, by means of wavelet series, functions with prescribed irregularity exponent. In Section 4, we recall already known results about the multifractal formalism for the pointwise anti-Hölderian irregularity. Section 5 is devoted to the question of the validity of this multifractal formalism: Using multifractal measures, we define a class of wavelet series for which the multifractal formalism holds. In the last section, we compare the two concepts of multifractal functions: The usual one and this new one related to anti-Hölderianity. We show that the two notions are clearly disconnected. Indeed, we exhibit an example of Davenport series which is multifractal for the usual pointwise regularity but monofractal for the pointwise irregularity.

Pointwise Hölderian regularity

We start by giving the definitions of the pointwise Hölderian regularity and anti-Hölderian irregularity. The concept of anti-Hölderian functions with exponent H has been introduced by C.Tricot in [START_REF] Tricot | Courbes et dimension fractale[END_REF]; he formalized a notion already used for investigating Weierstraß-type functions or sample paths properties of locally non deterministic Gaussian fields. Anti-Hölderian functions with exponent H were only defined in the case H ∈ (0, 1). A consistent definition is given here for H larger than 1. Since the anti-Hölderian condition is stronger than just negating the Hölderian condition, a weaker Hölderian regularity is obtained by negating the anti-Hölderian condition. Finally, discrete wavelet transform and multiresolution analysis are particularly efficient tools to study the Hölderian regularity of a function (see e.g. [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF]). The main results binding the regularity of a function and its wavelet coefficients are briefly reviewed at the end of this section.

Let us point out that the anti-Hölderian irregularity condition has also been considered in the measure setting (see e.g. [START_REF] Brown | On the multifractal analysis of measures[END_REF]); a review of this measure-based irregularity framework is presented in Section 5.1.

Weak and strong pointwise Hölderian regularity

We recall first the definition of the Hölderian regularity; this definition naturally leads to a notion of Hölderian irregularity. One will talk about Hölderian and anti-Hölderian functions. Finally, a weaker definition of pointwise smoothness is obtained by negating the condition related to the anti-Hölderian functions.

Definition 1 Let f : R d → R d ′ be a locally bounded function, let x 0 ∈ R d and α ≥ 0; f ∈ C α (x 0 ) if there exist C, R > 0 and a polynomial P of degree less than α such that f (x) -P (x) L ∞ (B(x0,r)) ≤ Cr α , ∀r ≤ R. (1)
Such a function is said Hölderian of exponent α at x 0 . The lower Hölder expo-

nent of f at x 0 is h f (x 0 ) = sup{α : f ∈ C α (x 0 )}. A function f is uniformly Hölderian of exponent α (f ∈ C α (R d )) if there exists C > 0 such that (1) is satisfied for any x 0 ∈ R d and R = ∞; f is uniformly Hölderian if there exists ε > 0 such that f ∈ C ε (R d ).
Recall that the lower Hölder exponent is simply denoted Hölder exponent in the literature. Since we are interested in introducing another concept of pointwise Hölderian regularity, the accustomed notation h is replaced here by h

The irregularity of a function can be studied through the notion of anti-Hölderianity. Recall that the finite differences of arbitrary order are defined as follows,

∆ 1 h f (x) = f (x + h) -f (x), ∆ n+1 h f (x) = ∆ n h f (x + h) -∆ n h f (x).
We use the following notation,

B h (x 0 , r) = {x : [x, x + ([α] + 1)h] ⊂ B(x 0 , r)}. Since condition (1) is equivalent to sup |h|≤r ∆ [α]+1 h f L ∞ (B h (x0,r)) ≤ Cr α , ∀r ≤ R (2)
(see e.g. [START_REF] Devore | Maximal functions measuring smoothness[END_REF][START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF][START_REF] Krantz | Lipschitz spaces, smoothness of functions, and approximation theory[END_REF]), the next definition is intuitive.

Definition 2 Let f : R d → R d ′ be a locally bounded function, let x 0 ∈ R d and α ≥ 0; f ∈ I α (x 0 ) if there exist C, R > 0 such that sup |h|≤r ∆ [α]+1 h f L ∞ (B h (x0,r)) ≥ Cr α , ∀r ≤ R. (3)
Such a function is said anti-Hölderian of exponent α at x 0 . Let us notice that the Whitney theorem asserts that I α1 (x 0 ) ⊂ I α2 (x 0 ) if α 1 ≤ α 2 (more precisely, it is a direct consequence of Proposition 1 of [START_REF] Clausel | Some prevalent results about strongly monoHölder functions[END_REF]). The upper Hölder exponent (or irregularity exponent) of f at x 0 is

h f (x 0 ) = inf{α : f ∈ I α (x 0 )}.
We will say that f is strongly Hölderian of exponent α at

x 0 (f ∈ C α s (x 0 )) if f ∈ C α (x 0 ) ∩ I α (x 0 ).
It follows from the definitions that if a function f is anti-Hölderian with exponent α, it cannot be Hölderian with exponent β if β > α. We thus have the following relation between the lower and upper exponents of f :

h f ≤ h f .
The statement (3) is stronger than just negating the Hölderian regularity since such a negation only yields the existence, for any C > 0, of a subsequence (r n ) n (depending on C) for which

sup |h|≤rn ∆ [α]+1 h f L ∞ (B h (x0,rn)) ≥ Cr α n .
We are naturally led to the following definition.

Definition 3 Let f : R d → R d ′ be a locally bounded function, let x 0 ∈ R d and α ≥ 0; f ∈ C α w (x 0 ) if f / ∈ I α (x 0 ), i.e.
for any C > 0 there exists a decreasing sequence (r n ) n such that

sup |h|≤rn ∆ [α]+1 h f L ∞ (B h (x0,rn)) ≤ Cr α n , ∀n ∈ N. (4)
Such a function is said weakly Hölderian of exponent α at x 0 .

Roughly speaking, a function is weakly Hölderian of exponent α at x 0 if for any C > 0, one can bound the oscillation of f over B(x 0 , r n ) by Cr α n for a remarkable decreasing subsequence (r n ) n of scales, whereas for an Hölderian function, the oscillation of f over B(x 0 , r) has to be bounded at each scale r > 0 by Cr α , for some C > 0.

Hölderian regularity and wavelet coefficients

Here, we review the wavelet criterion for strong Hölderian regularity and irregularity.

Let us briefly recall some definitions and notations (for more precisions, see e.g. [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF][START_REF] Meyer | Ondelettes et opérateurs[END_REF][START_REF] Mallat | A wavelet tour of signal processing[END_REF]). Under some general assumptions, there exists a function φ and 2 d -1 functions (ψ

(i) ) 1≤i<2 d , called wavelets, such that {φ(x -k)} k∈Z d ∪ {ψ (i) (2 j x -k) : 1 ≤ i < 2 d , k ∈ Z d , j ∈ Z} form an orthogonal basis of L 2 (R d ). Any function f ∈ L 2 (R d ) can be decomposed as follows, f (x) = k∈Z d C k φ(x -k) + +∞ j=1 k∈Z d 1≤i<2 d c (i) j,k ψ (i) (2 j x -k),
where c

(i) j,k = 2 dj R d f (x)ψ (i) (2 j x -k) dx,
and

C k = R d f (x)φ(x -k) dx.
Let us remark that we do not choose the L 2 normalization for the wavelets, but rather an L ∞ normalization, which is better fitted to the study of the Hölderian regularity. Hereafter, the wavelets are always supposed to belong to C r with r > α and the functions {∂ s φ} |s|≤r , {∂ s ψ (i) } |s|≤r are assumed to have fast decay.

A dyadic cube of scale j is a cube of the form

λ = [ k 1 2 j , k 1 + 1 2 j ) × • • • × [ k d 2 j , k d + 1 2 j ),
where k = (k 1 , . . . , k d ) ∈ Z d . In the sequel, we denote |λ| the scale of a dyadic cube |λ|. From now on, wavelets and wavelet coefficients will be indexed with dyadic cubes λ. Since i takes 2 d -1 values, we can assume that it takes values in {0, 1} d -(0, . . . , 0); we will use the following notations:

• λ = λ(i, j, k) = k 2 j + i 2 j+1 + [0, 1 2 j+1 ) d , • c λ = c (i) j,k , • ψ λ = ψ (i) (2 j x -k),
• e λ = k/2 j .

The pointwise Hölderian regularity of a function is closely related to the decay rate of its wavelet leaders.

Definition 4

The wavelet leaders are defined by

d λ = sup λ ′ ⊂λ |c λ |.
Two dyadic cubes λ and λ ′ are adjacent if they are at the same scale and if dist(λ, λ ′ ) = 0. We denote by 3λ the set of 3 d dyadic cubes adjacent to λ and by λ j (x 0 ) the dyadic cube of side 2 -j containing x 0 . Then

d j (x 0 ) = sup λ⊂3λj (x0) d λ .
The following theorem (Theorem 1 of [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF]) allows to "nearly" characterize the Hölderian regularity by a decay condition on d j as j goes to infinity.

Theorem 1 Let α > 0; if f ∈ C α (x 0 ), then there exists C > 0 such that

d j (x 0 ) ≤ C2 -αj , ∀j ≥ 0. (5)
Conversely, if [START_REF] Aouidi | Multifractal formalism for quasi self-similar functions[END_REF] holds and if f is uniformly Hölder, then there exist C, R > 0 and a polynomial P of degree less than α such that

f (x) -P (x) L ∞ (B(x0,r)) ≤ Cr α log 1 r , ∀r ≤ R.
To give necessary and sufficient conditions concerning the irregularity, we suppose that the wavelets are compactly supported and belong to C [α]+1 (R d ); such wavelets are constructed in [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF]. The result relies on the following lemma.

Lemma 1 Let f ∈ L ∞ loc (R d )
; the two following assertions are then equivalent: 1. there exists some β > 1 such that, for any C > 0, there exists a non decreasing sequence of integers (j n ) such that

sup j≤jn {2 j([α]+1) d j (x 0 )} ≤ C 2 jn([α]+1-α) j β n , (6)
2. there exists some β > 1 such that, for any C > 0, there exists a strictly increasing sequence of integers (j n ) such that, for any λ,

|c λ | ≤ C(θ(2 -|λ| ) + θ(|x 0 -e λ |)), ( 7
)
where θ is a non decreasing function such that, if j ∈ {j n , . . . , j n+1 -1} for some n ∈ N,

θ(2 -j ) = inf( 2 jn+1([α]+1-α) 2 -j([α]+1) j β n+1 , 2 -jnα j β n ).
Proof. Let us suppose that Property (7) holds. For any j, we have θ(2

• 2 -j ) ≤ 2 [α]+1 θ(2 -j ). Moreover, if λ ′ ⊂ 3λ j (x 0 ), one has |x 0 -e λ ′ | ≤ 4d2 -|λ ′ | = 2 -(|λ ′ |-2-log 2 d) . Therefore, 2 j([α]+1) d j (x 0 ) = 2 j([α]+1) sup λ ′ ⊂3λj (x0) |c λ ′ | ≤ C2 j([α]+1) sup λ ′ ⊂3λj (x0) θ(2 -|λ ′ | )(1 + 2 (2+log 2 d)([α]+1) ) ≤ C2 j([α]+1) 2 [α]+1 (1 + 2 (2+log 2 d)([α]+1) )θ(2 -j ),
Since 2 ([α]+1)j θ(2 -j ) ≤ 2 jn([α]+1-α) /j β n for any j ≤ j n , inequality (6) is satisfied. Conversely, if inequality (6) holds, we have d jn (x 0 ) ≤ C2 -jnα /j β n and therefore, since the sequence (d j (x 0 )) is non increasing, d j (x 0 ) ≤ C2 -jnα /j β n for any n ∈ N and any j ≥ j n . Moreover, we have

d j (x 0 ) ≤ 2 -j([α]+1) sup j ′ ≤jn+1 {2 j ′ ([α]+1) d j ′ (x 0 )} ≤ C 2 jn+1([α]+1-α) j β n 2 -j([α]+1)
for any n ∈ N, and any j ≤ j n+1 . These relations imply that the inequality d j (x 0 ) ≤ Cθ(2 -j ) is valid for any n ∈ N and any j ∈ {j n • • • , j n+1 -1}. Let us now fix λ ′ and set j = sup{m : λ ′ ⊂ 3λ m (x 0 )}. By definition of j, one has

|c λ ′ | ≤ d j (x 0 ) ≤ Cθ(2 -j ). If j = |λ ′ | or if j = |λ ′ | + 1
, using the fact that θ is non decreasing, one gets

|c λ ′ | ≤ Cθ(2 • 2 -j ) ≤ C2 [α]+1 θ(2 -|λ ′ | ) ≤ C2 [α]+1 (θ(2 -|λ ′ | ) + θ(|x 0 -e λ ′ |)). If j < |λ ′ |, θ(2 -j ) is larger than θ(2 -|λ ′ | ). Nevertheless, one has 2 -j-1 ≤ |x 0 -e λ ′ | and thus |c λ ′ | ≤ d j (x 0 ) ≤ Cθ(2 -j ) ≤ Cθ(|x 0 -e λ ′ |) ≤ C2 [α]+1 (θ(2 -|λ ′ | ) + θ(|x 0 -e λ ′ |)).
In any case, property [START_REF]Moments, continuité et analyse multifractale des martingales de Mandelbrot[END_REF] is recovered. This Lemma is indeed a two-microlocal characterization of Property [START_REF] Arneodo | The thermodynamics of fractals revisited with wavelets[END_REF]. We need it to prove the following wavelet characterization of pointwise irregularity:

Theorem 2 Let α > 0 and f ∈ L ∞ loc (R d ); 1. if there exists C > 0 such that d j (x 0 ) ≥ C2 -jα , (8)
for any j ≥ 0, then f ∈ I α (x 0 ), 2. conversely, suppose that f is uniformly Hölder; if f ∈ I α (x 0 ) then, for any β > 1, there exists C > 0 such that

sup j ′ ≤j 2 j ′ ([α]+1) d j ′ (x 0 ) ≥ C 2 j([α]+1-α) j β , (9)
for any j ≥ 0.

