Top: Original signal (left), superimposed with noisy version (right). Bottom: Denoising using the hard-thresholding wavelet (left) and Hölder exponent (right) schemes. Thresholding oversmooths and introduces oscillations (results obtained with FracLab).

Definition of the pointwise Hölder exponent

Definition: The pointwise Hölder exponent at x 0 of a continuous function f : R → R is the number α = α f (x 0 ) such that :

• ∀γ < α, lim h→0 |f (x 0 +h)-P (h)| |h| γ = 0,
• if α < +∞, ∀γ > α, lim sup h→0 |f (x 0 +h)-P (h)| |h| γ

= +∞

where P is a polynomial of degree not larger than the integer part of α.

(This definition has to be adapted if α is an integer.)

Particular case of continuous non-differentiable functions

When 0 < α < 1, equivalent definitions read:

α = sup{β : ∃c, ρ 0 > 0, ∀ρ < ρ 0 sup x,y∈B(x 0 ,ρ) |f (x) -f (y)| ≤ c ρ β } or α = lim inf h→0 log |f (x 0 + h) -f (x 0 )| log |h|
The intuitive notion of pointwise Hölder Exponent f (x 0 )

x 0 ±c|x -x 0 | α α has certain nice properties:

• (NP1) It captures to some extent the intuitive perception of regularity.

• (NP2) It is a versatile tool: The pointwise Hölder function of a continuous function f (i.e. the function x → α f (x)) is able to record fine changes in the regularity of f . 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5

Generalized Weierstrass Function t NP1: A signal with α(t) = α l (t) = t for all t in [0, 1] (signal obtained with FracLab).

NP2: The structure of the pointwise Hölder function Theorem (Daoudi/Lévy Véhel/Meyer; Jaffard):

Let s be a function from [0, 1] to [0, 1]. The following conditions are equivalent:

• s is the pointwise Hölder function of a continuous function.

• There exists a sequence of continuous functions (s n ) n≥1 such that:

s(x) = lim inf n→∞ s n (x)
Example: s = 1l Q (x) is the pointwise Hölder function of a continuous function, while 1 -s is not.

α p has certain undesirable properties:

• α p (x) does not contain all the regularity information.

Ex. :

"cusp function":

|x| γ γ > 0 α p (0) = γ "chirp function": |x| γ sin 1 |x| β γ > 0, β > 0 α p (0) = γ
• α p is not stable through the action of (pseudo-)differential operators. For instance, one can only ensure in general that: This means for instance that:

α X p ≤ α X p -1 (chirp : α X p (0) = α X p (0) -1 -β) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.1 -0.
• The multifractal spectrum does not give a complete description of the regularity structure of a function/measure.

• The multifractal spectrum of X cannot be deduced from the one of X.

-→ We need to supplement α p with additional regularity exponents.

The local Hölder exponent

Definition (Guiheneuf/Lévy Véhel):

Let

α l (x, ε) = sup{α : f ∈ C α (B(x, ε))}
where B(x, ε) is the ball centered at x with radius ε. Then the local Hölder exponent of f at x is

α l (x) = lim ε→0 α l (x, ε)
When 0 < α < 1, this is equivalent to:

α l (x 0 ) = sup{β : ∃c, ρ 0 > 0, ∀ρ < ρ 0 sup x,y∈B(x 0 ,ρ) |f (x) -f (y)| ≤ c |x -y| β }
The local Hölder Exponent: Pros and Cons

Advantages of the local Hölder exponent:

• α l is sensitive to oscillations:

α l (0) = γ 1+β for f (x) = |x| γ sin 1 |x| β • α l is stable through integro-differentiation
Drawback of the local Hölder exponent:

The evolution of the local Hölder function x → α l (x) is rather constrained: In particular that the local Hölder exponent will be constant for many "interesting" multifractal functions/measures.

The structure of the local Hölder function

Theorem (Guiheneuf/Lévy Véhel):

Let s l be a function from [0, 1] to [0, 1]. The following conditions are equivalent:

• s l is the local Hölder function of a continuous function.

• s l is lower semi-continuous.

Thus, one cannot detect a "regular" point in an "irregular" background with the help of α l

Relation between the pointwise and local Hölder functions

Theorem (Lévy Véhel/Seuret):

Let s be a function from [0, 1] to [0, 1]. Assume s is the liminf of a sequence of continuous functions. Let s l be a lower semi-continuous function. Assume that s l ≤ s. Then there exists a continuous function f such that:

• α l (x) = s l (x) ∀x • α(x) = s(x) ∀x / ∈ E
, where E is an everywhere dense set with Hausdorff dimension equal to 0.

Theorem (Lévy Véhel/Seuret):

Let f be a continuous function. Then its local and pointwise Hölder functions coincide at least on an everywhere dense set with Hausdorff dimension equal to 0. A signal with α(t) = α l (t) for all t outside a set of zero Hausdorff dimension (signal obtained with FracLab).

Other regularity exponents

The couple (α p , α l ) does not encompass all the local regularity information. Other exponents have been defined, which include:

• the oscillating exponent,

• the weak scaling exponent,

• and the chirp exponent.

The oscillating exponent Definition (Jaffard): Let f ∈ L 1 loc (R). Denote by α t (x 0 ) the pointwise Hölder exponent of a fractional integral of order t of f at x 0 . Then the oscillating exponent of f at x 0 , denoted β o , is defined as:

β o (x 0 ) = ∂ ∂t α t (x 0 ) t=0 + -1.
Thus β o + 1 measures the rate of increase of α through infinitesimal integration.

It is easy to show that β o is always non-negative.

"Nice" functions, such as the cusp |x| γ have β o = 0.

Functions that "oscillate", such as the chirp |x| γ sin 1 |x| β , have β o > 0.

A function f is said to be oscillating at

x 0 if β f o (x 0 ) > 0.
The weak scaling exponent Definition (Meyer)

Let f be a function in L 1 loc (R), and denote by f (-l) a primitive of f of order l. Then

β w (x 0 ) = sup{s : ∃n, f (-n) ∈ C s+n x 0 }.
This exponent thus characterizes the asymptotic behaviour of the function after a large number of integrations.

The chirp exponent

Definition (Meyer)

Let f be a function in L 1 loc (R), and denote by f

(-l) a primitive of f of order l. f is called a (h, β c )-type chirp at x 0 if ∀n ∈ N, f (-n) ∈ C h+n(1+β c ) x 0 .
This exponent measure, in a very precise sense, the oscillatory content of the function.

Examples • f (x) = |x| γ : β w (0) = α l (0) = α(0) and β c (0) = β o (0) = 0. • Weierstrass function: β w (x) = α l (x) = α(x) and β c (x) = β o (x) = 0 for all x. • f (x) = |x| γ sin 1 |x| β , γ > 0, β ≥ 0: α(0) = γ, β w (0) = ∞, β c (0) = β o (0) = β and α l (0) = γ 1+β .
• Fractal Interpolation Functions/Multiplicative Cascades:

Pointwise exponent: α is everywhere discontinuous and ranges in an interval [α m , α M ]. Furthermore, all the level sets of the function x → α(x) are everywhere dense.

β exponents:

β w = α, β c = β o = 0 at all points.
Local exponent: α l = α m at all points.

Note that only α l indicates that "something is going on", i.e. that α does not give exhaustive information: Fractal interpolation functions and cascades are not oscillating functions, but they are "multisingular" (see below for a definition) at each point.

Towards 2ml Analysis (I)

In some applications, no combination of these exponents gives sufficient information.

Example: Consider the exponents (α, β o ). This couple measures the current pointwise regularity, and how it will evolve under infinitesimal integration. What about finite order integration or differentiation ?

For instance, a function could be oscillation-free, but its derivative could be oscillating.

Example:

f (x) = x 4.1 + x 4.5 sin(1/x 2 ).

Towards 2ml Analysis (II)

The main point is that the set of exponents defined above do not allow to predict the pointwise Hölder exponent of (fractional) integrals or derivatives of f at a given point (except in simple cases such as the cusp/chirp).

2-microlocal analysis is a clever way to gather the knowledge which is necessary to compute such exponents.

Recalls: Littlewood-Paley analysis

As introduced by J.M. Bony, the concept of 2-ml spaces relies on a Littlewood-Paley analysis:

Let ϕ ∈ S(R), φ = 1 on -1 2 , 1 2 = 0 outside [-, 1] Let f ∈ S (R). The Littlewood-Paley analysis of f is:      S 0 f = ϕ * f ∆ j f = Ψ j * f j ∈ N where Ψ j (x) = 2 j+1 ϕ 2 j+1 x -2 j ϕ 2 j x
One has the fundamental decomposition:

f = S 0 f + ∞ j=0 ∆ j f 2ml Analysis: Original definition Definition (Bony): Let x o ∈ R, (s, s ) ∈ R 2 . f ∈ S (R) belongs to the 2ml space C s s x 0 if: ∃ C : |S 0 f (x)| ≤ C[1 + |x -x 0 |] -s ∀ j ∈ N |∆ j f (x)| ≤ C2 -js [1 + 2 j |x -x 0 |] -s
Fundamental property of 2-ml spaces Theorem (Bony)

f ∈ C s,s x 0 ⇐⇒ f (n) ∈ C s-n,s
x 0

In the sequel, we will be considering "local" versions of 2-ml spaces: By abuse of language, f belongs to the 2ml space C s s x 0 will mean that f coincides on a neighbourhood of x 0 with a distribution g that belongs to C s s x 0 .

Some basic properties of 2-microlocal spaces

∀x 0 ∈ R, • t ≤ s and t + t ≤ s + s ⇒ C s,s x 0 ⊂ C t,t x 0 . • ∀s > 0, C s x 0 ⊂ C s,-s x 0 . • ∀(s, s ) with s + s > 0, C s,s x 0 ⊂ C s x 0 .

Recalls: Wavelet analysis. Discrete case

There exists a wavelet-based characterization of 2-ml spaces. Before presenting it, we recall the definition of the wavelet transform.

Let ψ be a function in the Schwartz class

S(R) such that {ψ j,k (x) = 2 j/2 ψ(2 j x -k)} (j,k)∈Z 2 forms an orthonormal basis of L 2 (R).
The discrete wavelet coefficients of f are defined as:

d j,k = 2 j f (x)ψ(2 j x -k)dx.
One has:

f (x) = (j,k)∈Z 2 d j,k ψ(2 j x -k)
Recalls: Wavelet analysis. Continuous case

Let Ψ be an L 2 (R) function with zero mean, satisfying the "admissibility condition".

The continuous wavelet transform (or coefficients) of f is defined, for a > 0 and b ∈ R, by

C(a, b) = 1 a f (x)Ψ x -b a dx.

