Alexandre Boritchev
email: boritchev@math.polytechnique.fr

ESTIMATES FOR SOLUTIONS OF A LOW-VISCOSITY KICK-FORCED GENERALISED BURGERS EQUATION

Keywords: Burgers Equation, Viscous Conservation Law, Kick Force, Maximum Principle, Turbulence, Intermittency. 1

We consider a non-homogeneous generalised Burgers equation:

Here, ν is small and positive, f is strongly convex and satisfies a growth assumption, while η ω is a space-smooth random "kicked" forcing term.

For any solution u of this equation, we consider the quasistationary regime, corresponding to t ≥ 2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of u. These estimates are of the form Cν -β with the same values of β for bounds from above and from below. They depend on η and f , but do not depend on the time t or the initial condition.

Notation

Consider a zero mean value smooth function w on S 1 . For p ∈ [1, +∞], we denote its L p norm of by |w| p . The L 2 norm will be denoted by |w|, and •, • stands for the L 2 scalar product. From now on, L p , p ∈ [1, +∞] stands for the space of zero mean value functions in L p (S 1 ).

For a nonnegative integer n and p ∈ [1, +∞], W n,p stands for the Sobolev space of zero mean value functions w on S 1 with the norm |w| n,p = w (n) p , where w (n) = d n w dx n . In particular, W 0,p = L p for p ∈ [1, +∞]. For p = 2, we denote W n,2 by H n , and the corresponding norm is abbreviated as w n .

We recall a version of the classical Gagliardo-Nirenberg inequality (see [11, p. 125]). where m > β, and r is defined by

1 r = β + θ( 1 p -m) + (1 -θ) 1 q ,
under the assumption that θ = β/m if p = 1 or p = +∞, and β/m ≤ θ < 1 otherwise. Here C = C(m, p, q, β, θ) > 0 is a constant.

For a smooth function v(t, x) defined on [0, +∞) × S 1 , v t , v x , and v xx mean respectively ∂v ∂t , ∂v ∂x , and ∂ 2 v ∂x 2 .

Introduction

The generalised one-dimensional space-periodic Burgers equation

∂u ∂t + f (u) ∂u ∂x -ν ∂ 2 u ∂x 2 = 0, ν > 0 (1)
(the classical Burgers equation corresponds to f (u) = u 2 ) appears in different domains of science, ranging from cosmology to traffic modelling (see [START_REF] Bec | Burgers turbulence[END_REF]). It is sometimes called a viscous scalar conservation law.

Historically, it has drawn most attention as a model for the Navier-Stokes equation (NSE). Indeed, it has a nonlinear term analogous to the nonlinearity (u • ∇)u in the incompressible NSE. The dissipation term in (1) is also similar to the one in NSE. We note that the classical Burgers equation is explicitly solvable. This is done by the Cole-Hopf transformation (see [START_REF] Biryuk | Note on the transformation that reduces the Burgers equation to the heat equation[END_REF]).

In [START_REF] Biryuk | Spectral Properties of Solutions of the Burgers Equation with Small Dissipation[END_REF], A.Biryuk considered equation (1) with f strongly convex, i.e. satisfying f (x) ≥ σ > 0, x ∈ R.

(2) He studied the behavior of the Sobolev norms of solutions u for small values of ν and obtained the following estimates:

u 2 m ≤ Cν -(2m-1)/2 , 1 T T 0 u 2 m ≥ cν -(2m-1)/2 , m ≥ 1, ν ≤ ν 0 .
Note that exponents of ν in lower and upper estimates are the same. The quantities ν 0 , C, c, and T depend on the deterministic initial condition u 0 as well as on m. To get results independent from the initial data, a natural idea is to introduce random forcing and to estimate ensemble-averaged norms of solutions.