Proof. The first statement is a direct consequence of Theorem 4 of [START_REF] Clausel | Some prevalent results about strongly monoHölder functions[END_REF]. Let us prove the second statement by contrapositive. Assume that Property (9) does not hold, which is equivalent to assume that Property ( 6) is satisfied. We use Lemma 1 to prove that inequality ( 4) is satisfied for

r n = C2 -jn . Let x ∈ R d such that [x, x + ([α] + 1)h] ⊂ B(x 0 , 2 -jn ). We have ∆ M h f (x) = k C k ∆ M h φ k (x) + λ c λ ∆ M h ψ λ (x) = j≥0 ∆ M h f j , where f 0 (x) = k C k φ k (x) and f j (x) = i,k c (i) j,k ψ (i) (2 j x -k) if j ≥ 1.
Since f is assumed to be uniformly Hölder, there exists some ε > 0 such that max(sup

k |C k |, sup j (2 jε sup λ |c λ |)) < ∞. Let α ′ > α and define J n = [ α ′ j n ε ] + 1. The proof of Theorem 1 of [31] yields the following inequality j≥Jn ∆ [α]+1 h f j (x) ≤ C2 -Jnε0 ≤ C2 -jnα ′
for n sufficiently large.

Since the wavelets are assumed to be compactly supported, there exists ℓ 0 such that supp(ψ (i) ) ⊂ (-2 ℓ0 , 2 ℓ0 ). Let us give an upper bound of

jn+ℓ0 j=0 ∆ [α]+1 h f j (x). If |x 0 -e λ | ≥ 2 -j+ℓ0+1 then |x -e λ | ≥ 2 -j+ℓ0 and ψ λ (x) = 0. Therefore jn+ℓ0 j=0 ∆ [α]+1 h f j (x) = jn+ℓ0 j=0 λ, |x0-e λ |≤2 -j+ℓ 0 +1 c λ ∆ [α]+1 h ψ λ (x).
The regularity of the wavelets implies that

jn+ℓ0 j=0 λ, |x0-µ λ |≤2 -j+ℓ 0 +1 c λ ψ λ (x) belongs to C [α]+1 (R d ). Hence | jn+ℓ0 j=0 λ, |x0-e λ |≤2 -j+ℓ 0 +1 c λ ∆ [α]+1 h ψ λ (x)| ≤ |h| [α]+1 jn+ℓ0 j=0 λ, |x0-e λ |≤2 -j+ℓ 0 +1 c λ ψ λ (x) C [α]+1 (R d )
We now use Lemma 1 and the wavelet characterization of the spaces

C [α]+1 (R d ) to deduce that λ, |x0-e λ |≤2 -j+ℓ 0 +1 c λ ψ λ (x) C [α]+1 (R d ) ≤ sup λ, |x0-e λ |≤2 -j+ℓ 0 +1 (2 j[α]+1 |c λ |) ≤ C2 j[α]+1 (1 + 2 (ℓ0+1)([α]+1) )θ(2 -j )).
This leads to the following upper bound,

| jn+ℓ0 j=0 ∆ [α]+1 h f j (x)| ≤ |h| [α]+1 C(1 + 2 (ℓ0+1)([α]+1) ) jn j=0 2 j([α]+1) θ(2 -j ) = |h| [α]+1 C(1 + 2 (ℓ0+1)([α]+1) )2 jn([α]+1-α) ≤ C(1 + 2 (ℓ0+1)([α]+1) )2 -jnα .
Let us now give an upper bound of Jn j=jn+ℓ0 ∆

[α]+1 h f j (x). In that case, let us remark that if |x 0e λ | ≥ 2 -jn+1 then |xe λ | ≥ 2 -j+ℓ0+1 and ψ λ (x) = 0. We have

| Jn j=jn+ℓ0 ∆ [α]+1 h f j (x)| = | j=jn+ℓ0+1 λ, |x0-e λ |≤2 -jn+1 c λ ∆ [α]+1 h ψ λ (x)| ≤ ([α] + 1) Jn j=jn+ℓ0+1 λ, |x0-e λ |≤2 -jn+1 |c λ | sup j sup x∈R d |λ|=2 -j |ψ λ (x)| ≤ C([α] + 1)   sup j sup x∈R d |λ|=2 -j |ψ λ (x)|   Jn j=jn+ℓ0+1 λ, |x0-e λ |≤2 -jn +1 θ(2 -jn ) ≤ C([α] + 1)   sup j sup x∈R d |λ|=2 -j |ψ λ (x)|   J n 2 -jnα j β n ≤ C([α] + 1)   sup j sup x∈R d |λ|=2 -j |ψ λ (x)|   2 -jnα
for n sufficiently large. Gathering these relations, we obtain f ∈ C α w (x 0 ).

Note that we do not have a wavelet characterization of the property h f (x 0 ) = α. Indeed, it is proved in [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF] that, even up to a logarithmic correction, neither condition ( 8) is necessary, nor condition ( 9) is sufficient. Nevertheless, one can characterize the stronger property h f (x 0 ) = h f (x 0 ) = α using wavelets. Indeed, Theorems 1 and 2 lead to the following corollary that we will use in the sequel.

Corollary 1 Let α > 0 and suppose that f is uniformly Hölder. We have

h f (x 0 ) = h f (x 0 ) = α if and only if lim j→∞ log d j (x 0 ) -j log 2 = α.
Proof. The first point of Theorem 2 implies that if lim j log d j (x 0 )/j log 2 = α, we have

h f (x 0 ) = h f (x 0 ) = α.
Let us prove the converse result. Assume that for any ε > 0, we have f ∈ C α-ε (x 0 ) ∩ I α+ε (x 0 ). For any β > 1, the preceding Theorems imply the existence of a constant C > 0 such that

d j (x 0 ) ≤ C2 -j(α-ε) and sup j ′ ≤j 2 j ′ ([α]+1) d j ′ (x 0 ) ≥ 2 j([α]+1-α-ε) j -β /C, for any j ≥ 0. Let a > ε + 2ε/([α] + 1 -α + ε) and b = (β + ε)/([α] + 1 -α + ε). The previous relations lead to sup j ′ ≤j(1-a)-b log 2 j 2 j ′ ([α]+1) d j ′ (x 0 ) < C 2 j([α]+1-α-ε) -jε j β < C -1 2 j([α]+1-α-ε) j β and sup j(1-a)-b log 2 j≤j ′ ≤j 2 j ′ ([α]+1) d j ′ (x 0 ) ≥ C -1 2 j([α]+1-α-ε) j β ≥ C -1 2 j([α]+1-α-ε) j β ,
for any j ≥ 2 log 2 (C)/ε . Since the sequence (d j (x 0 )) is non increasing, we have

2 j([α]+1) d j(1-a)-b log 2 j (x 0 ) ≥ C -1 2 j([α]+1-α-ε) j β ,
for any j ≥ 2 log 2 (C)/ε. By setting ℓ = j(1a)b log 2 j, the preceding relation can be rewritten

2 1 1-a (ℓ+2b log 2 ℓ)([α]+1) d ℓ (x 0 ) ≥ C -1 2 j([α]+1-α-ε) j β ≥ C -1 2 ℓ 1-a ([α]+1-α-ε) ( ℓ 1-a ) β . for any ℓ ≥ 2 log 2 (C)(1 -a)/ε. Using the relation d j (x 0 ) ≤ C2 -j(α-ε) , we then obtain α -ε ≤ lim inf j→∞ log d j (x 0 ) -j log 2 ≤ lim sup j→∞ log d j (x 0 ) -j log 2 < α + ε.
Since this inequality holds for any ε > 0, the required result follows.

Construction of functions with prescribed lower and upper Hölder exponents

In this Section, we investigate in detail the structure of the irregularity exponent of a continuous function. In Section 3.1, we first prove, considering a Weierstrass-type function, that it is possible to construct a continuous function with prescribed pointwise Hölder exponent H provided that H satisfies "good properties". In Section 3.2 we focus on describing all the functions which are both the classical Hölder exponent and irregularity exponent of a continuous function. Finally, we study the case where classical pointwise Hölder exponent and irregularity exponent may differ. In this special case, we give a sufficient condition and a necessary condition for a couple of functions (H, H) to be respectively the pointwise Hölder exponent and the irregularity exponent of a continuous function.

A generic Weierstraß function with prescribed Hölder exponents

In the same spirit as in [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF], we consider the Weierstraß-type function

W (t) = +∞ j=0 λ -jH(t) sin(2πλ j t). (10)
and study its pointwise regularity.

Proposition 1 Let H be a β-Hölderian function from [0, 1] to [a, b] ⊂ (0, 1], satisfying sup t∈[0,1] |H(t)| < β.
If W is a function of the form [START_REF] Berman | Gaussian sample functions: uniform dimension and Hölder conditions nowhere[END_REF], where λ is an integer larger than 1, then

W ∈ C H(t) s (t) = C H(t) (t) ∩ I H(t) (t), ∀t ∈ [0, 1].
The proof of this proposition relies on the two following lemma, analogous to Lemma 14 and Proposition 15 of [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF].

Lemma 2 Let λ > 1 and (f j ) j∈N a sequence of bounded and Lipschitz functions on R for which there exists C > 0 such that

f j ∞ + f ′ j ∞ ≤ C. The function f (t) = +∞ j=0 λ -jH(t) f j (λ j t) belongs to C H(t) (t), for any t ∈ [0, 1]. Proof. Let t ∈ [0, 1]. For any j, |f j (t) -f j (s)| is bounded by C|s -t| or C. Let j 0 (t) = [-log |s -t|/(H(t) log λ)]. We have |f (t) -f (s)| ≤ C j0-1 j=0 |λ j t -λ j s|λ -jH(t) + C j≥j0 λ -jH(t) +C +∞ j=0 |λ -jH(t) -λ -jH(s) | ≤ C|t -s|Cλ j0(1-H(t)) + Cλ -j0H(t) +C| log λ||t -s| β +∞ j=0 jλ -ja ≤ C|t -s| H(t)
using the mean value theorem, β-Hölderianity of H and the fact that

H([0, 1]) ⊂ (0, 1]. Lemma 3 Let (f j ) j∈N be a sequence of 1-periodic C-Lipschitz functions from R to R and f (t) = ∞ j=0 λ -jH(t) f j (λ j t),
where λ is an integer larger than 1. Assume that there exists ℓ ∈ {-λ-1, • • • , λ-1} and an integer J such that

D = inf j≥J |f j ( ℓ λ ) -f j (0)| > 0. If Cℓ λ(λ -1) ≤ D then f ∈ I H(t) (t), ∀t ∈ [0, 1].
Proof. Since

f (t) -f (s) = ∞ j=0 λ -jH(t) (f j (λ j t) -f j (λ j s)) + ∞ j=0 (λ -jH(t) -λ -jH(s) )f j (s)
Proposition 15 of [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF] yields that for some

C 0 > 0, | ∞ j=0 λ -jH(t) (f j (λ j t) -f j (λ j s))| ≥ C 0 |t -s| H(t) . Moreover | ∞ j=0 (λ -jH(t) -λ -jH(s) )f j (s)| ≤ |t -s| β

Wavelet series-defined functions with similar lower and upper Hölder exponents at any point

The aim of this section is to prove a result analogous to Theorem 1 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF]. Here we extend the results stated in [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF] since we give a characterization of functions which are both the lower and the upper exponent of a continuous function and thus satisfy a stronger property than in Theorem 1 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF].