2-ml spaces: Wavelet-domain definition

Theorem (Jaffard) A function f : R → R belongs to C s,s x 0 if and only if its m th derivative exists around x 0 , and if there exist δ > 0, a polynomial P of degree smaller than [s] -m, and a constant C, that verify

f ∈ C s,s x 0 ⇐⇒ ∃ C : ∀ j, k : |x 0 -k2 -j | ≤ 1, |d j,k | ≤ C2 -js (1 + |k -2 j x 0 |) -s ⇐⇒ ∃ C : ∀ a > 0, ∀ b, |b -x 0 | ≤ 1, |C(a, b)| ≤ Ca s 1 + |b -x 0 | a -s 2
∂ m f (x) -P (x) |x -x 0 | [s]-m - ∂ m f (y) -P (y) |y -x 0 | [s]-m ≤ C|x -y| s+s -m (|x -y| + |x -x 0 |) -s -[s]+m for all x, y such that 0 < |x -x 0 | < δ, 0 < |y -x 0 | < δ.
Time-domain definition. Particular case of continuous non-differentiable functions When 0 ≤ s + s < 1, s ≥ 0, -1 ≤ s ≤ 1, the characterization takes the much simpler form:

|f (x) -f (y)| ≤ C|x -y| s+s (|x -y| + |x -x 0 |) -s
The 2-microlocal domain and the 2ml frontier

Stating that the Hölder exponent of f at x is α is the same as saying that α is the supremum of the real numbers s such that f belongs to the pointwise Hölder space C s x .

We seek a notion corresponding to α for C s,s spaces. This is more complicated, since there are two exponents, and there is no total order in R 2 .

Definition

The 2ml domain of f at x 0 is the set:

{(s, s ) ∈ R 2 : f ∈ C s,s x 0 } Proposition The 2ml domain is a convex subset of R 2 .

Definition

The 2ml frontier of f at x 0 is the curve

R → R s → s(s ) = sup{u : f ∈ C u,s x 0 }

Proposition

The 2ml frontier is non increasing, concave, with slope everywhere between -1 and 0. Through integration, the blue frontier is translated along the s-axis.

The new pointwise Hölder exponent is obtained by taking the intersection between the new frontier (in black) and the line s = -s .

2-ml frontier parameterization

Since the frontier may be vertical in the (s, s ) plane, one cannot use s as the free variable.

For reasons that will appear shortly, a convenient parameterization is to define the frontier as the curve:

s → σ(s ), where σ = s + s Examples of 2-ml frontiers • f (x) = |x| γ , γ > 0, x 0 = 0. s = -s γ s s • f (x) = |x| γ sin 1 |x| β , γ > 0, β > 0, x 0 = 0. ¡ s = -s γ 1+β (γ, -γ) s s • Weierstrass function W (x) = ∞ n=1 λ -nh sin λ n x. ¡ s = -s s s h ∀ x • Fractal Interpolation Function/Multiplicative cascade (determin- istic multinomial measure) ¡ ¢ £ ¤ ¥ s s α l α p (x) f h (α) α α l α p ∀ x s = -s 2-ml frontier (top), Multifractal spectrum (bottom)
The 2ml Formalism

The conditions to be verified for checking whether a distribution belongs to some 2-ml space are not simple.

It occurs that the 2-ml frontier of any distribution may be obtained as the Legendre transform of function which is easier to compute, called the 2-ml spectrum.

Two different 2-ml spectra may be defined, depending on whether one works in the wavelet or the time domain.

We concentrate here on the wavelet-based 2-ml spectrum.

The 2-ml spectrum: Intuitive idea The basic observation is that both the magnitude and the location of the wavelet coefficients are important to decide whether a distribution belongs or not to some 2-ml space.

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ a x 0 b |b| = a α : C(a, a α ) ∼ a χ(α) |b| = a
The three domains used to define the 2-ml spectrum: (I) "above" the point x 0 (red area), (II) all the curves |b| = a α for α ∈ (0, 1), and (III) the region "below" all such curves (black zone). On a curve |b| = a α , one one define χ(α) as the exponent such that the largest coefficients C(a, a α ) are "of the order" of a χ(α) , with a similar approach for the two other regions.

The 2-ml spectrum: Definition

Let, for 0 < α < 1:

χ ε (α) = lim inf a→0 inf α-ε≤β≤α+ε log |C(a, x 0 ± a β )| log a χ(α) = lim ε→0 + χ ε (α) χ(0) = lim α→0 + [sup{γ : ∃b 0 > 0, a α ≤ (b -x 0 ) < b 0 =⇒ |C(a, x 0 + b)| ≤ K α a γ }] χ(1) = lim α→1 - [sup{γ : ∃b 0 > 0, (b -x 0 ) < min(b 0 , a α ) =⇒ |C(a, x 0 + b)| ≤ K α a γ }] χ : [0, 1] → R + ∪ {+∞} is called the 2-ml spectrum of f at x 0 .

Basic properties of χ

Proposition

Let χ denote the convex envelop of χ. Then: χ(0) = χ(0) and χ(1) = χ(1).

Proposition

The function ρ → χ(ρ) is lower semi-continuous.

Examples of 2-ml spectra

f (x) = |x| γ : χ α ∞ γ 1 f (x) = |x| γ sin 1 |x| β : χ α γ 1+β ∞ 1 1 β+1
Weierstrass function:

α 1 h χ
Fractal interpolation function (the 2-ml spectrum is not convex here and the red line is its convex envelop): (αs + χ(α))

Remark: By inverse Legendre transform, one gets:

-χ(ρ) = inf s ∈R {ρs -σ(s )}.

Discrete-time 2ml spectrum and formalism

One can define a 2-ml spectrum using the discrete wavelet coefficients instead of the continuous ones. The procedure is the same, and one obtains again the 2-ml frontier as the Legendre transform of -χ. In addition, one can show that the spectrum so obtained does not depend on the orthonormal wavelet basis.

Note that χ contains more information that the 2-ml frontier, as χ does not have to be convex. Moreover, this additional information is intrinsic to the analyzed function.

The 2ml spectrum and the regularity exponents

A major advantage of χ is that it allows to compute the exponents very easily:

Proposition Assume α l (x 0 ) > 0. Then: 1. α l (x 0 ) = inf{χ x 0 (ρ) : ρ ∈ [0, 1]}. 2. α(x 0 ) = inf{ χ x 0 (ρ) ρ : ρ ∈ (0, 1]}. 3. β w (x 0 ) = χ x 0 (1) ∈ [0, +∞].
4. If α(x 0 ) < +∞, then β o (x 0 ) is the smallest number β that satisfies

χ x 0 1 β + 1 = α(x 0 ) β + 1 . 5. α l (x 0 ) = α(x 0 ) ⇒ α(x 0 ) = χ x 0 (1) and β o (x 0 ) = 0. 6. α l (x 0 ) = α(x 0 ) ⇒ χ x 0 (ρ) ≥ α(x 0 ), ∀ρ.
Item 2 shows how both the magnitude and the location of the large wavelet coefficients play a role in determining the pointwise Hölder exponent.

Item 4 allows to explain the "ringing" effect observed in wavelet hard-thresholding denoising.

Item 5 is crucial in the correct use of standard estimation procedures.

The 2ml formalism and the multifractal formalism

Recalls: Large deviation multifractal spectrum

For a function X, the large deviation spectrum is defined as:

f g (α) = lim ε→0 + f ε g (α)
where:

f ε g = lim inf n→∞ log N ε n (α) n N ε n (α) = #{k : α -ε ≤ α k n ≤ α + ε} α k n = log |Y k n | -n and: Y k n = increments of X in I k n = [k2 -n , (k + 1)2 -n ], or = oscillation of X in I k
n , or = wavelet coefficient of X at scale n and location k.

Weak multifractal formalism

The (weak) multifractal formalism, when it holds, states that f g is the Legendre transform of τ :

f g (α) = τ * (α)
where

τ (q) = lim inf n→∞ log S n (q) -n S n (q) = 2 n -1 k=0 |Y k n | q
Under mild conditions (for instance supp f g bounded) one always has:

τ (q) = (f g ) * (q)
2ml analysis = "multifractal analysis at one point"

X is said to be multifractal if:

• f g (α) ≥ 0 for more than one exponent α: This means that different regularity behaviours occur along time, and/or

• τ (q) is not a straight line: This means that it is not possible to predict the behaviour of S n (q) from the one of S n (q 0 ) (recall that, heuristically, S n (q) ∼ 2 -nτ (q) ).

Multisingularity

We say that X is multisingular at x if:

• χ x (α) < +∞ for more than one exponent: This means that different decays of wavelet coefficients occur on different curves (b) = x 0 + a α .

• σ x (s ) is not a straight line: This means that one cannot predict the variation of α p through integration of order ε, even if its variation through integration of order ε = ε is known.

Formal correspondences between 2-microlocal and multifractal analysis.

Multifractal Analysis 2-microlocal Analysis

q s or s τ (q) σ(s ) or s (s) α ρ f g (α) χ(ρ) dim B (Supp(µ)) -α l D q = τ (q) q-1 ∆(s ) = σ(s ) s +α Information Dimension α * = D 1 Information Exponent ρ * = σ (-α)
Note that ρ * = 1 1+β o . It has the following intuitive meaning: There always exists a single curve |b -x 0 | = a ρ * , with ρ * = σ (-α), that dictates the rate of increase of α, which equals 1 ρ * .

Graphical comparison between τ and σ, and f g and χ.

-dim

B q 0 τ σ s α l α * α f g χ α 1 α p -α p α l α * dim B

Further links between 2ml and multifractal analysis

Let E α = {x : α(x) = α}, and define the Hausdorff spectrum

f h of X as f h (α) = d H (E α )
, where d H is the Hausdorff dimension.

Proposition

Assume X ∈ C γ for some γ > 0. Then:

• sup x∈R χ x (0) = +∞ ⇒ Supp(f h ) unbounded. • E ∞ = {x : α(x) = +∞} is included in {x : X is monosingular at x}. • supp(f h ) bounded ⇒ X is everywhere multisingular.
Part 2: Stochastic 2-microlocal analysis. The fine regularity of Gaussian processes (joint work with Erick Herbin)

Motivations

• Stochastic processes are heavily used in applications as models for real phenomena : Internet traffic, financial records, medical and radar images, . . . . One would like to characterize the regularity of the models, and how it evolves under various processing.

• 2-ml analysis has been invented to track the singularities of solutions of PDEs. It seems natural to generalize it in the context of Stochastic Differential Equations.

Stochastic 2-microlocal analysis

For simplicity, we shall restrict to the case of continuous nowhere differentiable processes. Recall that in this case, 0 ≤ s + s < 1, s ≥ 0, -1 ≤ s ≤ 1, and the function f belongs to the space C s,s

x 0 iff:

|f (x) -f (y)| ≤ C|x -y| s+s (|x -y| + |x -x 0 |) -s
for all (x, y) in a neighbourhood of x 0 .

The stochastic exponents and the stochastic 2-ml frontier Let X = X t ; t ∈ R N + be a continuous Gaussian process.

Definition

Let t 0 ∈ R N + . The pointwise and local Hölder exponents of X at t 0 are defined as:

α(t 0 ) = sup    α; lim sup ρ→0 sup s,t∈B(t 0 ,ρ) |X t -X s | ρ α < ∞    α l (t 0 ) = sup    α; lim sup ρ→0 sup s,t∈B(t 0 ,ρ) |X t -X s | t -s α < ∞   
The 2-ml frontier of X at t 0 is the random function s → σ t 0 (s ), defined for s ∈ (-∞; 0):

σ t 0 (s ) = sup    σ; lim sup ρ→0 sup s,t∈B(t 0 ,ρ) |X t -X s | t -s σ ρ -s < ∞    For a given realization ω, X t (ω) ∈ C s,s t ⇐⇒ s + s < σ t 0 (s )
.