In this article we consider (1) with a random kick force in the righthand side. In Section 3 we recall classical existence and uniqueness results and introduce the probabilistic setting needed to define the kick force. Then, we estimate from above the moments of the W 1,1 norm of u. These estimates, valid after a certain damping time, are proved using ideas similar to those in [START_REF] Kruzkov | The Cauchy Problem in the large for Nonlinear Equations and for certain Quasilinear Systems of the First-order with several variables[END_REF]. Remarkably, this damping time and the estimate do not depend on the initial condition. This is the crucial result of this article.

Next, in Sections 4 and 5, this result allows us to obtain lower and upper estimates that are, up to taking the ensemble average, of the same type as in [START_REF] Biryuk | Spectral Properties of Solutions of the Burgers Equation with Small Dissipation[END_REF], for time t ≥ 2. These estimates will only depend on the function f and the forcing. Let us emphasise that, for t ≥ 2, we are in a quasi-stationary regime: all estimates hold independently of the initial condition. In Section 6, we give some additional estimates for the Sobolev norms.

In this paper, we use methods introduced by Kuksin in [START_REF] Kuksin | On Turbulence in Nonlinear Schrödinger Equations[END_REF][START_REF] Kuksin | Spectral Properties of Solutions for Nonlinear PDEs in the Turbulent Regime[END_REF], and developed by Biryuk in [START_REF] Biryuk | Spectral Properties of Solutions of the Burgers Equation with Small Dissipation[END_REF].

Equation ( 1) with ν 1 is a popular one-dimensional model for the theory of hydrodynamic turbulence. In Section 7, we present an interpretation of our results in terms of this theory.

Preliminaries

In this section, we review properties of solutions of (1) used in our proof.

Physically, t corresponds to the time variable, whereas x corresponds to the one-dimensional space variable, and the constant ν > 0 to a viscosity coefficient. The real-valued function u(t, x) is defined on [0, +∞) × R and is L-periodic in x. The function f is C ∞ -smooth and strongly convex, i.e. it satisfies the condition (2) for some constant σ. Moreover, we assume that f , as well as its derivatives, has at most polynomial growth, i.e.

∀m ≥ 0, ∃n ≥ 0, C m > 0 : |f (m) (x)| ≤ C m (1 + |x|) n , x ∈ R, (3)
where n = n(m). From now on, we fix L = 1, which amounts to studying the problem on [0, +∞) × S 1 . We note that L-periodic solutions of (1) with any L reduce, by means of scaling in x, to 1-periodic solutions with scaled f and ν.

Since we are mostly interested in the asymptotics of solutions of (1) as ν → 0 + , we assume that

ν ∈ (0, 1].
Moreover, it is enough to study the special case

S 1 u 0 (y)dy = 0. ( 4
)
Indeed, if the mean value of u 0 on S 1 equals b, we may consider

v(t, x) = u(t, x + bt) -b.
Then v satisfies (4) and is a solution of (1) with f (y) replaced with g(y) = f (y + b) -by.

Given a C ∞ -smooth initial condition u 0 = u(0, •), equation ( 1) has a unique classical solution u, C ∞ -smooth in both variables (see [START_REF] Kreiss | Initial-boundary value problems and the Navier-Stokes equations[END_REF]Chapter 5]). Condition (4) implies that the mean value of a solution for (1) vanishes identically in t.

Now provide each space W n,p (S 1 ) with the Borel σ-algebra. Consider a random variable ζ on a probability space (Ω, F, P) with values in L 2 (S 1 ), such that ζ ω ∈ C ∞ (S 1 ) for a.e. ω. We suppose that ζ satisfies the following three properties.

(i) (Non-triviality)

P(ζ ≡ 0) < 1.
(ii) (Finiteness of exponential moments for Sobolev norms) For every m ≥ 0 there are constants α = α(m) > 0,

β = β(m) such that E exp(α ζ 2 m ) ≤ β. In particular I m = E ζ 2 m < +∞, ∀m ≥ 0.
(iii) (Vanishing of the expected value)

Eζ ≡ 0.
It is not difficult to construct explicitly ζ satisfying (i)-(iii). For instance we could consider the real Fourier coefficients of ζ, defined for k > 0 by

a k (ζ) = √ 2 S 1 cos(2πkx)u(x), b k (ζ) = √ 2 S 1 sin(2πkx)u(x) (5)
as independent random variables with zero mean value and exponential moments tending to 1 fast enough as k → +∞. Now let ζ i , i ∈ N be independent identically distributed random variables having the same distribution as ζ. The sequence (ζ i ) i≥1 is a random variable, defined on a probability space which is a countable direct product of copies of Ω. From now on, this space will itself be called Ω. The meaning of F and P changes accordingly.