Theorem 3 Let f a continuous nowhere differentiable function defined on [0, 1] with similar lower and upper Hölder exponent at any point. There exists a sequence of continuous functions

(H j ) j∈N from [0, 1] to [0, 1] such that h(t) = h(t) = lim j→∞ H j (t) ∀t. Conversely, if H is a function from [0, 1] to [0, 1] such that H(t) = lim j→∞ H j (t),
where the (H j ) j∈N are continuous functions, then there exists a continuous function f defined on [0, 1] such that

h(t) = h(t) = H(t), ∀t.
The first part of Theorem 3 is straightforward. If one sets

ω x (t) = sup |h|≤t |f (x + h) -f (x)|,
the sequence (H j ) j∈N defined by

H j (t) = log(ω t (2 -j ) + 2 -2j ) -j log 2
satisfies the required conditions, since

h(t) = h(t) = lim j→∞ log ω t (2 -j ) -j log 2 .
Let us prove the converse assertion by means of wavelet series. We will need the following wavelet criterium for the pointwise regularity (see [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF]):

Proposition 2 Let α > 0 and assume that there exists C > 0 such that for any j ∈ N, sup

k |c j,k | ≤ C2 -j log j , (11)
and

d j (t) ≤ C2 -jα .
Then for any ε > 0, f ∈ C α-ε (t).

Remark 1 This Proposition is a reformulation of Proposition 4 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF] in terms of wavelet coefficients. Let us point out that in [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF], f is expanded in the Schauder basis. Then for any J (Lemma 1 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF]),

j≤J k c j,k Λ j,k
is a continuous piecewise affine function coinciding with f on dyadic numbers.

In the case of wavelet basis, this property does not hold. In order to prove Proposition 2, we thus need to assume [START_REF] Berman | Local Nondeterminism and Local Times of Gaussian Processes[END_REF] and use different arguments.

Proof. Since sup k |c j,k | ≤ C2 -j log j , the wavelet series converge uniformly on any compact. Let (f j ) j≥-1 be defined as follows,

f -1 (x) = k C k ϕ(x -k), f j (x) = k c j,k ψ j,k (x) (∀j ≥ 0)
As in [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF], if |β| ≤ [α], the series j ∂ β f j converges absolutely. Now, for any j ≥ -1, let us define

P j,t (x) = β≤[α] (x -t) β β! ∂f j (t).
If j 0 is the number such that

2 -j0-1 ≤ |x -t| ≤ 2 -j0 ,
and j 1 satisfies 2 -j 1 log j 1 ≤ 2 -αj0 , then as proved in [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF],

j≤j0 |f j (x) -P j,t (x)| ≤ Cj 0 |x -t| α and j≥j0 |P j,t (x)| ≤ C|x -t| α .
Moreover,

j1 j=j0 |f j (x)| ≤ Cj 1 |x -t| α .
Since for any ε and j 0 sufficiently large,

j 1 ≤ C ε 2 εj0 , one has j1 j=j0 |f j (x)| ≤ C ε |x -t| α-ε . Since j≥j1 |f j (x)| ≤ C j≥j1 2 -j log j ≤ C log 2 (j 1 )2 -j 1 log j 1 ≤ |x -t| α-ε ,
the proposition follows.

We will also use a slightly modified version of Lemma 2 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF]:

Proposition 3 Let H a function from [0, 1] to [0, 1] such that H(t) = lim j H j (t),
where (H j ) j∈N is a sequence of continuous functions. There exists a sequence (P j ) j∈N of polynomials such that

H(t) = lim j→∞ P j (t), ∀t ∈ [0, 1], P ′ j ∞ ≤ j, ∀j ∈ N. (12)
We can now define a wavelet series with the desired properties:

Proposition 4 Let H be a function from [0, 1] to [0, 1] such that H(t) = lim j H j (t)
, where the (H j ) j∈N is a sequence of continuous functions. Let (P j ) j∈N be a sequence of polynomials satisfying the relations [START_REF] Bousch | On oscillations of Weierstrass-type functions[END_REF]. For any

(j, k) ∈ N × {0, • • • , 2 j -1}, set H j,k = max( 1 log j , P j ( k 2 j )).
The function f defined as

f (x) = j∈N 2 j -1 k=0 2 -jH j,k ψ j,k (x). ( 13
)
satisfies the following relations,

h(t) = h(t) = H(t), ∀t ∈ [0, 1].
Let us look at a particular case.

Remark 2 If H is a continuous function, the wavelet series

j∈N 2 j -1 k=0 2 -jH j,k ψ j,k (x),
with H j,k = max(1/ log j, H(2 -j k)) has H both as lower and upper Hölder exponents.

Proof.

If λ ′ = λ ′ (j ′ , k ′ ) ⊂ 3λ j (t), then | k ′ 2 j ′ -t| ≤ C 1 2 -j .
Since the sequence (P j ) j∈N satisfies equalities [START_REF] Bousch | On oscillations of Weierstrass-type functions[END_REF], for any ε > 0, there exists an integer j 0 such that for any j ≥ j 0 and λ ′ ⊂ 3λ j (t),

|H j ′ ,k ′ -H(t)| ≤ |P ′ j (t) -P ′ j ( k ′ 2 j ′ )| + |H(t) -P ′ j (t)| ≤ j2 -j + |H(t) -P ′ j (t)| ≤ ε.
Then for any j ≥ j 0 , max(2

-j ′ log j ′ , 2 -j ′ ε 2 -j ′ H(t) ) ≤ |c ′ λ | ≤ max(2 -j ′ log j ′ , 2 j ′ ε 2 -j ′ H(t) ).
We deduce that lim

j log d j (t) -j log 2 = H(t), ∀t ∈ [0, 1]
and therefore

h(t) = h(t) = H(t), ∀t ∈ [0, 1].

Wavelet series-defined functions with different lower and upper Hölder exponents

Our main goal is to prove the following Theorem. where the (H j ) j∈N is a sequence of continuous functions. There exists a uniform Hölder function f from [0, 1] to R such that

h(t) = H(t) ≤ H(t) = h(t), ∀t ∈ [0, 1].

Remark 3

The second assertion of Theorem 4 is much weaker than the corresponding one of Theorem 3. In order to ensure the existence of a function f with prescribed lower and upper exponents at any point, we need stronger assumptions on H and H: indeed we assume that these functions take values in [a, b] ⊂ (0, 1) (and not in (0, 1)).

Remark 4 Let H and H be two continuous functions from

[0, 1] to [a, b] ⊂ (0, 1) satisfying ∀t ∈ [0, 1], H(t) ≤ H(t).
Define the sequence (H j ) j∈N as

H 2j = H, H 2j+1 = H, ∀j ∈ N.
Theorem 4 provides the existence of a function f with lower and upper exponent H and H at any point.

The proof of the direct part of Theorem 4 is exactly similar to this of Theorem 3 and is left to the reader. In order to prove the converse assertion we first need the following lemma.

Lemma 4 Let (H, H) a couple of functions defined from [0, 1] to [a, b] ⊂ (0, 1) such that H(t) = lim inf j→∞ H j (t), H(t) = lim sup j→∞ H j (t),
where the H j are continuous functions from [0, 1] to [0, 1]. For any (a ′ , b ′ ) ∈ (0, 1) 2 such that a ′ < a < b < b ′ , there exists a sequence (P j ) j∈N of polynomials from [0, 1] to [a ′ , b ′ ] and for any t ∈ [0, 1], there exists a strictly increasing sequence of integers (j n (t)) n∈N depending on t such that the three following properties hold simultaneously

• ∀t ∈ [0, 1], H(t) = lim inf j→∞ P j (t), H(t) = lim sup j→∞ P j (t), (14)
• ∀j ∈ N, P ′ j ∞ ≤ j, (15)
• ∀t ∈ [0, 1] ∀n ∈ N, ∀j ∈ {j n (t), • • • , j n+1 (t) -1}, jP j (t) ≥ sup(j n (t)(H(t) -ε), j + j n+1 (t)(H(t) -ε -1)). ( 16
)
Proof. Lemma 2 of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF] implies that there exists a sequence of polynomials (Q ℓ ) j∈N such that Conditions ( 14) and ( 15) both hold. Moreover in the construction of [START_REF] Daoudi | Construction of Continuous Functions with Prescribed Local Regularity Constr[END_REF], one may assume

Q ℓ -H ℓ ∞ ≤ 1 ℓ .
Then, for ℓ sufficiently large,

Q ℓ is onto [a ′ , b ′ ] ⊂ (0, 1). Set β 1 = [b ′ /a ′ ] + 1, β 2 = [(1 -a ′ )/(1 -b ′ )] + 1, β = β 1 β 2
and define the sequence (P j ) j∈N as follows

∀ℓ ∈ N, ∀β ℓ + 1 ≤ j ≤ β ℓ+1 , P j = Q ℓ .
We now prove that the sequence (P j ) j∈N satisfies the required properties. Let ε > 0, t ∈ [0, 1], (ℓ n (t)) n∈N a sequence such that

Q ℓn(t) ≥ H(t) -ε
and set j n (t) = β ℓn(t) β 2 . For any integer j we distinguish three cases.

• If there exists n ∈ N such that j n ≤ ℓ ≤ β 1 j n , then

P j = Q ℓn ≥ H -ε. Hence jP j ≥ j(H -ε) ≥ j n (H -ε), and
j(1 -P j ) ≤ j(1 -H + ε) ≤ jβ 1 (1 -H + ε)j n+1 (1 -H + ε).
• If there exists n ∈ N such that β 1 j n ≤ j ≤ j n+1 /β 2 , then

P j = Q ℓ with ℓ n + 1 ≤ ℓ ≤ ℓ n+1 -1.
Then

jP j ≥ ja ′ ≥ j β 1 H ≥ j n H, and j(1 -P j ) ≤ j(1 -a ′ ) ≤ jβ 2 (1 -H) ≤ j n+1 (1 -H).
• If there exists n ∈ N such that j n+1 /β 2 ≤ j ≤ j n+1 , then

P j = Q jn+1 ≥ H -ε. Hence jP j ≥ j(H -ε) ≥ j n (H -ε), and j(1 -P j ) ≤ j(1 -H + ε) ≤ j n+1 (1 -H + ε).
Thus, in any case we obtain the required property. Now we prove the following proposition where the (H j ) j∈N is a sequence of continuous functions. Let (P j ) j∈N a sequence of polynomials satisfying Properties ( 14),( 15) and ( 16) and consider the wavelet series defined by

f (x) = j∈N 2 j -1 k=0 2 -jPj ( k 2 j ) .
Then for any t ∈ [0, 1],

h(t) = H(t) < h(t) = H(t).
Proof. The assumption

P j ([0, 1]) ⊂ [a ′ , b ′ ] ⊂ (0, 1
) implies that f is uniform Hölder. Since for any j, P j satisfies Properties ( 14), ( 15) and ( 16), for any λ ∈ 3λ j (x 0 ), we have

2 -jPj (t) 2 -j 2 2 -j ≤ |c λ | ≤ 2 -jPj (t) 2 j 2 2 -j .
Thus, for any ε > 0, there exists j 0 sufficiently large such that

∀j ≥ j 0 , 2 -jPj (t) 2 -εj ≤ d j (x 0 ) ≤ 2 -jPj (t) 2 εj .
By definition of H,

lim inf j→∞ log d j (t) -log 2 = lim inf j→∞ P j (t) = H(t).
Hence, h(t) = H(t).

In the same way, h(t) ≤ H(t).

We now use Properties ( 14), ( 15) and ( 16). There exists a strictly increasing sequence of integers (j n ) n∈N such that ( 14), ( 15) and ( 16) hold. Then, ∀n ∈ N,

∀j ∈ {j n , • • • , j n+1 }, d j (t) ≤ 2 -jPj (t) 2 εj ≤ inf(2 -jn(H(t)-ε) , 2 -jn+1(H(t)-ε-1)-j ).
The wavelet criteria then provides

h(t) ≥ H(t).

Weak multifractal formalism

The aim of the multifractal analysis is to study "irregularly irregular" functions, i.e. functions whose Hölder exponent can jump from point to point. From a practical point of view, the numerical computation of the pointwise Hölder exponent of a signal is completely instable, and is indeed quite meaningless, especially for signals whose pointwise Hölder exponent can take very different values. Leaving this utopian view, one rather wishes to get global informations about the pointwise regularity: What are the values taken by the Hölder exponent? What is the "size" of the set of points E h where the Hölder exponent takes a given value h? First of all, one has to define this notion of size. Since the sets under consideration can be dense or negligible, by "size", we cannot mean "Lebesgue measure". The "fractal dimensions" are more fitted for this purpose. Once the right definition of dimension has been chosen, one still has to determine the spectrum of singularities of the function, i.e. the dimension of the sets E h . This is the purpose of the multifractal formalism. Naturally, all these definitions can be transposed for the upper Hölder exponent.

A notion of dimension

In multifractal analysis, the notion of dimension which is mainly used is the Hausdorff dimension. We recall here its definition. The Hausdorff dimension is defined through the Hausdorff measure (see [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF] for more details). The best covering of a set E ⊂ R d with sets subordinated to a diameter ε can be estimated as follows,

H δ ε (E) = inf{ ∞ i=1 |E i | δ : E ⊂ ∞ i=1 E i , |E i | ≤ ε},
where for any i,

|E i | denotes the diameter of E i .
Clearly, H δ ε is an outer measure. The Hausdorff measure is defined from H δ ε as ε goes to 0.