Deterministic versions of the local exponent, 2-microlocal spaces and 2-microlocal frontier

We seek to obtain results such as: X ∈ C s,s

x 0 almost surely. In that view, we set the following definitions.

Definition: Deterministic local exponent α l (t 0 ) = sup    α; lim sup ρ→0 sup s,t∈B(t 0 ,ρ) E [X t -X s ] 2 t -s 2α < ∞    Definition: Deterministic 2-microlocal space
A Gaussian process X is said to belong to C s,s t 0 for a fixed t 0 ∈ R N + and some s, s such that

     0 < σ = s + s < 1 s < 1 s < 0 if lim sup ρ→0 sup s,t∈B(t 0 ,ρ) E [X t -X s ] 2 t -s 2σ ρ -2s < +∞ Definition: Deterministic 2-microlocal frontier σ t 0 (s ) = sup    σ; lim sup ρ→0 sup s,t∈B(t 0 ,ρ) E [X t -X s ] 2 t -s 2σ ρ -2s < ∞    (1) = sup σ; X ∈ C σ-s ,s t 0
The almost sure 2-microlocal frontier: Results at one point Theorem (Herbin/Lévy Véhel)

Let X = X t ; t ∈ R N + be a continuous Gaussian process. For any t 0 ∈ R N + , the 2-microlocal frontier of X at t 0 is almost surely equal to the function s → σ t 0 (s ).

Corollary

For any t 0 ∈ R N + , the pointwise Hölder exponent of X at t 0 is almost surely equal to -inf{s : σ t 0 (s ) ≥ 0}, provided σ t 0 (0) > 0.

For any t 0 ∈ R N + , the local Hölder exponent of X at t 0 is almost surely equal to σ t 0 (0), provided σ t 0 (0) > 0.

Uniform almost sure results on R N + Theorem (Herbin/Lévy Véhel)

Let X = X t ; t ∈ R N
+ be a continuous Gaussian process. Assume that the function t 0 → lim inf u→t 0 α l (u) is positive. Then, almost surely

∀t 0 ∈ R N + ; lim inf u→t 0 α l (u) ≤ α l (t 0 ) ≤ lim sup u→t 0 α l (u)

Corollary

Let X = X t ; t ∈ R N + be a continuous Gaussian process. Assume that the function t 0 → α l (t 0 ) is continuous and positive. Then, almost surely

∀t 0 ∈ R N + ; α l (t 0 ) = α l (t 0 ) Corollary Let X = X t ; t ∈ R N
+ be a Gaussian process. Assume that the function t 0 → α l (t 0 ) is continuous and positive. Then, almost surely

∀t 0 ∈ R N + , ∀s < 0; α l (t 0 ) + s ≤ σ t 0 (s ) ≤ lim sup u→t 0 σ u (s )
These results yield only bounds for the uniform almost sure 2-microlocal frontier.

In specific cases, we are able to obtain a sharp result: This happens for instance for fractional Brownian motion and regular multifractional Brownian motion.

Application (I): Fractional Brownian motion (fBm)

Fractional Brownian motion was introduced by Kolmogorov and first studied in a fractal framework by Mandelbrot. Many results are known about its regularity (Adler, Kahane,. . . ) Definition: Fractional Brownian motion is the continuous zero-mean Gaussian process B H = B H t ; t ∈ R + such that for all s, t ∈ R + ,

E B H t -B H s 2 = |t -s| 2H (2)
where H ∈ (0, 1).

The 2-ml frontier of fBm A direct application of the results above yield:

Proposition

Almost surely, the 2-microlocal frontier at any t 0 of the fractional Brownian motion in the region -1 < s < 0 0 < σ < 1 + s is equal to the line σ = H + s .

Corollary

The local and pointwise Hölder exponents satisfy almost surely

∀t 0 ∈ R + ; α l (t 0 ) = α(t 0 ) = H Application (II): Multifractional Brownian motion (mBm)
The local regularity of fBm is constant along the paths.

This is boring.

A natural extension is to substitute the constant parameter H with a function t → H(t) taking values in (0, 1). This leads to the definition of multifractional Brownian motion. [(t -s) H(t)-1/2 -(-s) H(t)-1/2 ]dW (s) + t 0 (t -s) H(t)-1/2 dW (s), When H(t) = H for all t, mBm is just fBm of exponent H.

Covariance of mBm

Let X(t) be a standard mBm (i.e. such that the variance at time 1 is 1) with functional parameter H(t). Then: E(X(t)X(s)) = D(H(t), H(s))(t H(t)+H(s) + s H(t)+H(s) -|t -s| H(t)+H(s) ) where D(x, y) = Γ(2x + 1)Γ(2y + 1) sin(πx) sin(πy) 2Γ(x + y + 1)sin(π(x + y)/2)

Main properties of mBm

From the definition, one sees that mBm is a zero mean Gaussian process whose increments are in general neither independent nor stationary.

(recall: fBm has stationary correlated increments for H = 1/2).

The increments of mBm exhibit long range dependance for all H non constant (recall: The increments of fBm have long range dependance for H > 1/2).

Recalls: Long range dependance

Let Y (t) be a stationary second-order process. For all fixed t 0 and ρ > 0, there exist positive constants K(t 0 ) and L(t 0 ) such that, for all s, t ∈ B(t 0 , ρ): This approximation, together with the fact that mBm is Höldercontinuous but not differentiable, allows to compute the almost sure 2-microlocal frontier of the mBm at any point t 0 .

E [X t -X s ] 2 = K(t 0 ) t -
Pointwise almost sure 2-microlocal frontier of mBm

Proposition

The 2-microlocal frontier of the multifractional Brownian motion in the region -1 < s < 0 0 < σ < 1 + s is, at any fixed t 0 , almost surely equal to the "minimum" of the 2-ml frontier of H at t 0 and the line σ = H(t 0 ) + s : σ t 0 (s ) = (H(t 0 ) + s ) ∧ β t 0 (s ) where β t 0 (s ) denotes the 2-microlocal frontier of the deterministic function H at t 0 .

x

  An oscillation-free function whose derivative oscillates.

  -ml spaces: Time-domain definition Theorem (Kolwankar/Levy Vehel/Seuret) Let x 0 ∈ R, and s, s be two real numbers satisfying s+s > 0, s+s ∈ N, and s < 0 (and thus s ≥ 0). Let m = [s + s ].

2

  -ml frontier and exponents in the (s, s )-plane Theorem: Assume α l (x) > 0. Then, the regularity exponents defined above are as indicated on the figure below.

  Let f ∈ S (R). The 2-microlocal frontier of f at any x 0 is the Legendre transform of -χ, i.e.: σ(s ) = (-χ) * (s ) := inf α∈[0,1]

  mBm: Moving Average Representation Definition (Lévy Véhel/Peltier): Let H : [0, ∞) → [a, b] ⊂ (0, 1) be a Hölder function of exponent β > 0. For t ≥ 0, the following random function is called multifractional Brownian motion with functional parameter H

  The correlation function ρ(k) = E(Y (t)Y (t + k)) only depends on the lag k. A possible definition of long range dependence is: Definition Y (t) is said to have long range dependence if +∞ 0 |ρ(k)| = +∞ Long range dependance for non-stationary processes The increments of mBm are not stationary. The definition above maybe adapted as follows: Definition Let Y (t) be a second-order process. Y (t) is said to have long range dependence if ∀δ > 0, ∀s ≥ 0, +∞ 0 |cor Y (s, s + kδ)| = +∞ Behaviour of the covariance of mBm for small increments

  Example 3: Pointwise Hölder Exponent-based signal denoising

	1.2
										1.5
										1.4
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	1									1.2
										1.1
											1
										0.9
	0.8									0.8
										0.7
										0.6
	0	500	1000	1500	2000	2500	3000	3500	4000	4500
										0.5	0	500	1000	1500	2000	2500	3000	3500	4000	4500
	1.2									1.2
	1									1
	0.8									0.8
	0	500	1000	1500	2000	2500	3000	3500	4000	4500	0		500	1000	1500	2000	2500	3000	3500	4000	4500

At any t 0 , the pointwise and local Hölder exponents of the multifractional Brownian motion verify almost surely:

where β(t 0 ) and β(t 0 ) denote the pointwise and local Hölder exponents of H at t 0 .

Uniform almost sure local Hölder exponent of mBm

Under some assumptions on the function H or its regularity, uniform results hold.

Proposition

Let X = X t ; t ∈ R + be a multifractional Brownian motion such that the function t → β(t), where β(t) is the local Hölder exponent of H at t, is continuous on some open interval I. Then the local Hölder exponent of X satisfy almost surely

Uniform almost sure 2-ml frontier of the regular mBm Theorem Let X = X t ; t ∈ R + be a multifractional Brownian motion such that the function H satisfy, for some open interval I,

where β(t) is the local Hölder exponent of H at t. Then, almost surely, the 2-microlocal frontier at any t 0 ∈ I of X in the region

In particular, almost surely, for all t 0 ∈ I, α(t 0 ) = α l (t 0 ) = H(t 0 ).

Case where H verifies some global regularity condition Theorem Assume that, for all t 0 ∈ I,

Then, almost surely, for all t 0 ∈ I,

where β(t 0 ) and β(t 0 ) denote the pointwise and local Hölder exponents of H at t 0 .

Application (III): Linear Stochastic Differential Equations

Consider the simple linear stochastic differential equation (SDE):

where the deterministic functions µ and b satisfy the usual conditions ensuring existence and uniqueness of the solution. Then:

with

Almost sure frontier of the solution

It is easy to deduce the 2-ml frontier of λ and ψ from the ones of µ and b. Using the formula describing the variance of (X t -X s ), one gets:

Proposition

Assume µ(t) = 0 and b(t) = 0. Then, the 2-ml frontier at t of the solution X of the previous SDE verifies almost surely:


Jacques Lévy Véhel
email: jacques.levy-vehel@inria.fr

Claude Tricot
email: ctricot@wanadoo.fr

On Various Multifractal Spectra

Keywords:

We introduce two classes of multifractal spectra, called respectively dimension and continuous spectra. Dimension spectra offer an interesting alternative to the classical Hausdorff spectrum: They are much easier to estimate yet still give relevant information about the geometry of the Hölder function. Continuous spectra are a generalization of the large deviation spectrum that allow to obtain partition free results. Both classes of spectra allow to perform efficient multifractal analysis in an experimental framework.

Background and Notations

Multifractal analysis has developed in many directions since its introduction. Progress has been accomplished concerning the domain of validity of the multifractal formalism, both in the deterministic and random frameworks [START_REF] Arbeiter | Random self-similar multifractals[END_REF][START_REF]Continuity of the multifractal spectrum of a random statistically selfsimilar measure[END_REF][START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Lau | L q spectrum of the Bernouilli convolution associated with the golden ration[END_REF]. The analysis has been extended to functions [START_REF] Jaffard | Multifractal formalism for functions II: selfsimilar functions[END_REF] or sequences of capacities [START_REF] Tricot | Multifractal analysis of Choquet capacities[END_REF] in addition to measures. The paper by P. Mörters in this volume gives an account on the multifractal analysis of certain measures related to Brownian paths. More refined spectra and estimation procedures have been defined [START_REF] Chhabra | Direct determination of the f (α) singularity spectrum[END_REF][START_REF] Véhel | Numerical computation of the large deviation multifractal spectrum[END_REF].