For ω ∈ Ω and a time period θ > 0, the kick force η ω is a C ∞ -smooth function in the variable x, with values in the space of distributions in the variable t, defined by

η ω (x) = +∞ i=1 δ t=iθ ζ ω i (x),
where δ t=iθ denotes the Dirac measure at a time moment iθ.

The kick-forced version of (1) corresponds to the case where, in the right-hand side, 0 is replaced with the kick force. This means that for integers i ≥ 1, at the moments iθ the solution u(x) instantly increases by the kick ζ ω i (x), and that between these moments u solves (1). The equation is written as follows:

∂u ∂t + f (u) ∂u ∂x -ν ∂ 2 u ∂x 2 = η ω . (6)
Derivatives are taken in the sense of distributions. When studying solutions of ( 6), we will always assume that the initial condition u 0 = u(0, •) is C ∞ -smooth. Moreover, we normalise those solutions to be right-continuous in time at the kick moments iθ. Such a solution is uniquely defined for a given value of u 0 , for a.e. ω.

For a given initial condition u 0 , the function u(t, x) always will denote such a solution of (1). The value of u before the i-th kick will be denoted by u(iθ -, •), or shortly u - i . We will also use the notation u i = u(iθ, •) and denote the function u(t, •) by u(t). Finally, for a solution of (6), we consider time derivatives at the kick moments in the sense of right-sided time derivatives. Those derivatives are rightcontinuous in time.

Since space averages of the kicks vanish and u 0 (x) satisfies (4), the space average of u(t), t ≥ 0 vanishes identically. For the sake of simplicity, we normalise the kick period: from now on θ = 1.

We observe that, since the kicks are independent and between the kicks (6) is deterministic, the solutions of (6) make a random Markov process. For details, see [START_REF] Kuksin | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions[END_REF], where a kick force is introduced in a similar setting.

Agreements. All constants denoted C with sub-or super-indexes are strictly positive. Unless otherwise stated, they depend only on f , on the distribution of the kicks, as well as on the parameters a 1 , . . . , a k if they are denoted C(a 1 , . . . , a k ). u always denotes a solution of ( 6) with any initial condition u 0 . Averaging in ensemble corresponds to averaging in P. All our estimates hold independently of the value of u 0 .

We observe that for every integer i we have the following energy dissipation identity on the maximal kick-free intervals:

A i = |u i | 2 -u - i+1 2 , (7)
where

A i = 2ν (i+1) i u(t) 2 1 dt. (8)
Indeed, for any t

∈ (i, i + 1) u satisfies 2ν u(t) 2 1 = -2ν S 1 uu xx dx = -2 S 1 uf (u)u x dx -2 S 1 uu t dx.
The first term on the right-hand side vanishes since its integrand is a full derivative. The second term equals -d dt |u| 2 . Integrating in time we get [START_REF] Kruzkov | The Cauchy Problem in the large for Nonlinear Equations and for certain Quasilinear Systems of the First-order with several variables[END_REF]. We note that energy dissipation between kicks A i is always non-negative: energy can be added only at the kick points. We also note that an analogue of (7) holds on every kick-free time interval.

The following two lemmas are proved using the maximum principle in the same way as in [START_REF] Kruzkov | The Cauchy Problem in the large for Nonlinear Equations and for certain Quasilinear Systems of the First-order with several variables[END_REF].

Lemma 3.1. We have the estimate

u x (t, x) ≤ 2σ -1 , t ∈ [k + 1/2, k + 1), k ∈ N, x ∈ S 1 ,
where σ is the constant in the assumption (2).