Definition 5

The outer measure H δ defined as

H δ (E) = sup ε>0 H δ ε (E)
is a metric outer measure. Its restriction to the σ-algebra of the H δ -measurable sets defines the Hausdorff measure of dimension δ.

Since the outer measure H δ is metric, the algebra includes the Borelian sets. The Hausdorff measure is decreasing as δ goes to infinity. Moreover,

H δ (E) > 0 implies H δ ′ (E) = ∞ if δ ′ < δ.
The following definition is thus meaningful.

Definition 6 The Hausdorff dimension dim H (E) of a set E ⊂ R d is defined as follows dim H (E) = sup{δ : H δ (E) = ∞}.
With this definition, dim H (∅) = -∞.

From the strong multifractal formalism to the weak multifractal formalism

We first review the wavelet leaders based multifractal formalism as defined by Jaffard [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF], which is one of the two methods allowing to recover, in some particular cases, the entire spectrum of singularities (the second one is the wavelet transform of the maxima of the modulus method, introduced by Arneodo and his collaborators [START_REF] Arneodo | The thermodynamics of fractals revisited with wavelets[END_REF]). Other multifractal formalisms only give, at best, the increasing part of the spectrum (see [START_REF] Jaffard | Multifractal formalism for functions[END_REF]). These considerations on the strong Hölderian regularity can be transposed to the weak one.

The lower spectrum of singularities allows to characterize globally the regularity of a function through the lower Hölder exponents.

Definition 7 Let f be a locally bounded function; its lower isoHölder sets are the sets

E H = {x : h(x) = H}.
The lower spectrum of singularities of f is the function

d : R + ∪ {∞} → R + ∪ {-∞} H → dim H (E H ),
It is not always possible to compute the lower spectrum of singularities of a function. A multifractal formalism is a method that is expected to yield the function d through the use of a Legendre transform. These formalisms are variants of a seminal derivation which was proposed by Parisi and Frisch [START_REF] Parisi | On the singularity spectrum of fully developped turbulence[END_REF]. The wavelet leaders method (WLM) uses wavelet coefficients instead of L p norms, which are meaningless for negative values of p. The partition function is defined as follows S(j, p) = 2 -j λ:|λ|=2 -dj

d p λ .
By setting,

ω(p) = lim inf j→∞ log S(j, p) log 2 -j , ( 17
)
the spectrum of singularities d(h) is expected to be equal to

inf p {hp -ω(p) + d}. ( 18
)
The heuristic argument leading to the previous method is the following. The contribution of the dyadic cubes of side 2 -j containing a point whose lower Hölder exponent is h to the sum d p λ can be estimated as follows. By Theorem 1, the lower Hölder exponent h(x) of a function at x is

h(x) = lim inf j→∞ λ ′ ⊂3λ j (x) log d λ ′ log 2 -j ,
which allows us to write d λ ∼ 2 -hj . Moreover, the number of these dyadic intervals should be about 2 d(h)j , each of volume 2 -dj . Hence, the contribution is 2 (d(h)-d-hp)j . The dominating contribution is the one corresponding to the value h associated with the biggest exponent; by writing the equality (17) as d p λ ∼ 2 -ω(p)j , one can expect the following relation, -ω(p) = sup h {d(h)d -hp}. As -ω is a convex function, if d is concave, then -ω and -d are convex conjugate functions, so that d(h) = inf p {hpω(p) + d}. Let us remark that the preceding argument is far from being a mathematical proof; see [START_REF] Abry | Jaffard Wavelets Leaders in Multifractal Analysis, Wavelets Analysis and Applications[END_REF] and [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF] for a comparison between the WLM and other multifractal formalisms.

Although it can be shown that formula [START_REF] Clausel | Etude de quelques notions d'irrégularité : le point de vue ondelettes[END_REF] allows to recover the spectrum of singularities under additional assumptions (see [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF][START_REF] Jaffard | Multifractal formalism for functions[END_REF][START_REF] Aouidi | Multifractal formalism for quasi self-similar functions[END_REF] for instance), the validity does not hold in complete generality. Indeed, the only result valid in the general case is the following inequality [START_REF] Jaffard | Multifractal formalism for functions[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF],

d(h) ≤ inf p∈R * {hp -ω(p) + d}. ( 19
)
Once the lower spectrum of singularities has been introduced, the upper spectrum of singularities can be defined in a totally analogous way. A relation similar to the inequality [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF] holds.

The weak multifractal formalism is defined as follows.

Definition 8 Let f be a locally bounded function; its upper isoHölder sets are the sets

E H = {x : h(x) = H}.
The upper spectrum of singularities of f is the function

d : R + ∪ {∞} → R + ∪ {-∞} H → dim H (E H ).
The following theorem which can be found in [START_REF] Abry | The contribution of wavelets in multifractal analysis[END_REF] gives an upper bound for the upper spectrum of singularities.

Theorem 5 Let f a uniform Hölder function. The following inequality holds

d(h) ≤ inf p∈R * {hp -ω(p) + d}. (20)
Definition 9 Let f a uniform Hölder function and h > 0; if [START_REF] Devore | Maximal functions measuring smoothness[END_REF] is an equality, i.e. ∀h > 0, d(h) = inf p∈R * {hpω(p) + d} then function f is said to obey the weak multifractal formalism.

Construction of a class of wavelet series obeying the weak multifractal formalism

The aim of this Section is to exhibit a class of multifractal functions for pointwise irregularity. This question is in fact a non trivial one. A quite natural approach to solve this problem is to consider multifractal functions for the usual pointwise regularity.

Let us point out that if we want to define wavelet series that are multifractal both for the strong and weak Hölderian regularity point of views, we have to take into account that, except in the case where the lower and upper exponents coincide, there is no wavelet criteria for the pointwise irregularity.

In the same spirit as Barral and Seuret in [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF], we will define wavelets series built from a multifractal measure µ on [0, 1[ d in the following way,

F µ (x) = j≥0 |λ|=2 -j , λ⊂[0,1] d 2 -j(s0-d p 0 ) µ(λ) 1 p 0 ψ λ (x), (21)
∀x ∈ [0, 1] d , where the wavelets ψ i belongs to the Schwartz class on R d with all moments vanishing. This class of examples also proves that upper and lower spectra may coincide: Under specific assumptions detailed in section 5.3, we can obtain a class of functions obeying both the strong and the weak multifractal formalisms.

We begin by recalling some basic facts about the multifractal analysis of measures.

Some results about multifractal analysis of measures

Following Barral and Seuret, we adapt here the usual multifractal formalism of [START_REF] Brown | On the multifractal analysis of measures[END_REF]. The main difference lies in the definition of the isoHölder sets, since we just need a multifractal formalism associated with a dyadic grid. We first give some slightly modified versions of the usual definitions of lower and upper exponents of a given Borel measure µ at a point x 0 . For any σ ∈ {-1, 0, 1} d and any dyadic cube

λ = ℓ [ k ℓ 2 j , k ℓ 2 j +1 [, let us set λ σ = ℓ [ k ℓ + σ ℓ 2 j , k ℓ + 1 + σ ℓ 2 j [ and µ σ (λ) = µ(λ σ ).
We also define the quantities

α σ µ (x 0 ) = lim inf j→∞ log µ σ (λ j (x 0 )) -j log(2) , α σ µ (x 0 ) = lim sup j→∞ log µ σ (λ j (x 0 )) -j log(2) ,
and, in case of existence,

α σ µ (x 0 ) = lim j→∞ log µ σ (λ j (x 0 )) -j log(2) .
We will be concerned by the estimate of the Hausdorff dimension of the following isoHölder sets

E α (µ) = {x ∈ [0, 1[ d , α µ (x) = min σ∈{-1,0,1} d {α σ µ (x)} = α}.
The mapping

d µ : α ≥ 0 → dim H ( E α (µ)),
will be called the multifractal spectrum of the Borel measure µ. Recall that in the framework of [START_REF] Brown | On the multifractal analysis of measures[END_REF], the following isoHölder sets are used

E α (µ) = {x ∈ [0, 1[ d , lim j→+∞ log µ(λ j (x)) -j log(2) = α}.
Unfortunately, these isoHölder sets are not adapted to the study of the pointwise regularity of wavelet series F µ . Indeed, starting from lim j log µ(λ j (x))/j log(2), we cannot deduce the value of the upper pointwise Hölder exponent of the function F µ at x using wavelet criteria. We now recall well known results about upper bound of the upper multifractal spectrum, which can be found in [START_REF] Brown | On the multifractal analysis of measures[END_REF]. For any q in R, set

τ (q) = lim inf j→+∞ 1 log |λ| log * |λ|=2 -j µ(λ) q ,
where * means that the sum is taken over those λ such that µ(λ) > 0. As usual, τ * denotes the Legendre transform of the function τ , that is

∀α ≥ 0, τ * (α) = inf q∈R {αq -τ (q)}. Remark that, since α > 0, E α (µ) ⊂ E α (µ).
Using this inclusion, an upper bound for the multifractal spectrum of any Borel measure can be obtained from [START_REF] Brown | On the multifractal analysis of measures[END_REF]: Proposition 6 Let α ≥ 0 and µ a Borel measure. One has

dim H ( E α (µ)) ≤ τ * (α). Moreover, if τ * (α) < 0 then E α (µ) = ∅.
Definition 10 Let α 0 ≥ 0. One says that the Borel measure µ obeys the multifractal formalism at α = α 0 for the sets

E α (µ) if dim H ( E α0 (µ)) = τ * (α 0 ).

Wavelet series and multifractal measures

We want to define a wavelet series of the form (21) obeying the weak multifractal formalism for functions. First, we give an explicit relationship between the wavelet series F µ and the measure µ from the multifractal point of view.

A transference theorem

Theorem 6 Let µ a Borel measure and s 0 , p 0 two positive real numbers. Let F µ be the wavelet series defined by equality [START_REF] Demichel | Analysis of fractal sum of pulses[END_REF]. If the measure µ obeys the multifractal formalism at α 0 ≥ 0 for the sets E α (µ), then F µ obeys both the strong and weak multifractal formalisms at

H = s 0 - d p 0 + α 0 d p 0 , and
d(H) = d(H) = d µ (α 0 ).
Remark 5 Let us notice, as in [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF], that if x 0 ∈ supp(µ), there exists some j 0 such that, ∀j ≥ j 0 , d j (x 0 ) = 0.

Thus, in this special case, h Fµ (x 0 ) = +∞.

Proof. The proof mimics the one of Theorem 1 of [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF]. It relies on the following Lemma:

Lemma 5 For any α ≥ 0, the following inclusion holds:

E α (µ) ⊂ E H (F µ ) ∩ E H (F µ ),
where H = s 0 -d p 0 + αd p 0 .

Proof. Remark that, for all λ d λ = 2 -j(s0-d p 0

) µ(λ)

1 p 0 .
where j =log(|λ|)/ log(2).

For any given x 0 ,

d j (x 0 ) = 2 -j(s0-d p 0 ) max σ∈{-1,0,1} d {µ σ (λ j (x 0 ))} 1 p 0 .
Assume that x 0 ∈ E α (µ). Since for any σ ∈ {-1, 0, 1} d , α σ (x 0 ) ≥ α, for all ε > 0 there exists an integer j 0 (ε, σ) such that

∀j ≥ j 0 (ε, σ), µ σ (λ j (x 0 )) ≤ 2 -j d p 0 (α-ε) .
Hence,

∀j ≥ max σ {j 0 (ε, σ)}, d j (x 0 ) ≤ 2 -j(s0-d p 0 ) 2 -j d p 0 (α-ε) ≤ 2 -j(H-ε) and lim inf j→+∞ log d j (x 0 ) log 2 -j ≥ H.
Furthermore, since for some σ 0 ∈ {-1, 0, 1} d we have α σ0 (x 0 ) ≤ α, for all ε > 0 there exists an integer j 0 (ε) such that ∀j ≥ j 0 (ε), µ σ0 (λ j (x 0 )) ≥ 2 -jd(α+ε) .

Then, ∀j ≥ j 0 (ε), d j (x 0 ) ≥ 2 -j(H+ε) and lim sup j→+∞ log d j (x 0 ) log 2 -j ≤ H.

Hence, using Corollary 1,

E α (µ) ⊂ {x ∈ [0, 1] d , h Fµ (x) = h Fµ (x) = H} = E H (F µ ) ∩ E H (F µ ).
Since µ obeys the multifractal formalism at α = α 0 for the sets

E α (µ), dim H ( E α0 (µ)) = τ * (α 0 ) ≤ dim H (E H (F µ )) with H = s 0 - d p 0 + α 0 d p 0 .
Remark then that for any p ∈ R,

ω * f (p) = p(s 0 - d p 0 ) -τ ( p p 0 ).
Since for any 0 < H < ∞ and any locally bounded function one has

dim H (E H (F µ )) ≤ inf p∈R {pH -ω * f (p) + d} = τ * (α 0 ) = dim H ( E α0 (µ)),
one can conclude that F µ obeys the weak multifractal formalism. A similar approach proves that F µ also obeys the strong multifractal formalism.