The numerical computation of a multifractal spectrum on sampled data remains however a challenging task. We introduce in this work two new classes of spectra, called dimension spectra and continuous spectra. Our aim is to facilitate the estimation problem, so as to obtain meaningful numerical results even when no assumption on the data structure is made (i.e., in a non-parametric frame).

It is well-known that the Hausdorff multifractal spectrum f h (see section 2 for definitions) is very hard to calculate in general. Apart from very restricted classes of mathematical models, the exact value of the Hausdorff dimension is difficult to obtain theoretically, and almost impossible to estimate on experimental data. The sets E α which form a partition of the support are practically inaccessible in a discrete framework. To the contrary, the box dimension ∆ can be estimated in experimental situations. But ∆ is of no use here since in all interesting situations the E α are either empty, or dense in an interval: In this case ∆(E α ) = 1, and the spectrum f ∆ is trivial. One can try to estimate f h with the help of the large deviation spectrum f g , which is most of the times easier to work with. The inequality f h ≤ f g is always true. However, f g measures an information which is essentially different from f h : The Hausdorff spectrum is defined as a dimension function which emphasizes the geometric structure of the singularities of a function or measure, whilst f g yields statistical information. We introduce in this paper two spectra, denoted f lim d and f lim sup d where d is any dimension. As is the case for f h or f ∆ , they are defined by using set dimensions. On the other hand, they share many properties with f g . For instance, f lim d and f lim sup d are upper semi-continuous functions, and, conversely, every upper semi-continuous function is the f lim d or f lim sup d spectrum for some signal. The major motivation for defining these dimension spectra is that f lim sup ∆ is both easy to evaluate in practical situations and "more precise" than f g , i.e. one always has f h ≤ f lim sup ∆ ≤ f g . Another path consists in focusing on the large deviation spectrum f g . Although originally introduced as a way to estimate f h , it has soon been realized that f g is of independent interest, specially in applications (see for instance [START_REF] Véhel | Introduction to the multifractal analysis of images[END_REF][START_REF] Fisher | Fractional Brownian motion and data traffic modeling: The other end of the spectrum[END_REF] for applications in image processing and Internet traffic modelling). A drawback of the large deviation spectrum is that its very definition relies on an arbitrary partitioning of the support of the signal. Different partitions will in general lead to different spectra. We introduce two variants of f g . Both are continuous spectra, and thus allow to get rid of the discretization. As a consequence, these partitionfree spectra contain a more intrinsic multifractal information. Moreover, one of them, denoted f c g , is defined using only one limit (η → 0), instead of the two limiting operations (n → ∞ and ε → 0) used for f g . We define the corresponding continuous Legendre spectra, and show that they are, under mild conditions, the concave envelopes of the continuous large deviation spectra.

To gain generality, we introduce our spectra for abstract set functions, rather than for Hölder exponents of measures or functions of a real variable, as is done classically. More precisely, let X([0, 1]) be the metric set of all closed sub-intervals (including the singletons) of [0, 1] (the extension to R and R n is straightforward). We shall base our multifractal analysis on the study of a function A : X([0, 1]) -→ R + ∪ {+∞}. The interpretation of A in the classical frame is as follows (the length of an interval u is denoted by |u|):

• For the analysis of a Borelian measure µ, take A(u) = log µ(u)/ log |u|;

• For the analysis of a function z, take A(u) = log v z (u)/ log |u|, where v z (u) measures the "variation" of z in u. or f lim sup d is equal to F ) is solved in Section 4 in two cases: d is σ-stable (like the Hausdorff dimension), or d = ∆. Section 5 shows explicit computations of dimension spectra. Finally, we define and study the continuous spectra f c g and f c g and the corresponding Legendre spectra in Section 6.

New dimension spectra

For x ∈ [0, 1], u n (x) denotes the dyadic interval u k n = [k2 -n , (k + 1)2 -n ] which contains x (take the right one if there are two such intervals). For any real number α, N α (ε, n) denotes the number of 2 -n -dyadic intervals such that |A(u) -α| ≤ ε, and I α (ε, n) their union. Recall the definition of the large deviation spectrum f g :

(1)

f g (α) = lim ε→0 (lim sup n→∞ log N α (ε, n) n log 2 ),
with the convention that log

N α (ε, n)/n log 2 = -∞ if N α (ε, n) = 0.
Prior to defining the dimension spectra, let us clarify our notion of a dimension.

Definition 1. We call dimension a function d

: P([0, 1]) → R + ∪ {-∞}, such that (i) E ⊂ F ⇒ d(E) ≤ d(F ) (ii) d(∅) = -∞, d({x}) = 0 for any x, and d(E) = 1 for any E such that |E| > 0.
A dimension may have the following properties:

Definition 2. The dimension d is stable if d(E ∪ F ) = max{d(E), d(F )} for all sets E, F in [0, 1]. It is σ-stable if d( ∪ n E n ) = sup{d(E n )}
for any countable set family (E n ).

Let us now define our new spectra. For any real number x in [0, 1], set α n (x) = A(u n (x)). For any real α, let

E α (ε, N ) = {x, n ≥ N ⇒ |α n (x) -α| ≤ ε}.
Note that E α (ε, N ) increases with N , so that ∪ N E α (ε, N ) may be written as

sup N E α (ε, N ). Also, E α (ε, N ) = ∩ n≥N I α (ε, n). Let E α (ε) = sup N E α (ε, N ) = {x, ∃N such that n ≥ N ⇒ |α n (x) -α| ≤ ε} = lim inf N →∞ I α (ε, N ).
Since the sets E α (ε) decrease with ε, one may define

E α = lim ε→0 E α (ε) = {x, α n (x)→ n→∞ α}.

Definition 3. For any dimension d and any real α, define the following spectra:

(2)

f d (α) = d(E α ) = d(lim ε→0 sup N E α (ε, N )) (3) f lim d (α) = lim ε→0 d(E α (ε)) = lim ε→0 d(sup N E α (ε, N )) (4) f lim sup d (α) = lim ε→0 sup N d(E α (ε, N )).
When d is the Hausdorff dimension h, f d = f h is the usual Hausdorff spectrum.

Properties and relationships

Let us describe a few basic properties verified by dimension spectra.

Domain of definition

It is clear that the spectra defined above all range in [0, 1]∪{-∞}. Let D = Im(A) be the closure of the image of the function A. For every α ̸ ∈ D, there exists

ε 0 such that ε ≤ ε 0 =⇒ E α (ε, N ) = ∅. Therefore f d (α) = f lim d (α) = f lim sup d (α) = -∞. Also, N α (ε, n) = 0, so that f g (α) = -∞.
Thus, while all the spectra are defined on R, their "support" (i.e. the set of α for which the spectrum belongs to [0, 1]) is included in D.

Inequalities

Since E α ⊂ E α (ε), we have (5) f d (α) ≤ f lim d (α) for all α. Also, d(E α (ε, N )) ≤ d(sup N E α (ε, N )) implies that (6) f lim sup d (α) ≤ f lim d (α
). We will see that there is no relationship in general between f d and f lim sup

d . If d is σ-stable, then d(sup N E α (ε, N )) = sup N d(E α (ε, N )), so that f lim d is identical to f lim sup d .
In this particular case, inequalities (5) and (6) reduce to [START_REF] Jaffard | Multifractal formalism for functions II: selfsimilar functions[END_REF]

f d (α) ≤ f lim d (α) = f lim sup d (α).
The spectra f d and f g cannot be compared without specifying the dimension d.

Let us take for d the box dimension, which is defined for any bounded set E as

∆(E) = lim sup n→∞ log ω(2 n , E) n log 2 where ω(2 n , E) denotes the number of 2 n -dyadic intervals covering E. This dimen- sion is stable, but not σ-stable. For all n ≥ N , ω(2 n , E α (ε, N )) ≤ N α (ε, n), so that for all α, ε, N , ∆(E α (ε, N )) ≤ f g (α). Therefore (8) f lim sup ∆ (α) ≤ f g (α).
There is no relationship in general between f g and f lim ∆ . Lemma 1. Let d 1 , d 2 , be such that for all E ⊂ [0, 1], d 1 (E) ≤ d 2 (E). Then for every A and α:

f d1 (α) ≤ f d2 (α) , f lim d1 (α) ≤ f lim d2 (α) , f lim sup d1 (α) ≤ f lim sup d2 (α).
We leave the proof to the reader. If h denotes the Hausdorff dimension, it is well known that h(E) ≤ ∆(E) for all E. Gathering previous results, we get the following sequence of inequalities: Proposition 1. For any set function A,

(9) f h (α) ≤ f lim sup h (α) = f lim h (α) ≤ f lim sup ∆ (α) ≤ min(f lim ∆ (α), f g (α)
). When the (strong) multifractal formalism holds, f h (α) = f g (α) for all α, so that all the above spectra coincide, with the possible exception of f lim ∆ . Incidentally, this result explains the a priori unexpected fact that a naive numerical estimation of f h on a multinomial measure yields acceptable results: Indeed, estimating simply the box dimension of the sets E α (ε, N ) gives a correct approximation in this case.

More generally, i.e. without assuming the multifractal formalism, [START_REF] Véhel | Numerical computation of the large deviation multifractal spectrum[END_REF] shows that f lim sup ∆ (α) is always a better approximation to f h than f g . In addition, it is not more difficult to estimate.

Maximum of a spectrum

Since d([0, 1]) = 1, every spectrum has a maximum not larger than 1. In general, the upper bound depends on A and d. To make this precise, let us introduce new sets. Let

S(ε, N ) = {x, m ≥ N, n ≥ N ⇒ |α m (x) -α n (x)| ≤ ε}. Note that S(ε, N ) increases with N , so that ∪ N S(ε, N ) = sup N S(ε, N ). Let S(ε) = sup N S(ε, N ) = {x, ∃N such that m ≥ N, n ≥ N ⇒ |α m (x) -α n (x)| ≤ ε}.
Since the sets S(ε) decrease as ε → 0, one may define

S = lim ε→0 S(ε).
These constructions are similar to those of E α (ε, N ), E α (ε), E α , except that they are independent of α. Note that for all ε ′ < ε,

S(ε) ⊂ {x, lim sup α n (x) -lim inf α n (x) ≤ ε} ⊂ S(ε ′ )
and S = {x, α n (x) converges}. Now define the dimensional indices:

d 0 = d(S) = d(lim ε→0 sup N S(ε, N )). d lim 0 = lim ε→0 d(S(ε)) = lim ε→0 d(sup N S(ε, N )) d lim sup 0 = lim ε→0 sup N d(S(ε, N )).
Proposition 2. For every α, Proof 1. We first show that there exists

(10) f d (α) ≤ d 0 , f lim d (α) ≤ d lim 0 , f lim sup d (α) ≤ d lim sup 0 . Proof Use the inclusions E α ⊂ S, E α (ε) ⊂ S(2ε), E α (ε, N ) ⊂ S(2ε, N ).
α 1 ∈ D such that d lim sup 0 ≤ f lim sup d (α 1 ). Let N ∈ N, ε > 0. For any x ∈ S(ε, N ) and any n ≥ N , |α n (x) -α(x)| ≤ ε, where α(x) = 1 2 (lim inf α n (x) + lim sup n α n (x)). Therefore x ∈ E α(x) (ε, N ). This implies that S(ε, N ) ⊂ ∪ α∈R E α (ε, N ). For every α there exists k ∈ Z such that [α-ε, α+ε] ⊂ [2(k-1)ε, 2(k+1)ε], so that ∪ α∈R E α (ε, N ) ⊂ ∪ k∈Z E 2kε (2ε, N ).
If D is bounded, then all sets E 2kε (2ε, N ) are empty but a finite number of them. Using the stability of d, we deduce that

d(S(ε, N )) ≤ max k∈Z d(E 2kε (2ε, N )).
From this inequality it follows that for all ε, N there exists a real number

β(ε, N ) such that d(S(ε, N )) ≤ d(E β(ε,N ) (2ε, N )). Since the distance from β(ε, N ) to D is not more than 2ε, the sequence (β(ε, N )) N has a limiting value β(ε) (for instance β(ε) = lim sup N β(ε, N )). Let N k be a sub- sequence such that β(ε, N k ) → β(ε). If k is large enough, [β(ε, N k ) -2ε, β(ε, N k ) + 2ε] ⊂ [β(ε) -3ε, β(ε) + 3ε], so that E β(ε,N k ) (2ε, N k ) ⊂ E β(ε) (3ε, N k ). Therefore d(S(ε, N k )) ≤ d(E β(ε) (3ε, N k )).
Since the two sides of this inequality increase as k → +∞, we obtain

sup N d(S(ε, N )) ≤ sup N d(E β(ε) (3ε, N )).
The function β(ε) has a limiting value α 1 in D (for instance

α 1 = lim sup ε→0 β(ε)). Let ε i be a sequence such that β(ε i ) converges to α 1 . Let η > 0. For i large enough, ε i ≤ η and |β(ε i ) -α 1 | ≤ η. Therefore [β(ε i ) -3ε i , β(ε i ) + 3ε i ] ⊂ [α 1 -4η, α 1 + 4η].
This implies that for all N ∈ N,

E β(εi) (3ε i , N )) ⊂ E α1 (4η, N ), so that sup N d(S(ε i , N )) ≤ sup N d(E α1 (4η, N )).
When i tends to ∞, the left hand side tends to d lim sup 0 . When η tends to 0, the right hand side tends to f lim sup d (α 1 ). This proves the required inequality.

Let us now show that there exists α

2 ∈ D such that d lim 0 ≤ f lim d (α 2 )
. The proof goes along the same lines, but it is somewhat simpler.

First check that S(ε

) ⊂ ∪ α∈R E α (ε) ⊂ ∪ k∈Z E 2kε (2ε). Deduce that d(S(ε)) ≤ max k d(E 2kε (2ε)). Now choose γ(ε) such that d(S(ε)) ≤ d(E γ(ε) (2ε)), and a limiting value α 2 ∈ D of γ(ε). Let ε i → 0 be such that γ(ε i ) → α 2 . Let η > 0. If i is large enough, show that E γ(εi) (2ε i ) ⊂ E α2 (3η). This implies that d(S(ε i )) ≤ d(E α2 (3η)). Deduce that d lim 0 ≤ f lim d (α 2 ). Particular cases: 1. If d is σ-stable, then the spectra f lim
then sup α f d (α) ≤ d 0 ≤ sup α fd (α).
This result is less precise than those of Proposition 3. The following example shows that the difference between f d and the other spectra may be as large as possible.

Example: Consider the generalized Weierstrass function

W (x) = ∞ ∑ k=1 λ -kx sin(λ k x),
with λ > 1 and x ∈ [0, 1]. It is proved in [START_REF] Tricot | Courbes et dimension fractale[END_REF] that α(x) = x for all x. Setting A(u) = log v W (u)/ log |u|, where v W (u) is the oscillation of W in u, it is easy to check that the support of all spectra is [0, 1]. Since E α contains only one point for every

α ∈ [0, 1], f d (α) = 0 identically in [0, 1]. On the other hand, α n (x) → x for all x ∈ [0, 1], thus d 0 = 1 and f lim sup d (α) = f lim d (α) = f g (α) = 1 for all α in [0, 1].

Semi-continuity

We show in this section that f lim sup d , f lim d and f g share a semi-continuity property. Once again f d does not have this property in general (for a study of the structural properties of f d with d = h, see [START_REF] Tricot | Multifractal analysis of Choquet capacities[END_REF]).

Recall that a function f

: D ⊂ R → R is upper semi-continuous if for all x ∈ D, for all sequences (x n ) of D converging to x, (11) lim sup n→∞ f (x n ) ≤ f (x).
Another way to express this property is as follows:

∀x ∈ D, ∀ε > 0, ∃η(x, ε) : |x -y| ≤ η ⇒ f (y) ≤ f (x) + ε. Proposition 4. The functions f g , f lim d , f lim sup d are upper semi-continuous. Proof 1. Let (α k ) be a sequence in D converging to α. Let ε > 0. For any η, 0 < η < ε, there exists K(η) such that k ≥ K(η) ⇒ [α k -η, α k + η] ⊂ [α -ε, α + ε], so that for any n, N α k (η, n) ≤ N α (ε, n). Therefore, k ≥ K(η) ⇒ lim sup n→∞ log N α k (η, n) n log 2 ≤ lim sup n→∞ log N α (ε, n) n log 2 .
This gives

lim sup k→∞ (lim sup n→∞ log N α k (n, η) n log 2 ) ≤ lim sup n→∞ log N α (n, ε) n log 2 .
Since N α k (n, η) decreases as η tends to 0, we deduce that lim sup

k→∞ ( lim η→0 (lim sup n→∞ log N α k (n, η) n log 2 )) ≤ lim sup n→∞ log N α (n, ε) n log 2 .
Let ε tend to 0 to get

(12) lim sup k→∞ f g (α k ) ≤ f g (α).
2. Similarly, for any N :

k ≥ K(η) ⇒ E α k (η, N ) ⊂ E α (ε, N ).
Letting N tend to ∞, and using the increasing property of d:

k ≥ K(η) ⇒ d(E α k (η)) ≤ d(E α (ε)).
Therefore, lim sup Inequality [START_REF] Echelard | Some properties of the 2-microlocal and large deviation multifractal spectra[END_REF] and Proposition 4 imply that for all α:

k→∞ d(E α k (η)) ≤ d(E α (ε)). Since d(E α k (η)) decreases as η → 0, lim sup k→∞ ( lim η→0 d(E α k (η))) ≤ d(E α (ε)).
(13) fd (α) ≤ f lim d (α). In particular, one always has fh (α) ≤ f lim h (α) ≤ f g .

Spectrum of a maximum Proposition 5.

Let A, B be two functions : X([0, 1]) → R, and C = max{A, B}. Let f g (α, A), f g (α, B), f g (α, C) be the corresponding spectra. Then, for all α,

(14) f g (α, C) ≤ max{f g (α, A), f g (α, B)} Proof Let α n (x) = A(u n (x)), β n (x) = B(u n (x)), γ n (x) = C(u n (x)
). The relation

|γ n (x) -α| ≤ ε ⇒ either |α n (x) -α| ≤ ε or |β n (x) -α| ≤ ε,
valid for all α, n, x, implies the following (with obvious notations):

N C α (n, ε) ≤ N A α (n, ε) + N B α (n, ε), hence relation (14). Proposition 6. If d is stable, the same result as (14) holds for f d , f lim d , f lim sup d .
Proof Using similar notations, we observe that

E C α (ε, N ) ⊂ E A α (ε, N ) ∪ E B α (ε, N ). The stability of d implies d(E C α (ε, N )) ≤ max{d(E A α (ε, N )), d(E B α (ε, N ))}, so that f lim sup d (α, C) ≤ max{f lim sup d (α, A), f lim sup d (α, B)}.
The proof for the other spectra uses similar arguments.

The inverse problem for spectra

We have seen that the three spectra f lim sup d , f lim d and f g are upper semi-continuous. It is natural to enquire whether the converse holds, i.e. whether any u.s.c. function F is the spectrum of some function A. The main result of this section answers by the affirmative for f lim sup d and f lim d . The case of f g is treated in [START_REF] Echelard | Some properties of the 2-microlocal and large deviation multifractal spectra[END_REF].

Generalities on u.s.c. functions

Let us recall some well-known facts. For E ⊂ R, one has: E is closed ⇔ 1 E is u.s.c, where 1 E is the characteristic function of E. If f is an upper semi-continuous function defined on a compact set D and ranging in R, then: -f is bounded from above and reaches its maximum.

-f can be written as f = inf{g ≥ f, g continuous on D}.

We shall need the following result:

Lemma 2. The function f : D → R is u.s.c. iff there exists a countable set E dense in D such that 1. The restriction f |E is u.s.c. 2.
In D -E, f can be obtained by semi-continuity:

∀x ∈ D -E, f (x) = lim ε→0 sup{f (y) : |x -y| ≤ ε, y ∈ E}.
Proof ⇒ Assume that f is u.s.c.. Construct E as follows: In every dyadic interval [k 2 -n , (k + 1)2 -n ], choose a point x k,n where f reaches its maximum. Take

E = {x k,n : k ∈ Z, n ∈ N}. The restriction f |E is u.s.c.. Let x ∈ D -E. For ev- ery n, there exists x k,n in the interval u n (x) such that f (x) ≤ f (x k,n ). Therefore f (x) ≤ sup{f (y)/y ∈ E, |x -y| ≤ 2 -n }. Letting n → +∞, we obtain f (x) ≤ lim sup ε→0 {f (y)/y ∈ E, |x -y| ≤ ε}.
Since f is u.s.c., we obtain an equality. ⇐ It suffices to show that, if (x n ) is a sequence of D -E converging to x (which is necessarily in D), then relation ( 11) is verified. For all x ′ ∈ D and ε > 0, let:

s(x ′ , ε) = sup{f (y)/|x ′ -y| ≤ ε, y ∈ E}.
The function ε → s(x ′ , ε) is non decreasing by construction. Using Assumption 2, s(x n , ε) converges to f (x n ) as ε → 0. Thus, for all n, there exists ε small enough to ensure that f

(x n ) ≤ s(x n , ε) ≤ f (x n ) + 1/2n. There exists y n ∈ E such that |x n -y n | ≤ min{ε, 1/n} and f (y n ) ≤ s(x n , ε) ≤ f (y n ) + 1/2n. Then |f (y n ) - f (x n )| ≤ 1/n. The sequence (y n ) tends to x. If x ∈ E, then f (x) ≥ lim sup n f (y n ) from Assumption 1. If x ∈ D -E, the same result stems from Assumption 2. Then f (x) ≥ lim sup n f (y n ) = lim sup n f (x n ).
Lemma 2 shows that, as is the case for continuous functions, a u.s.c. function f is fully determined by its values on a countable set E. However, contrarily to continuous functions, the set E here depends on f .