A class of wavelet series obeying both the strong and the weak multifractal formalisms

The aim of this section is to exhibit a class of multifractal measures obeying the multifractal formalism at any α ≥ 0 for the sets E α (µ), yielding an example of wavelet series satisfying both the strong and weak multifractal formalisms.

To this end we first give some examples of multifractal measures obeying the multifractal formalism for sets E α using Theorem 2 of [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF] for sets E α . Indeed, even if we consider slightly different iso-Hölder sets Theorem 2 still holds : the proof is exactly the same that this of [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF].

We give two canonical examples of measures satisfying the conditions above. By construction, any multinomial measure is quasi-Bernoulli.

In the following, we consider only continuous quasi-Bernoulli measure, that is without atom. Recall that any continuous quasi-Bernoulli measure satisfies the assumptions of Theorem 2 of [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF] and thus obeys the multifractal formalism at any α > 0 for the sets E α .

Hence we have the following Theorem.

Theorem 7 If µ is a continuous quasi-Bernoulli measure, then the wavelet series F µ defined by ( 21) obeys both the strong and the weak multifractal formalisms at any H > s 0d/p 0 .

The case of b-adic random multiplicatives cascades

Let b an integer larger than 2 and d = 1. Canonical random cascades were introduced by Mandelbrot in [START_REF] Mandelbrot | Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier[END_REF] and their multifractal properties have been widely studied, mainly in the setting of b-adic grid (see e.g. [START_REF] Kahane | Produits de poids aléatoires et indépendants et applications[END_REF][START_REF] Holley | Multifractal dimensions and scaling exponents for strongly bounded random cascades[END_REF][START_REF] Collet | Large deviations for multiplicative chaos[END_REF][START_REF] Molchan | Scaling exponents and multifractal dimensions for independent random cascades[END_REF][START_REF]Moments, continuité et analyse multifractale des martingales de Mandelbrot[END_REF][START_REF]Continuity of the multifractal spectrum of a random statistically selfsimilar measures[END_REF]).

We first recall the construction of these measures. Let W a non negative random variable, not almost surely constant, satisfying E(W ) = 1/b. We thus consider (W w ) w∈A * a sequence of independent copies of W and µ n the random measure whose density with respect to the Lebesgue measure on any dyadic interval is constant and equals

b n W w1 • • • W w1•••wn .
Almost surely, this sequence of measures converges weakly to a measure µ as n goes to infinity. Recall that if µ is a b-adic random multiplicative cascade, then almost surely on J, µ satisfies the assumptions of Theorem 2 of [START_REF] Barral | From Multifractal Measures to Multifractal Wavelet Series[END_REF] and thus obeys the multifractal formalism for any q ∈ J at α = τ ′ (q) for the sets E α .

Then similarly, to the case of quasi-Bernoulli measure, we have the following result, Theorem 8 Let W be an almost surely positive random variable. Let µ a b-adic random multiplicative cascade such that τ ′ (1) = -1log b (E(W )) > 0. Then the wavelet series F µ defined by [START_REF] Demichel | Analysis of fractal sum of pulses[END_REF] obeys almost surely both the strong and the weak multifractal formalisms at any H > 0.

A multifractal function whose lower and upper spectra of singularities differ

Although similar results hold for both the strong and the weak multifractal formalisms, there is no direct relation between d and d. We introduce here a function defined as p-adic Davenport series whose upper multifractal spectrum is reduced to two single points, while its lower multifractal spectrum is linear on an interval.

A p-adic Davenport series (p ≥ 2) is a series of the form

f (x) = ∞ j=0 a j {p j t},
where {x} is the sawtooth function

{x} = x -[x] - 1 2 .
We will assume here that (a j ) j ∈ l 1 , so that the series is normally convergent. The function f is thus continuous at every non p-adic rational number and has left and right limit at every p-adic rational kp -l (k ∧ p = 1) with a jump of amplitude m≥l a m . Recent results on Davenport series can be found in [START_REF] Jaffard | On Davenport expansions[END_REF]. Let β > 1; the functions f β we will study is defined by

f β (x) = l∈N {2 l x} 2 lβ .
6.1 The lower spectrum of singularities of f β

The functions f β are derived from the famous Lévy's function (which can be seen as a special case, where β = 1). The properties of the lower spectrum of singularities of this function have already been investigated in [START_REF] Jaffard | Multifractal formalism for functions[END_REF]; Propositions 8 and 9 together can be seen as a generalization of Proposition 4 of [START_REF] Jaffard | Multifractal formalism for functions[END_REF]. Proposition 12 of [START_REF] Jaffard | On Davenport expansions[END_REF] implies that d is linear on [0, β].

To determine explicitly the lower isoHölder sets of f β , we will use the following notations. Let p ∈ N, p > 1; for a sequence of integers (x l ) l∈N satisfying 0 ≤ x j < p, we will write (0; x 1 , . . . , x l , . . .) p [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF] to denote one expansion in basis p of the real number

x = l∈N x l p l .
If there is no k such that x l = p -1 for all l ≥ k, [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF] is the proper expansion of x in basis p. If (0; x 1 , . . .) p is the proper expansion of x, we define

θ p (x) = inf{l : x l = 0} -1.
Let δ(k) = sup{l : ∀l ′ ≤ l, x k+l ′ = x k } and let (m l ) l∈N be the sequence defined recursively, m 1 = inf{l :

x l = 0 or x l = p -1}, m k = inf{l ≥ m k-1 + δ(m k-1 ) : x l = 0 or x l = p -1} (k > 1)
. One also defines the sequence (δ k ) k∈N by δ k = δ(m k ). Finally, ρ p (x) = lim sup k→∞ δ k /m k ; if x is a p-adic rational, one sets ρ p (x) = ∞. The number ρ p (x) defines, in some way, the rate of approximation of the number x by p-adic rationals, since we have the following obvious result.

Proposition 7 If x is not a p-adic rational, the equation (depending on k and l)

|x - k p l | ≤ ( 1 p l ) φ (k ∧ p = 1)
has an infinity of solutions if and only if φ ≤ ρ p (x) + 1.

We will denote by φ(x) the critical exponent φ(x) = ρ 2 (x)+1. The lower Hölder exponents of f β only depend on φ.

Proposition 8

The lower Hölder exponents of f β are given by

h(x) = β φ(x) .
Proof. As a corollary of Theorem 21 of [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF], we have the following equalities: if x is not a dyadic rational,

h(x) = lim inf j→∞ -βj log 2 dist(x, 2 -j Z) ; (23)
otherwise, h(x) = 0. We can suppose that x ∈ (0, 1) is not a dyadic rational. For a given j ∈ N, let ε j = dist(x, 2 -j Z). One has θ 2 (ε j ) = j + 1 + δ(j + 1) and thus ε j ∼ 2 -(j+δ(j+1)+1) . Then ( 23) can be rewritten

h(x) = lim inf j→∞ βj j + 1 + δ(j + 1) = β 1 + ρ 2 (x)
.

The lower isoHölder sets are now characterized.

Corollary 2

The lower isoHölder sets of the function f β are the sets

E H = {x : φ(x) = β H } (0 < H ≤ β).
The set E 0 is the set of the dyadic rationals.

To conclude this study on the strong Hölder regularity, we have the following result.

Proposition 9

The lower spectrum of singularities of f β is

d(h) = h β if h ∈ [0, β] -∞ otherwise .
Proof. The main idea is the same as in Proposition 4 of [START_REF] Jaffard | Multifractal formalism for functions[END_REF]. If α ≥ 1/β, let

F α = lim sup j→∞ k [k2 -j -2 -jαβ , k2 -j + 2 -jαβ ]. Using (23), h(x) = H means x ∈ γ>H F 1/γ - γ<H F 1/γ . (24)
Clearly, dim H (F α ) ≤ 1/αβ; let us show that the converse inequality holds. Let (j l ) l∈N be a sequence satisfying j l = 2 j l-1 , let I k (l) = [k2 -j l -2 -j l αβ , k2 -j l + 2 -j l αβ ] and

G α = l k I k (l).
A probability measure µ supported by G α can be obtained as follows. If l = 1, we put on each interval I k (1) the same mass 2 -j1 . If each of these intervals contains n intervals of type I k (2), on each of these intervals, we put the measure 2 -j1 /n. This construction can be iterated to obtain, at the limit, a probability measure µ supported by G α . One easily checks that µ([xh, x + h]) ≤ Ch 1/αβ ∀x ∈ G α .

Moreover, Proposition 4.9 of [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF] implies that

H 1/αβ (G α ) > 0
and thus, since G α ⊂ F α , dim H (F α ) = 1/αβ, which, thanks to [START_REF] Hardy | Weierstrass's non differentiable function[END_REF], is sufficient to conclude.

The upper spectrum of singularities of f β

We show here that from the weak Hölder regularity point of view, the function f β only displays two kinds of singularities: it is discontinuous at dyadic rationals and has an upper Hölder exponent equal to β at non dyadic rationals.

Let

Ω α = lim inf j→∞ {x ∈ R : ∃k ∈ Z such that |x - k 2 j | ≤ 2 -αj }.
We have the following relation between the sets Ω α and h(x 0 ).

Proposition 10 If α > 1, then

x 0 / ∈ Ω α ⇒ h(x 0 ) ≥ β α .
Proof. Let ε > 0 be a given real number. We want to prove that for any C > 0, there exists a strictly decreasing sequence (r n ) n of real positive numbers, such that sup for all n ≥ n 0 . We have h(x 0 ) ≥ β/αε, which is sufficient to conclude. The sets Ω α are explicitly known whenever α ≥ 1.

Proposition 11 If α > 1, then

Ω α = { k 2 j : (k, j) ∈ Z × N}.
Moreover, Ω 1 = R.

Proof. The case α = 1 is trivial. Let α > 1 and let x ∈ (0, 1) be a non dyadic rational. If x = (0; x 1 , . . .) 2 , one has, for j sufficiently large,

min k∈Z |x - k 2 j | ≥ 2 -n(j)+1 ,
where n(j) is the first index greater or equal to j such that x n(j) = 1. We then have, for j = n(j) -1 sufficiently large,

min k∈Z |x - k 2 j | ≥ 2 -j > 2 -αj .
Therefore, x / ∈ Ω α . Thus, we have a lower bound for the upper Hölder exponent.

Corollary 3 If x 0 is not a dyadic rational, then h(x 0 ) ≥ β.

Let us now prove the converse inequality. The following proposition is similar to Lemma 1 of [START_REF] Jaffard | Multifractal formalism for functions[END_REF].

Proposition 12 Let f be a function defined on R, continuous everywhere except on a dense countable set of points and admitting a left and a right limit at every point. Let also x 0 ∈ R be a point of continuity of f and (r n ) n a sequence of points of discontinuity converging to x 0 . Finally, let s n (n ∈ N) be the jump of f at r n . If there exists a strictly increasing function ψ satisfying ψ(0) = 0 such that ∃r 0 : ∀r ≤ r 0 , ∃r n : Since F n has only a countable set of discontinuities, one can find h arbitrarily close to zero such that [r n , r n + ([α] + 1)h] ⊂ B(x 0 , t n ) and such that F n is continuous at h. Therefore,

ψ(t n ) ≤ s n = |f (r + n ) -f (r - n )| ≤ | [α]+1 k=1 (-1) k [α] + 1 k f (r n + kh) + f (r - n )| +| [α]+1 k=1 (-1) k [α] + 1 k f (r n + kh) + f (r + n )| ≤ 2(|∆ [α]+1 h f (r - n )| + |∆ [α]+1 h f (r + n )|) ≤ 4Ct α n .
This inequality implies log ψ(t n ) log t n ≥ α + log 4C log t n .

As t n tends to zero, lim sup r→0 log ψ(r) log r ≥ α, and the result follows. Since f β is continuous except at dyadic rationals and since Ω 1 = R, Proposition 12 and Corollary 3 imply the following result.

Theorem 9 If x 0 is not a dyadic rational, h(x 0 ) = β, if x 0 is a dyadic rational, h(x 0 ) = 0.

  with β > H(t). It provides the required conclusion. By applying Lemma 2 and Lemma 3 to f j = sin(2π•), C = 1, ℓ = [λ/4] and D = | sin(ℓ/λ)|, Proposition 1 is then straightforward.

5. 3 . 1 Example 1

 311 Quasi-Bernoulli measures Let b an integer larger than 2. Let us recall that a Borel positive measure on [0, 1] d µ is said quasi-Bernoulli if for some C > 0 and for any v, w ∈ (A d ) * , 1 C µ(I v )µ(I w ) ≤ µ(I vw ) ≤ Cµ(I v )µ(I w ). A classical example of quasi-Bernoulli measures is the well-known example of multinomial measures: Let b an integer larger than 2 and let (m 0 , • • • , m b-1 ) ∈ (0, 1) b such that, a sequence of probability measures (µ n ) n∈N on [0, 1) d as follows. For any integer n, define a probability measure µ n on [0, 1) d such that for any w ∈ A nd , µ n (I w ) = nd ℓ=1 m w ℓ . This sequence has a weak limit µ called multinomial measure of base b with weight (m 0 , • • • , m b-1 ).