Construction of A when d is σ-stable

Let a compact set D ⊂ R, and a semi-continuous function F : D -→ [0, 1] ∪ {-∞} such that sup D F = 1, be given. We want to construct a function A, the f lim d or f lim sup d spectrum of which is equal to F . By semi-continuity, the set F -1 ([0, 1]) is closed. Then we may assume that F (D) ⊂ [0, 1] without loss of generality. The function F can be extended to R by defining F (x) = -∞ if x / ∈ D. Lemma 2 shows that it suffices to match the values of F on a countable set E = {α k } dense in D. On D -E, F is obtained by semi-continuity. We can assume that F (α 0 ) = 1, and that α i ̸ = α j if i ̸ = j. We will construct a function A :

X([0, 1]) -→ R such that (15) f d (α k ) = f lim d (α k ) = f lim sup d (α k ) = F (α k )
for all k ≥ 0. This implies that (16)

f lim d (α) = f lim sup d (α) = F (α) for all α ∈ D. Let (I k ) be a family of closed, non degenerate, disjoint intervals in [0, 1]. For all k ≥ 1, let C k ⊂ I k be a compact set such that d(C k ) = F (α k ). Let C 0 = [0, 1] - ∪ ∞ k=1 C k . Since C 0 contains an open set, d(C 0 ) = 1. For any dyadic interval u, define A(u) = α l(u) , where l(u) = { 0 if u ⊂ C 0 min{k ≥ 1/u ∩ C k ̸ = ∅} otherwise
Since the intervals I k are disjoint, there exists for every k ≥ 1 an integer N k such that for all n ≥ N k , for all

x ∈ C k , u n (x) ⊂ C 0 ∪ C k . Then A(u n (x)) = α k . Therefore E α k = C k and f d (α k ) = F (α k ). For all ε > 0, (17) E α k (ε) = ∪ {C i : |α i -α k | ≤ ε}, so that d(E α k (ε)) ≥ sup{d(C i ) : |α i -α k | ≤ ε}. Since d is σ-stable, this is an equality, so that f lim d (α k ) = lim ε→0 sup{F (α i ) : |α i -α k | ≤ ε}.
The right hand side member is equal to F (α k ). Finally, the set

C k is included in E α k (ε, N ) for all N ≥ N k , so that F (α k ) ≤ f lim sup d (α k ). Using (6), we obtain (15) for k ≥ 1. For k = 0, f lim d (α 0 ) = f lim sup d (α 0 ) = F (α 0 ) = 1.

Construction of A when d = ∆

We will use the same A as before, with extra conditions on the sequences (I k ) and (C k ).

Lemma 3. Given a dyadic interval [0, 2 -N ], and a real number δ ∈ (0, 1), there exists a compact set C such that ∆(C) = δ, and for all n ≥ N :

(18) ω(2 n , C) ≤ 2 (n-N )δ .
Proof For any sequence (ω n ) of integers such that

(19) ω N = 1 , ω n ≤ ω n+1 ≤ 2ω n ,
there are infinitely many ways to construct a compact set C as the limit of embedded coverings by ω n dyadic intervals; such a set verifies ω(2

n , C) = ω n . Let us define ω n = 2 E[(n-N )δ]
, where E[.] denotes the integer part. It is clear that ω n ≤ ω n+1 . On the other hand,

δ ≤ 1 =⇒ (n -N + 1)δ ≤ (n -N )δ + 1, so that E[(n -N + 1)δ] ≤ E[(n -N )δ] + 1, which gives ω n+1 ≤ 2ω n .
Then the conditions given in (19) are fulfilled. This proves the existence of the set C. Finally, (18) is also verified, and

∆(C) = lim sup log ω n n log 2 = δ. Lemma 4. Let I k = [2 -2k , 2 -2k+1 ], δ k ∈ [0, 1], and C k be a compact set included in I k such that ∆(C k ) = δ k and for all n ≥ 2 2k , (20) ω(2 n , C k ) ≤ 2 (n-2k)δ k .
Then for any strictly increasing sequence (k i ) of integers:

(21) ∆( ∪ i C ki ) = sup i ∆(C ki ).
The existence of C k is proved in Lemma 3.

Proof Let δ = sup i ∆(C ki ). The inequality ∆( ∪ i C ki ) ≥ δ is trivial.
For the reverse inequality, choose for any n the integer i n such that

2k in ≤ n ≤ 2k in+1 . Then ∪ i≥in+1 C ki ⊂ [0, 2 -n ],
and

ω(2 n , ∪ i C ki ) ≤ 1 + in ∑ i=1 ω(2 n , C ki ). Using (20), i ≤ i n ⇒ ω(2 n , C ki ) ≤ 2 (n-2ki)δ ≤ 2 nδ . Since i n ≤ n, ω(2 n , ∪ i C ki ) ≤ 1 + n2 nδ . Thus ∆( ∪ i C ki ) ≤ lim sup log(1 + n2 nδ ) n log 2 = δ.
Let us now take (I k ) and (C k ) as in Lemma 4, with δ k = F (α k ), and come back to Equation (17). Using (21) we obtain:

∆(E α k (ε)) = sup{∆(C i )/|α i -α k | ≤ ε}, so that f lim ∆ (α k ) = lim ε→0 sup{F (α i )/|α i -α k | ≤ ε} = F (α k ).
We conclude as before.

Examples of spectra

In this section, we provide various examples of computation of the new spectra. They are meant to show that the inequalities between the spectra may be strict. We choose for d the box dimension, as it is the one most often used in applications.

Example 1: f ∆ (α) < f lim ∆ (α) for all α. Let A([a, b]) = a for all [a, b] ⊂ [0, 1]. Then D = [0, 1], and for all x ∈ [0, 1], α(x) = x. For all N , E α (ε, N ) = E α (ε) = [α -ε, α + ε] , E α = {α}. Therefore ∆(E α ) = 0, ∆(E α (ε)) = ∆(E α (ε, N )) = 1. Finally, N α (ε, n) ≃ 2ε/2 -n . For all α ∈ D, we obtain (22) f ∆ (α) = 0 , f lim ∆ (α) = f lim sup ∆ (α) = 1 , f g (α) = 1.
It is not possible to find a function or a measure whose Hölder regularity is exactly A above. However, it is easily checked that the set function defined on intervals by C([a, b]) = |b -a| a extends to a Choquet capacity on the Borel subsets of [0, 1], to which a multifractal analysis may be applied (see [START_REF] Tricot | Multifractal analysis of Choquet capacities[END_REF]). Alternatively, one may relax the condition A([a, b]) = a. Indeed, the computations above still apply when, for all u ⊂ [0, 1], ( 23)

A(u) + O( 1 log |u| ) ∈ u
uniformly with respect to |u|. This situation is illustrated by the generalized Weierstrass function W (see Section 3.3), with A(u) = log v W (u)/ log |u| and v W (u) the oscillation of W in u. There exists constants C 1 ̸ = 0 and C 2 such that:

∀ u = [a, b], C 1 |u| b ≤ v W (u) ≤ C 2 |u| a .
Then ( 23) is verified, and the spectra take the values shown in (22).

Example 2:

f lim sup ∆ (α) < f lim ∆ (α) for some α. This example shows that f lim sup d may be different from f lim d when d is not σ-stable. Let γ > 0 and F = {k -γ : k ≥ 1}. Then ∆(F ) = 1/(γ +1). The left extremity of u is denoted by u min . Let p : [0, 1] -→ R be a strictly increasing, continuous function such that p(0) = 0. For all u ∈ X([0, 1]), let A(u) = { 0 if u ∩ F ̸ = ∅ and p(|u|) < u min 1 otherwise If x ∈ F , u n (x) ∩ F ̸ = ∅. There exists a smallest integer N (x) such that n ≥ N (x) ⇒ p(2 -n ) ≤ u n (x) min , so that α n (x) = 0. For all ε = 2 -n , the set E 0 (ε, N ) = {k -γ /N (k -γ ) ≤ N } is finite, and E 0 (ε) = F . If x ̸ ∈ F ,
and n is large enough, α n (x) = 1. Let us take two real numbers c < d and an integer K such that

2 -γ < c -2 -K ≤ d + 2 -K < 1. For all x ∈ [c, d] and n ≥ K, then u n (x) ∩ F = ∅, so that α n (x) = 1. Therefore [c, d] ⊂ E 1 (2 -K , N ) for all N ≥ K. We deduce that for all ε < 2 -K , sup N ∆(E 1 (ε, N )) = 1. Therefore, f lim sup ∆ (α) = { 0 if α = 0 1 if α = 1
and

f lim ∆ (α) = f ∆ (α) = { ∆(F ) if α = 0 1 if α = 1 .

These results do not depend on the function p.

To illustrate this case in the frame of classical multifractal analysis, we deal with a slightly more complex but similar situation, and consider the measure defined as

µ = L + ∞ ∑ k=1 a k δ k -γ
where L is the Lebesgue measure on [0, 1], δ x is the Dirac mass at x and (a k ) is a sequence of real numbers decreasing to 0 and such that

∑ ∞ k=1 a k converges. Let A(u) = log µ(u)/ log |u|. If x = k -γ , then µ(u n (x)) = 2 -n + a k if n is large enough. Since a k > 0, α(u n (k -γ
)) → 0. But this convergence is not uniform with respect to k, so that for every N the set E 0 (ε, N ) is finite as before. If x / ∈ F , and x ̸ = 0, then µ(u n (x)) = 2 -n when n large, so that α(u n (x)) → 1. The results on the spectra are the same.

Example 3: f lim ∆ (α) ̸ = f g (α)
for some value of α. We need to show that there is no general relationship between these two spectra. Let β, γ, δ, ω be such that 0 < δ < min{β, 1} and 0 < ω < γ/(γ + 1). Let F = {k -γ } as in Example 2. For all u ∈ X([0, 1]), let

A(u) =    0 if |u| β ≤ u min ≤ |u| δ , or |u| ω < u min and u ∩ F ̸ = ∅ 1 otherwise. The function p(x) of Example 2 is replaced by x ω . Results on the spectra f ∆ , f lim ∆ , f lim sup ∆ are unchanged. Let us compute f g .
The number of dyadic intervals which do not meet F is of the order of 2 n . Therefore f g (1) = 1.

Let us evaluate the number

K n of dyadic intervals u n such that u n ∩F ̸ = ∅ and (u n ) min > 2 -nω . Note that the minimum distance between two points of F in [x, 1] is ρ(x) ≃ k -γ -(k + 1) -γ ≃ k -γ-1 , where k -γ ≃ x. Therefore ρ(x) ≃ x (γ+1)/γ . Letting x = 2 -nω , ρ(2 -nω ) is equivalent to 2 -nω(γ+1)/γ which is larger than 2 -n since ω < γ/(γ +1). We deduce that K n ≃ Card(F ∩[2 -nω , 1]). Since K -γ n ≃ 2 -nω , then K n ≃ 2 nω/γ . Since δ < 1, the number of dyadic intervals u n such that 2 -nβ ≤ (u n ) min ≤ 2 -nδ is K ′ n ≃ 2 n (2 -nδ -2 -nβ ) ≃ 2 n(1-δ) . The number of u n verifying A(u n ) = 0 is equivalent to K n + K ′ n . Therefore f g (α) = { max{ ω γ , 1 -δ} if α = 0 1 if α = 1 .
This result does not depend on β. For a numerical application, take γ = 1, ω = 1/3, δ = 2/3: Then f g (0) = 1/3, f lim ∆ (0) = 1/2. Let now γ = 1, ω = 1/3, δ = 1/3: Then f g (0) = 2/3, f lim ∆ (0) = 1/2. The spectrum f lim ∆ can thus be larger or smaller than f g for some values of α. To exhibit a function whose Hölder regularity is similar to the above function A, we shall use a technique based on wavelets. Fix a wavelet ψ in the Schwartz class such that the functions t → ψ j,k (t) = 2 j/2 ψ(2 j t -k), j, k ∈ Z form an orthonormal basis of L 2 , and define a function z by its wavelet coefficients in this basis, c j,k = 2 j ∫ ψ j,k (t)z(t)dt (note that we use here an L ∞ normalization rather than an L 2 one). We set:

c j,k =    1 if 2 -nβ ≤ k2 -n ≤ 2 -nδ , or 2 -nω < k2 -n and u k n ∩ F ̸ = ∅ 2 -n otherwise where u k n = [k2 -n , (k + 1)2 -n ]. One immediately checks that z is in L 2 if β < γ. Choose now as is usual A(u k n ) = | log(c n,k )|/n
. This yields the desired behaviour. Note that z depends on the wavelet ψ.

Continuous spectra

The definition of the large deviation spectrum f g has two drawbacks: First it depends on the choice of the interval partitions (usually the dyadic intervals), and second it uses two limiting operations (n tends to +∞, then ε tends to 0) which makes it difficult to evaluate from a given set of data (see [START_REF] Véhel | Numerical computation of the large deviation multifractal spectrum[END_REF] for more on this topic). This section introduce variants of f g , denoted by f c g and f c g , and called the continuous large deviation spectra. They are independent of any interval partition. Moreover f c g uses only one limiting operation. These spectra are helpful in numerical applications of multifractal analysis.

As before, A is a function X([0, 1]) -→ R + ∪ {+∞}. Recall that for any set family F, ∪F denotes the union of all sets in F. For any measurable set E of the line, |E| is its Lebesgue measure. Let η ∈ (0, 1). We introduce families of intervals:

R η = {u ∈ X([0, 1]) such that |u| = η}
f c g (α) = lim ε→0 lim sup η→0 log ( | ∪ R ε η (α)|/η ) | log η| = lim ε→0 lim sup η→0 ( 1 - log | ∪ R ε η (α)| log η ) f c g (α) = lim sup η→0 log (| ∪ R η (α)|/η) | log η| = lim sup η→0 ( 1 - log | ∪ R η (α)| log η )
The integer N α (ε, n) used in the definition of f g has been replaced by a mean number | ∪ R ε η (α)|/η in the definition of f c g . In many applications, A is continuous and R η (α) is non empty for a whole range of values of α: This allows to define the spectrum f c g , which avoids the use of the "ε-tolerance".

Recall that, when the so-called weak multifractal formalism holds, f g can be obtained as the Legendre transform of the following function:

τ (q) = lim inf n→∞ log S n (q)
-n where, for all q ∈ R, S n (q) is defined with dyadic intervals as follows:

(24)

S n (q) =

2 n -1 ∑ k=0 2 -nqA(u k n )
with the convention 0 q = 0. One denotes by f l the Legendre transform of τ :

f l (α) := τ * (α) = inf q∈R (qα -τ (q)).
The function f l is called the Legendre multifractal spectrum, and the equality f l = f g , when it holds, is essentially a consequence of Ellis theorem (see [START_REF] Ellis | Large Deviations for a General Class of Random Vectors[END_REF]).

Let us introduce similar notions in a continuous framework. An interval family is a packing if all of its intervals are disjoint. For any q ∈ R and for any interval family R, let

H q (R) = sup{ ∑ u∈R ′ |u| qA(u) , R ′ ⊂ R, R ′ is a packing}.
By convention H q (∅) = 0. Here are some basic properties of H q (R):

Lemma 5. Let R 1 , R 2 be two families of interval. 1. R 1 ⊂ R 2 =⇒ H q (R 1 ) ≤ H q (R 2 ); 2. H q (R 1 ∪ R 2 ) ≤ H q (R 1 ) + H q (R 2 ); 3. If |u| = η and α ≤ A(u) ≤ β for all u ∈ R, then (25) ∀q ≥ 0, 1 2 | ∪ R| η βq-1 ≤ H q (R) ≤ | ∪ R| η αq-1 .
Proof Property 1 comes from the fact that for any packing R 3 ⊂ R 1 we have

R 3 ⊂ R 2 . For Property 2, take a packing R ′ in R 1 ∪ R 2 . Let R ′ 1 = R ′ ∩ R 1 and R ′ 2 = R ′ ∩ R 2 . Then ∑ u∈R ′ |u| qA(u) ≤ ∑ u∈R ′ 1 |u| qA(u) + ∑ u∈R ′ 2 |u| qA(u) ≤ H q (R 1 ) + H q (R 2 ).
For Property 3, it suffices to consider the case where ∪ R is an interval. Let M η be the maximum number of intervals of R covering ∪ R. Show that

ηM η ≤ | ∪ R| ≤ 2ηM η .
Then use the inequalities η βq ≤ |u| qA(u) ≤ η αq for all u ∈ R and q > 0.

To define a continuous counterpart to S n (q), we simply take H q η := H q (R η ). It can be written as

H q η = sup{ ∑ u∈R ′ η qA(u) , R ′ is a packing of [0, 1] by intervals of length η}.
The relevant quantity that corresponds to f c g is:

J q η = sup α H q (R η (α)) = sup α sup{ ∑ u∈R ′ η αq , R ′ is a packing such that |u| = η and A(u) = α}. Definition 5. Let τ c (q) = lim inf η→0 log H q η log η and τ c (q) = lim inf η→0 log J q η log η .
The continuous Legendre spectra are defined as

f c l = (τ c ) * and f c l = (τ c ) * .
Here are some obvious properties of

f c g , f c g , f c l , f c l . Proposition 7. 1. f c l and f c l are concave. 2. ∀ α, f c g (α) ≤ f c g (α) and f g (α) ≤ f c g (α). 3. If µ is a multinomial measure, f c g (α) = f c g (α) = f c l (α) = f c l (α) = f g (α)
. A fundamental property of the continuous spectrum is that, under a mild restriction, f c l is the concave envelope of f c g : Proposition 8.

For all

α, f c l (α) ≥ (f c g (α)) * * and f c l (α) ≥ ( f c g (α)) * * . 2. Assume that f c g = -∞ outside a compact interval. Then: ∀α, f c l (α) = (f c g (α)) * * .
For part 2, we will need a corollary of the next Lemma, which is of independent interest: Lemma 6. Let X (R) be the family of all closed intervals in R. Let F : X (R) → R ∪ {-∞} be such that I ⊂ J ⇒ F (I) ≤ F (J).

For any α ∈ R, define f (α) = lim ε→0 F ([α -ε, α + ε]). Then, for any compact interval J and for any real q, (26) lim

ε→0 inf α∈J {α q -F ([α -ε, α + ε])} = inf α∈J {α q -f (α)}. Proof Fix q ∈ R. Let l ε = inf α∈J {qα -F ([α -ε, α + ε])}, l = inf α∈J {qα -f (α)}. Since F ([α -ε, α + ε]
) decreases to f (α) as ε → 0, l ε increases and is bounded by l. Therefore we must show that l ≤ lim ε l ε . For all ε > 0, there exists α(ε) ∈ J such that (27) q α(ε) -F ([α(ε) -ε, α(ε) + ε]) ≤ l ε + ε.

Let β be a limit value of the sequence (α(1/n)) in the compact J, and r > 0. Let N ≥ 2/r. There exists n ≥ N such that |α(1/n) -β| ≤ r/2. Since 1/n ≤ r/2,

[α( 1 n ) - 1 n , α( 1 n ) + 1 n ] ⊂ [β -ε, β + ε]
so that F ([α(1/n) -1/n, α(1/n) + 1/n]) ≤ F ([β -r, β + r]). We obtain q β -F ([β -r, β + r]) ≤ q(α(

1 n ) + r 2 ) -F ([α( 1 n ) - 1 n , α( 1 n ) + 1 n ]) (28) ≤ l 1/n + 1 n + q r 2 . ( 29
)
Since n can be taken to be arbitrarily large, inequality (29) implies

q β -F ([β -r, β + r]) ≤ lim ε→0 l ε + q r 2 .
When r → 0, F ([β -r, β + r]) tend to f (β), so that q β -f (β) ≤ lim ε→0 l ε .

This result may be applied directly to f g (α, ε) or to f c g (α, ε), when these functions have compact support. Therefore: Corollary 1. If there exists a compact interval J such that f g (α, ε) = -∞ for all α / ∈ J and ε ≤ 1, then

lim ε→0 inf α {qα -f g (α, ε)} = inf α {qα -f g (α)}.
If there exists a compact interval J such that f c g (α, ε) = -∞ for α / ∈ J and ε ≤ 1:

lim ε→0 inf α {qα -f c g (α, ε)} = inf α {qα -f c g (α)}.

Remark:

The corollary remains true even when f c g (α) (resp. f g (α)) does not have a compact support (see [START_REF] Echelard | Some properties of the 2-microlocal and large deviation multifractal spectra[END_REF]).

Proof of Proposition 8

We treat the case q ≥ 0. 1. We shall prove the equivalent statements: τ c (q) ≤ inf α (qα -f c g (α)) and τ c (q) ≤ inf α (qα -f c g (α)).

For all α and ε > 0, R ε η (α) ⊂ R η , so that (30) H q (R ε η (α)) ≤ H q η . Relation (25) implies that (31)

H q (R ε η (α)) ≥ 1 2 | ∪ R ε η (α)| η q(α+ε)-1 .
For all δ > 0, there exists ε 0 > 0 such that for all ε ≤ ε 0 ,

f c g (α) ≤ lim sup η→0 log(| ∪ R ε η (α)|/η) | log η| + δ 2 .
For all η 0 > 0, there exists η ≤ η 0 such that

f c g (α) ≤ log(| ∪ R ε η (α)|/η) | log η| + δ, so that (32) | ∪ R ε η (α)| ≥ η -f c g (α)+δ+1 .
Gathering (30), (31), (32), we get: H q η ≥ 1 2 η q(α+ε)-f c g (α)+δ . Taking the logarithm on both sides and using the definition of τ c as a lim inf: τ c (q) ≤ qα -f c g (α) + qε + δ. Let ε → 0, then δ → 0 to obtain τ c (q) ≤ inf α (qα -f c g (α)). Let us now consider f c g (α). For all α, J q η ≥ 1 2 H q (R η (α)) ≥ 1 2 | ∪ R η (α)| η qα-1 . We deduce that for all δ > 0, η 0 > 0, there exists η ≤ η 0 such that

J q η ≥ 1 2 η qα-f c g (α)+δ .
Therefore τ c (q) ≤ qα -f g (α) + δ. 2. Let ε > 0. Let L ε (q) = inf α {qα -f c g (α, ε)}. For every α there exists η(α, ε) < 1 such that

η ≤ η(α, ε) ⇒ log(| ∪ R ε η (α)|/η) | log η| ≤ f c g (α, ε) + ε.
Using (25):

H q (R ε η (α)) ≤ | ∪ R ε η (α)|η qα-|q|ε-1 ≤ η qα-f c g (α,ε)-ε(1+|q|)
≤ η Lε(q)-ε(1+|q|) .