2

 2 h f β (x) L ∞ (B h (x0,rn)) ≤ Cr β/α n . m M m {2 l (x + mh)}.(25)If x 0 / ∈ Ω α , then there exists a strictly increasing sequence of integers(j n ) n such that |x 0 -k 2 jn | ≥ 2 -αjn , ∀n ∈ N, k ∈ Z. Let r n = 2 -αjn ; the interval [x, x + ([β/α] + 1)h] ⊂ B(x 0 , r n ) does not contain any dyadic rational of the form k2 -l , with l ≤ j n -1. This implies m M m {2 l (x + mh)} = 0.Relation[START_REF] Heurteaux | Estimations de la dimension inférieure et de la dimension supérieure des mesures[END_REF] leads to the following inequality, lβ ≤ C ′ r β/α n .Let now n 0 be an integer such that C ′ r

  |r nx 0 | ≤ r, |s n | ≥ ψ(r) , then h(x 0 ) ≤ lim sup r→0 log ψ(r) log r . Proof. Let α > 1; if f ∈ C α w (x 0 ), for any C > 0 there exists a sequence (t n ) n such that sup h>0 ∆ [α]+1 h f (x) L ∞ (B h (x0,tn)) ≤ Ct α n .Let (r n ) n a sequence such that, for any integer n sufficiently large,|r nx 0 | < t , |s n | ≥ ψ(t n ).Let F n be the function defined on R by F n (h) = (f (r n + h), . . . , f (r n + ([α] + 1)h)).

Acknowledgment. The authors would like to thank A. Ayache, Y. Heurteaux and S. Jaffard for several stimulating discussions.


Marianne Clausel

Samuel Nicolay
email: s.nicolay@ulg.ac.be.

A weak local irregularity property in S ν spaces

Although it has been shown that, from the prevalence point of view, the elements of the S ν spaces are almost surely multifractal, we show here that they also almost surely satisfy a weak uniform irregularity property.

Introduction

The uniform regularity defined from the Hölder spaces C α (R d ) is one of the most popular concepts for the uniform regularity. It has been introduced to study smoothness properties of functions such as the Weierstraß function (see [START_REF] Hardy | Weierstrass's non differentiable function[END_REF]). Indeed, many "historical" functions share the same property (see [START_REF] Jaffard | Pointwise smoothness of space-filling functions[END_REF]): there exist H ∈ (0, 1) and a constant C > 0 such that the function f satisfies on some interval I,

∀x, y ∈ I, |f (x) -f (y)| ≤ C|x -y| H , (1)
and ∀x, y ∈ I, sup

(u,v)∈[x,y] 2 |f (x) -f (y)| ≥ 1 C |x -y| H . (2)
It has been shown in [START_REF] Clausel | Some prevalent results about strongly monoHölder functions[END_REF] that this behavior is the typical behavior of the functions belonging to C H (R d ), in the sense of prevalence (the notion of prevalence will be defined in the sequel, see section 2.3). In other words, almost every function of C H (R d ) satisfies Relations (1) and [START_REF] Arneodo | Wavelet-based multifractal formalism: Applications to DNA sequences, satellite images of the cloud structures and stock market data[END_REF]. However, in many cases, the functions do not satisfy Relations (1) and [START_REF] Arneodo | Wavelet-based multifractal formalism: Applications to DNA sequences, satellite images of the cloud structures and stock market data[END_REF]. It is in particular the case of the so-called multifractal functions (see Definition 5), originally introduced in the context of turbulence and now used in many fields of science (see e.g. [START_REF] Abry | Lois d'échelle, Fractales et Ondelettes[END_REF][START_REF] Arneodo | Wavelet-based multifractal formalism: Applications to DNA sequences, satellite images of the cloud structures and stock market data[END_REF][START_REF] Halsey | Fractal measures and their singularities: The characterization of strange sets[END_REF][START_REF] Jaffard | Wavelets: Tools for Science and Technology[END_REF][START_REF] Sreenivasan | Fractals and multifractals in turbulence[END_REF]). It can be shown that in several functional spaces, almost every function (in the sense of prevalence) is multifractal (see [START_REF] Fraysse | How smooth is almost every function in a Sobolev space?[END_REF][START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF]).

In this paper, we aim at investigating the typical irregularity properties of the elements of the S ν spaces. These functional spaces, defined in [START_REF] Jaffard | Beyond Besov spaces, part I: Distribution of wavelet coeficients[END_REF][START_REF] Aubry | Topological properties of the sequences spaces S ν[END_REF], give rise to an efficient multifractal formalism (see section 2), i.e. an heuristic method to study the pointwise regularity of a function from a global point of view. It has been shown in [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF] that almost every element of S ν is multifractal. We show here that, although they are multifractal, the elements of S ν also satisfy almost surely a weak uniform irregularity property. To this end, we introduce a concept of weak uniform irregularity, define the irregularity exponent H(x 0 ) of a function at a given point x 0 and prove that, in the sense of prevalence, there exists α (depending on S ν ) such that almost every function of S ν satisfy H(x) = α, for any x.

The paper is organized as follows. In the next section, we recall some definitions about the multifractal analysis and the S ν spaces. Next, we define the local irregularity exponent. To prove our main result, we will need wavelet criteria for the irregularity; these are stated in Section 3. The prevalent result is obtained in the last section.

Multifractal models

The multifractal analysis aims to study the smoothness of very irregular signals.

For such a function, it is meaningless to try to characterize its regularity at a given point, since the pointwise regularity can abruptly change. One rather tries to determine the so-called spectrum of singularities, which gives the size of the set of points that share the same pointwise regularity. We first have to precise what is meant by "pointwise regularity" and "size of set". The S ν spaces have been introduced in order to provide an efficient multifractal formalism, i.e. a method that allows the computation of the multifractal spectrum in many practical cases. Although this technique does not always lead to the right spectrum, it has been shown that the S ν -based multifractal formalism gives the right answer for almost every element of S ν . Here the term "almost every" has to be clearly stated, since one can not use the usual Lebesgue measure for the infinite dimensional settings. Finally, we introduce another index of regularity, called the local irregularity exponent.

Pointwise regularity, Hausdorff dimension and multifractal spectrum

The notion of the pointwise regularity which we will use here is based on the characterization of the Hölder spaces C α (x 0 ).

Definition 1 Let x ∈ R d and α > 0. A locally bounded function f : R d → C belongs to C α (x 0 )
if there exist C > 0 and a polynomial P of degree strictly lower than α such that, in a neighborhood of x 0 ,

|f (x) -P (x -x 0 )| ≤ C|x -x 0 | α .
The Hölder exponent of f at x 0 is the quantity

H(x 0 ) = sup{α > 0 : f ∈ C α (x 0 )}.
The multifractal analysis provides a description of the collection of the Hölder exponents of a function by way of the multifractal spectrum (also called the singularity spectrum), which associates to a value h the size of the set of points for which the Hölder exponent is h. By size, one usually means Hausdorff dimension; this notion of dimension is usually preferred because it relies on a measure. We will only give the necessary definitions; the interested reader is referred to e.g. [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF][START_REF] Mattila | Geometry of Sets and Measures in Euclidian Spaces[END_REF].

Definition 2 Let S be a Borelian subset of R d and Γ ǫ (S) be the collection of all the countable ǫ-coverings of S, i.e. the collection of all the countable coverings of S by sets whose diameter is lower than ǫ. The δ-dimensional Hausdorff measure of S is

m δ (S) = lim ǫ→0 inf (Sj )j ∈Γǫ(S) j |S j | δ ,
where |S j | denotes the diameter of the set S j .

It is easy to check that if m δ0 (S) is finite, then m δ (S) = ∞ if δ < δ 0 and m δ (S) = 0 if δ > δ 0 . We are then naturally led to the following definition.

Definition 3

The Hausdorff dimension dim S of a non-empty Borelian subset S of R d is given by dim S = inf{δ : m δ (S) = 0}.

One sets dim ∅ = -∞.

If S is empty or uncountable, one has dim S = sup{δ : m δ (S) = ∞}. We are now able to introduce the multifractal spectrum of a function.

Definition 4

The multifractal spectrum of a locally bounded function f : R d → C is the application

d : (0, ∞] → [-∞, d] h → dim{x ∈ R d : H(x) = h}.
Finally, let us introduce some usual denominations.

Definition 5 A function is called multifractal if the associated multifractal spectrum takes more than one real value on a support with non-zero Hausdorff dimension. A function that is not multifractal is a monofractal function. If the spectrum only takes one value, the function is said to be monoHölder.

Bases of wavelets and S ν spaces

One of the remarkable aspects of the wavelets is that they provide bases of the space L 2 (R d ). Historically, the first orthonormal wavelet basis is the Haar basis, constructed long before the introduction of the term "wavelet". In [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF], it is shown that, under some suitable conditions, the multifractal spectrum of a function can be estimated from its wavelet coefficients. The S ν spaces were introduced in [START_REF] Jaffard | Beyond Besov spaces, part I: Distribution of wavelet coeficients[END_REF] to improve the classical wavelet-based multifractal formalism. The S ν spaces are closely related to the Besov spaces. Let us briefly recall some definitions and notations (for more precisions, see e.g. [START_REF] Meyer | Ondelettes et opérateurs[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Mallat | A wavelet tour of signal processing[END_REF]). Under some general assumptions, there exist a function φ and 2 d -1 functions (ψ (i) ) 1≤i<2 d called wavelets such that

{φ(x -k) : k ∈ Z d } ∪ {ψ (i) (2 j -k) : 1 ≤ i < 2 d , k ∈ Z d , j ∈ N} form a basis of L 2 (R d ). A function f ∈ L 2 (R d ) can be decomposed as follows, f = k∈Z d C k φ(• -k) + ∞ j=1 k∈Z d 2 d -1 i=1 c (i) j,k ψ (i) (2 j • -k),
where

C k = R d f (x)φ(x -k) dx and c (i) j,k = 2 dj R d f (x)ψ (i) (2 j x -k) dx. (3)
Let us remark that we do not choose the (usual) L 2 normalization for the wavelets, but rather an L ∞ normalization, which is better fitted to the study of the Hölderian regularity. Expressions such as (3) can make sense in more general settings (e.g. if f is a distribution). Hereafter, we will assume that the wavelets belong to C γ (R d ) with γ ≥ α + 1, and that the functions {∂ s φ} |s|≤γ , {∂ s ψ} |s|≤γ have fast decay. Moreover, for the sake of simplicity, when dealing with the S ν spaces, we will suppose that the application f is defined on the torus

T d = R d /Z d . Let Λ = (i, j, k) : 1 ≤ i < 2 d , j ∈ N, k ∈ {0, . . . , 2 j -1} d . If (i, j, k) ∈ Λ, the periodized ψ (i) j,k = l∈Z d ψ 2 j (• -l) -k
form a basis of the one-periodic functions in L 2 ([0, 1] d ). We will denote (c

(i) j,k ) (i,j,k)∈Λ or (c λ ) λ∈Λ the wavelet coefficients of a function in L 2 ([0, 1] d ).
Let us now introduce the S ν spaces.

Definition 6 For a sequence c = (c λ ) λ∈Λ , C > 0 and α ∈ R, we define

E j (C, α)[c] = {(i, k) : |c (i) j,k ≥ C2 -jα }.
The wavelet profile ν c of c is defined as

ν c (α) = lim ǫ→0 + lim sup j→∞ log 2 #E j (1, α + ǫ)[c] j , with α ∈ R. If c represents the wavelet coefficients of a function f , one sets E j (C, α)[f ] = E j (C, α)[c] and ν f = ν c .
Clearly, ν c is non-decreasing, right-continuous and non-negative (lower than d) when not equal to -∞. It gives, in some way, the asymptotic behavior of the number of coefficients of c that have a given order of magnitude.

Definition 7 Let ν : R → {-∞} ∪ [0, d] be a non-decreasing, right-continuous function such that there exists α 0 for which ν

(α) = -∞ if α < α 0 and ν(α) ∈ [0, d] otherwise. A sequence distribution f belongs to S ν if its wavelet coefficients satisfy ν f (α) ≤ ν(α) for any α ∈ R.
These spaces are robust, in the sense that their definition does not depend on the choice of the wavelet basis (see [START_REF] Jaffard | Beyond Besov spaces, part I: Distribution of wavelet coeficients[END_REF]). Roughly speaking, a function belongs to S ν if for each scale j and every α ∈ R, there are about 2 ν(α)j coefficients c

(i) j,k
that are larger than 2 -αj . The following definition is equivalent to the preceding one (see [START_REF] Aubry | Topological properties of the sequences spaces S ν[END_REF]).