Let α 1 , ..., α K be a finite sequence such that J ⊂ ∪ i [α i -ε, α i + ε]. For η ≤ min i {η(α i , ε)} : H q η ≤ K η Lε(q)-ε(1+|q|) . Taking logarithms: log H q η log η ≥ L ε (q) -ε(1 + |q|) + log K log η .

Now let η and ε tend to 0 to get τ c (q) ≥ lim ε→0 L ε (q). Use corollary 1 to conclude. The case q < 0 goes along the same lines.

Remark:

The same technique shows that f l = (f g ) * * when f g has compact support.

We end this section with some easily proved properties of τ c and τ c .

d and f lim sup d . 1

 d1 We study their main properties in Section 3: Domain of definition, relationships, maximum. The inverse problem (given an upper semi-continuous function F , find A whose spectrum f lim d This choice requires care, since the resulting spectra will depend on the analyzing wavelet.

Proposition 3 .

 3 If the set D is bounded, and d is a stable dimension, then f lim sup d and f lim d reach the above upper bounds. In other words, and d lim 0 ∈ f lim d (D).

d. 2 .= d lim sup 0 = 1 .

 201 and f lim sup d are the same and d lim 0 = d lim sup 0 If |S| > 0, then |S(ε)| > 0 for all ε and S(ε, N )| > 0 for N large enough. In this case, d 0 = d lim 0 Both f lim d and f lim sup d reach the value 1. This is also the maximum of f g . Remark: Regarding the spectrum f d , one can show the following: If fd denotes the spectrum fd (α) = lim ε→0 d({x/α n (x) converges and | lim α n (x) -α| ≤ ε}),

  Let ε tend to 0 to get lim sup k→∞ f lim d (α k ) ≤ f lim d (α). The same type of arguments hold for f lim sup d . Notation For any f : R → R, we denote by f the upper semi-continuous envelope of f , that is f (α) = lim ε→0 (sup{f (β), |β -α| ≤ ε}).

RDefinition 4 .

 4 η (α) = {u ∈ X([0, 1]) such that |u| = η and A(u) = α} R ε η (α) = {u ∈ X([0, 1]) such that |u| = η and |A(u) -α| ≤ ε}. By convention log 0/ log η = +∞. The continuous large deviation spectra are

Proposition 9. 1. τ c and τ c are increasing and concave.

τ

with µ a probability measure, then τ c (1) = τ c (1) = 0. 4. For any sequence (η n ) tending to zero such that lim n→∞ log η n / log η n+1 = 1, τ c (q) = lim inf n→∞ log H q ηn / log η n and τ c (q) = lim inf n→∞ log J q ηn / log η n . The last property is important for numerical applications, since it allows to evaluate τ c and τ c using discrete sequences such as η n = 2 -n .


J L Fouquet

ON MEASURES OF THE EDGE UNCOLORABILITY OF GRAPHS WITH MAXIMUM DEGREE THREE

Keywords: Mathematics Subject Classification. 05C15. Key words and phrases. Cubic graph, Edge-coloring, 1

Kochol study three invariants of graphs measuring how far a graph is from having a proper 3-edge-coloring. We show here how to get his main result easily.

Introduction

Throughout this note we shall be concerned with connected graphs with maximum degree 3.

Let ϕ : E(G) → {α, β, γ, δ} be a proper edge-coloring of G. It is often of interest to try to use one color (say δ) as few as possible. When an edge coloring is optimal, following this constraint, we shall say that ϕ is δ -minimum. Since any two δ-minimum edge-coloring of G have the same number of edges colored δ we shall denote by s(G) this number (the color number as defined in [START_REF]Measurements of edge-uncolorability[END_REF]).

In [START_REF] Fouquet | Graphes cubiques d'indice chromatique quatre[END_REF] we gave without proof (in French, see [START_REF]Tools for parsimonious edge-colouring of graphs with maximum degree three[END_REF] for a translation) results on δ-minimum edge-colorings of graphs with maximum degree three.

An edge coloring of G with {α, β, γ, δ} is said to be δ-improper whenever we only allow edges colored with δ to be incident. It must be clear that a proper edge coloring (and hence a δ-minimum edge-coloring ) of G is a particular δ-improper edge coloring. For a proper or δ-improper edge coloring ϕ of G, it will be convenient to denote E ϕ (x) (x ∈ {α, β, γ, δ}) the set of edges colored with x by ϕ.

A strong matching C in a graph G is a matching C such that there is no edge of E(G) connecting any two edges of C, or, equivalently, such that C is the edge-set of the subgraph of G induced on the vertex-set V (C).

Results

The proof of the following lemma is given in [START_REF] Fouquet | On parsimonious edge-colouring of graphs with maximum degree three[END_REF] in the context of simple graphs. We extend this result here to graphs with multiple edges (without loops). Lemma 2.1. [START_REF] Fouquet | Contribution à l' étude des graphes cubiques et problèmes hamiltoniens dans les graphes orientés[END_REF] Let ϕ be δ-improper coloring of G. Suppose that E ϕ (δ) contains two edges uv and uw. Then there exists a δ-improper edge coloring

ϕ ′ of G such that E ϕ ′ (δ) = E ϕ (δ) -uv or E ϕ ′ (δ) = E ϕ (δ) -uw Proof If v = w,
u and v are joined by two edges and one color, α, β or γ, at least is not incident to both vertices. By giving this color to one of the edges joining u and v we get a new δ-improper edge coloring ϕ ′ satisfying our conclusion.

We can thus assume that v ̸ = w. If some color in {α, β, γ} is missing in u and v, we can give this color to uv and the result follows. For the same reason, we can assume that the 3 above colors are incident to u and w. Without loss of generality, we can consider that the two colors α and β are incident to v (as well as to w) while u is incident to γ.

Let P be the path alternately colored with α and γ with u as end. If P does not end with v, then we can exchange the colors α and γ on this path leading to a δ-improper edge coloring such that the color γ is missing in u and v. The color γ could be now given to the edge uv leading to a δ-improper edge coloring ϕ ′ satisfying our conclusion. If P ends with v, let P ′ be the path alternately colored with α and γ with w as end. This path does not end with u and an exchange of colors on this path leads to a a δ-improper edge coloring such that the color α is missing in u and w. The color α could be now given to the edge uw leading to a δ-improper edge coloring ϕ ′ satisfying our conclusion.

As in [START_REF] Kochol | Three measures of edge-uncolorability[END_REF] denote by ρ(G) the minimum number of vertices that must be deleted from G so that the resulting graph is 3-edge colorable. We give here a short proof of the main result of Kochol [START_REF] Kochol | Three measures of edge-uncolorability[END_REF].

Theorem 2.2. [5] Let G be a graph with maximum degree 3 then s(G) = ρ(G)

Proof Let X ⊆ V (G) such that G-X is 3-edge colorable and let ϕ : E(G-X) → {α, β, γ} be a 3-edge coloring. We can extend ϕ in a δ-improper edge coloring ϕ ′ of the edge set of G by giving the color δ to the edges incident to the vertices of X. By repeated application of Lemma 2.1, we can find a proper edge coloring of G ϕ

′′ such that E ϕ ′′ (δ) ⊆ E ϕ ′ (δ). Hence |X| ≥ |E ϕ ′′ (δ)| ≥ s(G).
Conversely let ϕ be a δ-minimum edge-coloring of G.

By Lemma 2.1 E ϕ (δ) is a matching. Let X ⊆ V (G)

be a minimal set of vertices intersecting each edge of E ϕ (δ). Then G -X is 3-edge colorable and ρ(G) ≤ |X| ≤ s(G).

The following theorem was first proved by Payan in [START_REF] Payan | Sur quelques problèmes de couverture et de couplage en combinatoire[END_REF].

Theorem 2.3. [2] [6]

Let G be a graph with maximum degree at most 3. Then G has a δ-minimum edge-coloring ϕ where E ϕ (δ) is a strong matching and, moreover, any edge in E ϕ (δ) has its two ends of degree 3 in G.

In [START_REF] Steffen | Classifications and characterizations of snarks[END_REF] Steffen showed that it is always possible to find an independent set of size at most s(G) in a graph with maximum degree 3 whose deletion leaves a 3-edge colorable graph. It can be noticed that this is an immediate corollary of Theorem 2.3. Denote by ρ α (G) the minimum number of vertices of an independent set that must be deleted from G so that the resulting graph is 3-edge colorable.

Theorem 2.4. Let G be a graph with maximum degree

3 then s(G) = ρ α (G) Proof Obviously we have ρ(G) ≤ ρ α (G) and hence s(G) ≤ ρ α (G) by Theorem 2.2.
Conversely let ϕ be a δ-minimum edge-coloring of G such that E ϕ (δ) is a strong matching (Theorem 2.3) and let X ⊆ V (G) be a minimal set of vertices intersecting each edge of E ϕ (δ). Then X is an independent set and G -X is 3-edge colorable. Hence ρ α (G) ≤ |X| ≤ s(G). By Theorem 2.3 an optimal set satisfying Theorem 2.4 can be found with all vertices of degree 3. In fact Theorem 2.4 says that we can find at least 2 s(G) such independent sets (when s(G) > 0).

In [START_REF]Measurements of edge-uncolorability[END_REF] Steffen showed that s(G) ≤ n 8 when G is a bridgeless cubic graph with at least 16 vertices. In fact this result can be easily extended to a subclass of graphs with maximum degree 3. Proposition 2.5. Let G be a bridgeless graph with maximum degree 3 and k ̸ = 1 vertices of degree 2 with at least 16 vertices. Then s(G) ≤ n 8 . Proof By the result of Steffen [START_REF]Measurements of edge-uncolorability[END_REF], we can assume that G has k ≥ 2 vertices of degree 2. Let v 1 . . . v k be these vertices. Let G ′ be a copy of G and v ′ 1 . . . v ′ k be the corresponding vertices of degree 2. Let H be the graph obtained from these two copies by joining the vertices v i and v ′ i (i = 1 . . . k) by an edge. It is an easy matter to see that H is a bridgeless cubic graph. By the result of Steffen [START_REF]Measurements of edge-uncolorability[END_REF], we have s(H) ≤ 2n 8 . Let p (q respectively) be the number of edges colored with δ in the copy of G (G ′ respectively). We obviously have p ≤ q and thus p ≤ 2n 16 . That is s(G) ≤ n 8 as claimed.

Corollary 2.6. Let G be a bridgeless graph with maximum degree 3 and at least 16 vertices. Assume that G has k ̸ = 1 vertices of degree 2. Then s(G) = ρ(G) = ρ α (G) ≤ n