Definition 8 If ν is a function as given in Definition 7, f belongs to S ν if for any α ∈ R, ǫ > 0 and C > 0, there exists J such that j ≥ J implies

#E j (C, α)[f ] ≤ 2 (ν(α)+ǫ)j .
A topological framework for the spaces S ν will also be needed (further details can be found in [START_REF] Aubry | Topological properties of the sequences spaces S ν[END_REF][START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF]). The ancillary spaces are useful to obtain a structure of complete metric space on S ν .

Definition 9 Let m, n ∈ N; f belongs to E m,n if there exist C > 0 such that #E j (C, α m )[f ] ≤ C2 ν(αm)+ǫn)j , (4)
for any j, where (α m ) m is any dense sequence in R and (ǫ n ) n is decreasing to zero.

If for f ∈ E m,n , the infimum of the constants C satisfying the inequality (4) is noted d m,n (f, 0), the distance d m,n (f, g) = d m,n (f -g, 0) makes (E m,n , d m,n ) a metric space. It can be shown that S ν = ∩ m,n E m,n
and that with the distance

d(f, g) = m,n≥0 2 -m-n d m,n (f, g) 1 + d m,n (f, g) ,
the space (S ν , d) is a complete separable metric space. The distance d may depend on the sequences chosen in (4), but the induced topology does not. The Borel σ-algebra relative to this topology will be denoted B(S ν ).

Prevalence of multifractal functions in S ν

In a finite dimensional space, a property "holds almost everywhere" if the set of points for which this property is not satisfied vanishes for the Lebesgue measure. The Lebesgue measure has a preponderant role, as there is no other σ-finite translation invariant measure. Unfortunately, there is such measure in the infinite dimensional Banach spaces. The notion of prevalence provides the analogue of "Lebesgue measure zero" in complete metric vector spaces.

In [START_REF] Christensen | On sets of Haar measure zero in Abelian Polish groups[END_REF], to recover the notion of "almost every" in infinite vector spaces, a wellknown characterization of Lebesgue measure zero subsets of R d is generalized: In R d , a Borel set B has Lebesgue measure zero if and only if there exists a compactly supported probability measure µ such that µ(B + x) = 0 for any x ∈ R d . This characterization can be turned into a definition in more general settings and leads to the concept of Haar-null set, which provides the analogue of "Lebesgue measure zero" set for infinite dimensional spaces.

Definition 10 Let E be a complete metric vector space. A Borel set B ⊂ E is Haar-null if there exists a Borel probability measure, strictly positive on some compact set K ⊂ E such that µ(B + x) = 0 for any x ∈ E. A subset of E is Haar-null if it is included in a Haar-null Borel subset of E. The complement of a Haar-null set is called a prevalent set.

In [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF], the prevalent behavior of almost every function of S ν has been studied and it has been proved that almost every function of S ν is multifractal. More precisely, an upper bound of the spectrum d of any f ∈ S ν is first given: if α max = inf h≥α0 {h/ν(h)} and

d ν (h) = h sup h ′ ∈(0,h] {ν(h ′ )/h ′ } if h ≤ α max 1 otherwise , one has, for all f ∈ S ν and h ∈ R, d(h) ≤ d ν (h). Then it is shown that the sets {f ∈ S ν : d(h) = d ν (h) if h ≤ α max and d(h) = -∞ otherwise}
and {f ∈ S ν : the pointwise regularity is almost everywhere α max } are prevalent. The "typical elements" (in the sense of the prevalence) of S ν are thus multifractal and do not satisfy the classical law of the iterated logarithm almost everywhere.

Local regularity exponents

We introduce here another notion of local regularity, which will be used to state that a weak irregularity condition is satisfied for almost every multifractal function of S ν . To this end, we first need to recall different concepts of global regularity, based on the global Hölder regularity.

If Ω is an open subset of R d , for any h ∈ R d , we will denote by Ω h the set

Ω h = {x ∈ R d : [x, ([α] + 1)h] ⊂ Ω},
where the value α > 0 will be implied by the context and [α] denotes the greatest integer lower than α. We will need the classical notion of finite difference.

Definition 11 Let x, h ∈ R d and f : R d → C; the first order difference of f is

∆ 1 h f (x) = f (x + h) -f (x).
For n ≥ 2, the difference of order n is defined by

∆ n h f (x) = ∆ n-1 h ∆ 1 h f (x).
We can now introduce the Hölder spaces C α (Ω).

Definition 12

Let Ω be an open subset of R d and α > 0; a bounded function f defined on a subset of R d belongs to C α (Ω) if there exist C, r 0 > 0 such that for any r ≤ r 0 , sup

|h|≤r ∆ [α]+1 h f (x) L ∞ (Ω h ) ≤ Cr α .
The function f is said to be uniformly Hölderian on Ω if for some α > 0, f belongs to C α (Ω).

We will also need a notion of uniform irregularity.

Definition 13

Let Ω be an open subset of R d , α > 0 and β ∈ R; a bounded function f defined on a subset of R d belongs to I α (Ω) if there exist C, r 0 > 0 such that for any r ≤ r 0 ,

sup |h|≤r ∆ [α]+1 h f (x) L ∞ (Ω h ) ≥ Cr α .
A function that belongs to I α (Ω) is said to be uniformly irregular with exponent α.

Let us remark that the statement f ∈ I α (Ω) is not equivalent to f / ∈ C α (Ω). In the latter case, for any C > 0, there exists a sequence (r n ) n (depending on C) decreasing to zero for which

sup |h|≤rn ∆ [α]+1 h f L ∞ (Ω h ) ≥ Cr α n .
We are thus naturally led to the following definition.

Definition 14

Let Ω be an open subset of R d , α > 0 and β ∈ R; a bounded function belong to C α w (Ω) if f is defined on Ω and if for any C > 0, there exists a decreasing sequence (r n ) n decreasing to zero such that

sup |h|≤rn ∆ [α]+1 h f L ∞ (Ω) ≤ Cr α n ,
for any n ∈ N. The set C α β,w (Ω) is simply denoted C α w (Ω). A function that belong to C α w (Ω) is said to be weakly uniformly Hölder with exponent α on Ω.

Definition 15

The upper uniform Hölder exponent of a weakly uniformly Hölder function f on an open set Ω is defined as

H(Ω) = sup{α > 0 : f ∈ C α w (Ω)}.
We now define the local irregular exponent, using the same approach as in [START_REF] Lévy | The local Hölder function of a continuous function[END_REF].

Definition 16 A sequence (Ω n ) n of open subsets of R d is decreasing to x 0 ∈ R d if • m < n implies Ω n ⊂ Ω m , • |Ω n | → 0 as n → ∞, • ∩ n Ω n = {x 0 }.
The following lemma is needed.

Lemma 1 If (Ω n ) n and (Ω ′ n ) n are two sequences of open sets that decrease to x 0 , then sup n∈N {H(Ω n )} = sup n∈N {H(Ω ′ n )}. Proof. Let us suppose that sup n∈N {H(Ω n )} > sup n∈N {H(Ω ′ n )}. There exists an index n 1 such that H(Ω n1 ) > sup n∈N {H(Ω ′ n )}. Now let r > 0 be such that B(x 0 , r) ⊂ Ω n1 ; since (Ω ′ n ) n is decreasing to x 0 , there exists an index n 2 such that Ω ′ n2 ⊂ B(x 0 , r). One thus have H(Ω ′ n2 ) ≥ H(Ω n1 ) > sup n∈N {H(Ω ′ n )},
which leads to a contradiction.

Definition 17 If f is a bounded function, the local irregularity exponent of f at x 0 is H(x 0 ) = sup n {H(Ω n )},
where (Ω n ) n is a sequence of open sets decreasing to x 0 .

Wavelet criteria for the uniform irregularity and characterization of the local irregularity exponent

In this section, we first give necessary and sufficient conditions for a function to belong to I α (Ω). Next we characterize the local irregularity exponent in terms of wavelet coefficients.

Wavelet criteria for the uniform irregularity

In what follows, we will assume that the multiresolution analysis is compactly supported (see [START_REF] Daubechies | Orthonormal bases of compactly supported wavelets[END_REF]). The following result is shown in [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF]: in R, if the wavelet basis belongs to C M (R), there exist a fast decaying function Ψ M such that ψ = ∆ M 1/2 Ψ M . Furthermore, the function Ψ M can be picked compactly supported with support included in this of ψ. In R d , we will use the tensor product wavelet basis (see [START_REF] Meyer | Ondelettes et opérateurs[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]),

ψ (i) (x) = Ψ (1) (x 1 ) • • • Ψ (d) (x d ),
where Ψ (i) (i ∈ {1, . . . , d}) are either ψ or φ, but at least one of them must equal ψ. We will also use the following notations: given j ∈ N, Ω an open subset of R d and a family of wavelets ψ (i) j,k , we set

I j = {(i, k) : supp(ψ (i) j,k ) ⊂ Ω} and c (•) j,• ℓ ∞ (Ω) = sup Ij |c (i) j,k |.
The uniform regularity of a function is related to the decay rate of its wavelet coefficients (see [START_REF] Meyer | Ondelettes et opérateurs[END_REF]). Let f be a bounded function and α ∈ (0, 1); f belongs to Ċα (Ω) (where Ċα (Ω) denotes the homogeneous version of the Hölder space C α (Ω)) if and only if there exists C > 0 such that for any (j, k) such that for any j ≥ 0, c

(•) j,• ℓ ∞ (Ω) ≤ C2 -αj . (5)
The following result gives a sufficient condition to belong to I α (Ω).

Theorem 1 Let α > 0, f ∈ C α (Ω) and set M = [α] + 1. If there exists C > 0 and γ > 1 such that, for any j ≥ 0, max{ sup

j≤l≤j+log 2 j c (•) l,• ℓ ∞ (Ω) , 2 -jM sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (Ω) )} ≥ C2 -jα j γ , ( 6
) then f ∈ I α (Ω).
Now, if f belongs to I α (Ω), there exist C > 0 and β ∈ (0, 1) such that for any integer j ≥ 0, max{ sup

j≤l≤j+log 2 j c (•) l,• ℓ ∞ (Ω) , 2 -jM j β sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (Ω) )} ≥ C2 -jα . (7)
Proof. To prove the first part of the theorem, let us suppose that f ∈ C α w (R d ) and let C > 0. As shown in [START_REF] Clausel | Some prevalent results about strongly monoHölder functions[END_REF], there exists some increasing sequence of integers (j n ) n such that for any n ∈ N and any j ≥ j n , sup

|h|≤2 -j ∆ M h f L ∞ (Ω h ) ≤ C2 -jnα . ( 8
)
Let us show that this inequality leads to a contradiction.

For the sake of simplicity, let us suppose that Ψ (1) = ψ (let us recall that one of the Ψ (i) is ψ). We then have, by definition of the wavelet coefficients, for any j ≥ 0 and any (i, k) ∈ I j ,

c (i) j,k = 2 jd R d f (x)Ψ (1) (2 j x 1 -k 1 ) • • • Ψ (d) (2 j x d -k d ) dx = 2 jd R d f (x)∆ M 1/2 Ψ M (2 j x 1 -k 1 ) • • • Ψ (d) (2 j x d -k d ) dx = 2 jd R d ∆ M 1/2 j+1 e1 f (x)Ψ M (2 j x 1 -k 1 ) • • • Ψ (d) (2 j x d -k d ) dx,
where e 1 = (1, 0, . . . , 0). Using the assumptions on the support of Ψ M and the definition of I j , one has, for any n ∈ N, any j ≥ j n and any (i, k) ∈ I j ,

|c (i) j,k | ≤ 2 jd Ω |∆ M 1/2 j+1 e1 f (x)||Ψ M (2 j x 1 -k 1 ) • • • Ψ (d) (2 j x d -k d )| dx ≤ C2 jd 2 -jnα R d |Ψ M (2 j x 1 -k 1 ) • • • Ψ (d) (2 j x d -k d )| dx = C2 -jnα Ψ M ⊗ • • • ⊗ Ψ (d) L 1 (R d ) ,
by using Relation [START_REF] Clausel | Some prevalent results about strongly monoHölder functions[END_REF].

For n ∈ N, let l n = j n + γ[log 2 j n ]; for n sufficiently large, one has l nlog 2 l n ≥ j n . Therefore, the following relations hold for n sufficiently large (and any (i, k) ∈ I j ), sup

ln≤j≤ln+log 2 ln |c (i) j,k | ≤ C2 -jnα ≤ C2 -lnα l γ ′ n and sup ln-log 2 ln≤j≤ln 2 jM |c (i) j,k | ≤ C2 lnM 2 -jnα ≤ C2 lnM 2 -lnα l γ ′ n ,
which is in contradiction with the relation [START_REF] Clausel | Quelques notions d'irrégularité uniforme et ponctuelle : le point de vue ondelettes[END_REF].

To prove the second part of the theorem we will use the following result (see [START_REF] Meyer | Ondelettes et opérateurs[END_REF]

): Let f ∈ C γ (Ω); since C γ (Ω) ⊂ B 0 ∞,∞ (Ω) ∩ Ċγ (Ω)
, the wavelet characterizations of these two functional spaces lead to the existence of a constant C > 0 that does not depend on the function such that for any h ∈ R d and any x ∈ Ω h , one has

|∆ [γ]+1 h f (x)| ≤ C sup j∈N c (•) j,• l ∞ (Ω) , (9)
and

|∆ [γ]+1 h f (x)| ≤ C|h| γ sup j∈N {2 jγ c (•) j,• l ∞ (Ω) }. (10)
Let us assume f ∈ I α (Ω) and that Property ( 7) is not satisfied. In this case, for any C > 0 and any β ∈ (0, 1), there exists an increasing sequence of integers (j n ) n such that, for any n ∈ N, max{ sup

jn≤l≤jn+log 2 jn c (•) l,• ℓ ∞ (Ω) , 2 -jnM j β sup jn-log 2 jn≤l≤jn (2 lM c (•) l,• ℓ ∞ (Ω) )} ≤ C2 -jnα . (11)
Let us fix C > 0, x ∈ Ω h and let

n 0 ∈ N, h ∈ R d be such that |h| ≤ 2 -jn0
. By definition of I α (Ω), we have to show that f ∈ C α w (Ω). We will use the following notations:

f -1 = k∈Z d C k φ(• -k), f j = 2 d -1 i=1 k∈Z d c (i) j,k ψ(2 j • -k),
with j ≥ 0. Since f is uniformly Hölder, f j and j≥-1 f j converge uniformly on any compact set and

∆ M h f = j≥-1 ∆ M h f j .
We first consider the function

g 1 = jn 0 j=-1 f j . Let us fix γ ∈ ([α] + 1 - β, [α] + 1
). The regularity of the wavelets and property [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] imply the existence of C > 0 not depending on n 0 , x and h such that

|∆ M h g 1 (x)| ≤ C|h| γ sup l≤jn 0 2 lγ c (•) l,• ℓ ∞ (Ω) . (12)
Since f ∈ C α (Ω), there exists a constant such that, for any n ∈ N,

sup l≤jn 0 -log 2 jn 0 (2 lγ c (•) l,• ℓ ∞ (Ω) ) ≤ C2 (jn 0 -log 2 jn 0 )(γ-α) = C ′ 2 jn 0 (γ-α) j γ-α n0 .
On the other hand, we have, using relation [START_REF] Dispa | Beyond Besov spaces, S ν spaces: Topology and prevalent properties[END_REF],

sup jn 0 -log 2 jn 0 ≤l≤jn 0 (2 lγ c (•) l,• ℓ ∞ (Ω) ) = sup jn 0 -log 2 jn 0 ≤l≤jn 0 (2 l(γ-M) 2 lM c (•) l,• ℓ ∞ (Ω) ) ≤ C2 (jn 0 -log 2 jn 0 )(γ-M) 2 jn 0 (M-α) /j β n0 = C ′ j M-γ n0 2 jn 0 (γ-α) /j β n0 ≤ C ′ 2 jn 0 (γ-α) .
This implies that for n 0 sufficiently large (since 0 < Mγ ≤ β),

sup l≤jn 0 (2 lγ c (•) l,• ℓ ∞ (Ω) ) ≤ C2 jn 0 (γ-α) ,
and hence, using inequality [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF],

|∆ M h g 1 (x)| ≤ C2 -jn 0 α , (13)
since h has been chosen adequately.

Let us now consider g 2 = j>jn 0 f j . Property (9) applied to g 2 directly gives the following relation,

|∆ M h g 2 | ≤ C sup l>jn 0 ( c (•) l,• l ∞ (Ω) ). (14)
Once again, since f ∈ C α (Ω), we have sup

l≥jn 0 +log 2 jn 0 c (•) l,• ℓ ∞ (Ω) ≤ C2 -(jn 0 +log 2 jn 0 )α = C ′ 2 -jn 0 α j α n0 . Now, Relation (11) implies sup jn 0 ≤l≤jn 0 +log 2 jn 0 c (•) l,• ℓ ∞ (Ω) ≤ C2 -jn 0 α .
These last inequalities lead to the following relation for n 0 sufficiently large,

|∆ M h g 2 (x)| ≤ C2 -jn 0 α , (15)
thanks to relation [START_REF] Halsey | Fractal measures and their singularities: The characterization of strange sets[END_REF]. Putting relations ( 13) and ( 15) together, we obtain

|∆ M h f (x)| = |∆ M h (g 1 + g 2 )(x)| ≤ C2 -jn 0 α
for n 0 sufficiently large, that is f ∈ C α w (Ω), which is impossible, since f ∈ I α (Ω).

Characterization of the local irregularity exponent

The preceding result leads to the following characterization of the irregularity exponent.

Corollary 1 Let α > 0; if f ∈ C α (Ω), then the irregularity exponent of f on Ω equals α if and only if

lim j→∞ log 2 max{ sup j≤l≤j+log 2 j c (•) l,• ℓ ∞ (Ω) , 2 -jM sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (Ω) )} -j = α,
where M = [α] + 1.

Proof. Theorem 1 directly yields that if f ∈ C α (Ω), the irregularity exponent of f on Ω equals α if and only if lim sup j→∞ log 2 max{ sup

j≤l≤j+log 2 j c (•) l,• ℓ ∞ (Ω) , 2 -jM sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (Ω) )} -j = α.
We have to prove that the lim sup can be replaced by a limit. Since f ∈ C α (Ω), the relation [START_REF] Christensen | On sets of Haar measure zero in Abelian Polish groups[END_REF] implies that lim inf j→∞ log 2 max{ sup

j≤l≤j+log 2 j c (•) l,• ℓ ∞ (Ω) , 2 -jM sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (Ω) )} -j = α.
is always larger than lim inf j→∞ log 2 max{sup j≤l≤j+log 2 j 2 -lα , 2 -jM sup j-log 2 j≤l≤j 2 l(M-α) } -j = α, which is sufficient to conclude.

We can now state the local version of the previous result.

Theorem 2 Let α > 0; if f ∈ C α (Ω) and x 0 ∈ R d , then H(x 0 ) = α if and only if lim r→0 lim j→∞ log 2 max{ sup j≤l≤j+log 2 j c (•) l,• ℓ ∞ (B(x0,r)) , 2 -jM sup j-log 2 j≤l≤j (2 lM c (•) l,• ℓ ∞ (B(x0,r)) )} -j = α,
where M = [α] + 1.

A prevalent result on the S ν spaces

Prevalence supplies a natural definition of "almost every" which is translation invariant and where no specific measure plays a particular role. To prove our prevalent result, we first need to introduce the method we will use. We will also need some properties obtained in [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF].

The stochastic process technique

Our result concerning the prevalence relies on the stochastic process technique.

Let us recall that a random element X on a complete metric space E is a measurable mapping X defined on a probability space (Ω, A, P ) with values in E. For a random element on E, one can define a probability on E by the formula

P X (A) = P {X ∈ A}.
Replacing the measure µ in Definition 10 of a Haar-null set with µ = P X , we see that in order to prove that a set is Haar-null, it is sufficient to check that for any f ∈ E, P X (A + f ) = 0.

The stochastic process that we will use here is a random wavelet series associated in a proper to ν. To this end, for each j ≥ 0, let us define as in [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF],

F j (α) = 0 if α < α min , 2 -jd sup{j 2 , 2 jν(α) } if α ≥ α min . ( 16
)
Since F j is non-decreasing and piecewise continuous, it is the repartition function of some probability distribution associated to a probability law ρ j supported on [α min , ∞], where α min = inf{α : ν(α) ≥ 0}.

The following remark is made in [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF]. If ρ j is the probability distribution whose repartition function F j is defined by ( 16), then there exists some sequence of random numbers (c λ ) λ∈Λ with independent phase and moduli such that for any j, ρ j is the common law oflog 2 |c (i) j,k |/j and satisfies the two following conditions:

∀α ∈ R, lim j→∞ 2 jd ρ j ((-∞, α]) j = ν(α), (17)
and ∀α ≥ α min ,

2 jd ρ j ((-∞, α]) ≥ j 2 . ( 18
)
Starting from these results, we will use the following random wavelet series associated to ν,

X ν = λ∈Λ c λ ψ λ . (19)
It is shown in [START_REF] Aubry | Topological properties of the sequences spaces S ν[END_REF] that the metric topology d(•, •) on S ν makes it a Polish space, which is a very good framework for prevalence. Moreover, as proved in [START_REF] Aubry | Prevalence of multifractal functions in S ν spaces[END_REF], the measure P X is a Borel measure (relatively to this topology). In other words, we can use the stochastic process technique with X ν .

Prevalent irregularity properties in S ν

We are now ready to prove the following result. From this point of view, the "typical elements" of S ν are multifractal and satisfy a weak uniform irregularity property.

From what precede, it is sufficient to show the following result.

Proposition 1 Let X ν be the random wavelet series defined by [START_REF] Jaffard | Wavelet techniques in multifractal analysis, fractal geometry and applications[END_REF]. Then, for any f ∈ S ν , the local irregularity exponent H(x) of f + X ν at x is equal to α min almost surely. (1 -1 -2 -2 d(j-m)exp(-2 -md 2 jν(αmin+1/n) )

2 md
) < ∞.

Therefore, the Borel-Cantelli lemma implies that for any n, there exists m 0 ∈ N such that, for any m ≥ m 0 , any r and any j ≥ m, the inequality

Theorem 3

 3 The following set is prevalent in S ν , {f : ∀x ∈ T d , the local irregularity exponent of f at x is H(x) = α min }, where α min = inf{α : ν(α) ≥ 0}.

  Proof. Since S ν ⊂ C αmin (T d ) then for any f ∈ S ν , f + X ν ∈ C αmin (T d ) almost surely. Let us now fix m ∈ N and define, for any r ∈ Z d , that T d = ∪ r T r,m . We aim at showing that the equality H(T r,m ) = α min holds almost surely for any r, m; in this case, Theorem 2 directly yields the required result.For λ ∈ Λ, we will denote as usual c λ the wavelet coefficients associated to X ν and d λ the wavelet coefficients associated to f ∈ S ν . Let us first remark that if for some fixed λ ∈ Λ, ℜ(c λ d λ ) ≤ 0 i.e. c λ is in the complex half-plane opposite to d λ , then|c λd λ | ≥ |c λ |Therefore, for any λ ∈ Λ,P (f : |c λd λ | ≥ |c λ |) ≥ P (f : ℜ(c λ d λ ) ≤ 0) ≥ 1/2. (20)If we define for any N ∈ N and λ = (i, j, k) ∈ Λ,A N,λ = {f : (∃k ′ ∈ k + [0, N ] d : |c (i) j,k ′d (i) j,k ′ | ≥ |c (i) j,k ′ |)},then, thanks to the independence of the wavelet coefficients of X ν and inequality (20), we haveP (A N,λ ) ≥ 1 -2 -N dfor any N and λ. Now, for N ∈ N, let us setB r,m,n = {f : c (•) j,• ℓ ∞ (Tr,m) ≥ 2 -j(αmin+1/n) }.If for any i, we have supp(ψ (i) ) ⊂ [0, M ] d , then supp(ψ λ ) ⊂ T r,m if and only if, for any l ∈ {1, . . . , d}, we have r l 2 j-m ≤ k l ≤ (r l + 1)2 j-m -M . Therefore, for any fixed j ≥ m,P (B r,m,n ) = 1 -(1 -2 j(ν(αmin+1/n)-d) ) 2 d(j-m) ≥ 1exp(-2 -md 2 jν(αmin+1/n) ).Let us choose N = 2 j-m ; for any m, n and any j ≥ m, we haveP (f : ℓ ∞ (Tr,m) ≥ 2 -j(αmin+1/n) )≥ P (f : ((i,k)∈Ij A 2 j-m ,λ ) ∩ B 2r,m+1,n ) 2 m≥ 1 -2 -2 d(j-m)exp(-2 -md 2 jν(αmin+1/n) )2 md.Moreover, for any n, m j≥m

  ℓ ∞ (Tr,m) ≥ 2 -j(αmin+1/n)[START_REF] Mallat | A wavelet tour of signal processing[END_REF] holds almost surely. Using inequality[START_REF] Mallat | A wavelet tour of signal processing[END_REF], we see that for any n ∈ N, ℓ ∞ (Tr,m) } is larger than 2 -j(αmin+1/n) for any r, m almost surely. Moreover, since X ν +f ∈ C αmin (T d ) almost surely, we also almost surely have that, for any j and any r, m, ℓ ∞ (Tr,m) } is lower than 2 -jαmin . These two bounds allows us to say that for any n ∈ N, ℓ ∞ (Tr,m) } -j almost surely belongs to [α min , α min + 1/n], for any r, m. Since this relation is valid for any n, Corollary 1 allows to conclude.