Proof. Consider the equation ( 6) on the kick-free time interval [0, 1 -] for arbitrarily small and differentiate it once in space. We get

∂u x ∂t + f (u)u 2 x + f (u) ∂u x ∂x -ν ∂ 2 u x ∂x 2 = 0. ( 9
)
Consider v(t, x) = tu x (t, x). For t > 0, v verifies

∂v ∂t + t -1 (-v + f (u)v 2 ) + f (u) ∂v ∂x -ν ∂ 2 v ∂x 2 = 0. ( 10
)
Now observe that, if v > 0 somewhere on the domain S = [0, 1 -] × S 1 , then v attains its maximum M on S at a point (t 1 , x 1 ) such that t 1 > 0. At (t 1 , x 1 ) we have ∂v ∂t ≥ 0, ∂v ∂x = 0, and ∂ 2 v ∂x 2 ≤ 0. Therefore, [START_REF] Kuksin | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions[END_REF]

yields that t -1 1 [-v(t 1 , x 1 ) + f (u(t 1 , x 1 ))v 2 (t 1 , x 1 )] ≤ 0. Since, by (2), f ≥ σ > 0, then -M + σM 2 ≤ 0,
and therefore

M ≤ σ -1 .
Thus we have proved that v ≤ σ -1 everywhere on S for every > 0.

In particular, by definition of v and S , we get that

u x (t, x) ≤ 2σ -1 , x ∈ S 1 , t ∈ [1/2, 1).
Repeating the same argument on all the intervals [k, k + 1), k ∈ N we get the lemma's assertion.

Lemma 3.2. There are constants C , C such that

E exp(C sup t∈[k,k+1) max u x (t, •)) ≤ C, k ≥ 1.
Proof. Fix k ≥ 1. Since the W 1,∞ norm is dominated by the H 2 norm, then for C > 0 we get

exp(C u x (k, x)) ≤ exp(C u x (k -, x) + C ζ k 2 ), x ∈ S 1 .
The same inequality holds when we maximise in x. Now denote by X k the random variable max u x (k, •).

By Lemma 3.1 and Property (ii) of the kicks, for C = α(2) we get

E exp(C X k ) ≤ exp(2C σ -1 )E exp(C ζ k 2 ) ≤ C, (11)
for some constant C. Now consider the equation ( 9). An application of the maximum principle to the function u x , which cannot be negative everywhere, yields

max u x (t, •) ≤ max u x (k, •), t ∈ [k, k + 1) .
Therefore, in [START_REF] Nirenberg | On elliptic partial differential equations[END_REF], we can replace X k by sup t∈[k,k+1) max u x (t, •). This proves the lemma's assertion.

Corollary 3.1. For the same C , C as in Lemma 3.2 we have

E exp C 2 sup t∈[k,k+1) |u(t)| 1,1 ≤ C, k ≥ 1.
Proof. Since the mean value of u x (t) is 0, then

S 1 |u x (t)| = 2 S 1
max(u x (t), 0).

Corollary 3.2. For the same C , C as in Lemma 3.2 we have

E exp(C sup t∈[k,k+1) |u(t)| p ) ≤ C, k ≥ 1, p ∈ [1, +∞].
Note that C and C do not depend on p.

Lower estimates of H m norms

For a solution u of ( 6), the first quantity that we estimate from below is the expected value of

1 N N +1 1 u(t) 2 1 = 1 N (2ν) -1 N i=1 A i , (12)
where N is a fixed natural number chosen later, and A i is the same as in [START_REF] Kuksin | On Turbulence in Nonlinear Schrödinger Equations[END_REF].

Lemma 4.1. There exists a natural number N ≥ 1, independent from u 0 , such that

1 N N +1 1 E u(s) 2 1 ≥ Cν -1 .
Proof. For N ≥ 1 we have

E u - N +1 2 ≥ E u - N +1 2 -u - 1 2 = E N i=1 |u - i+1 | 2 -|u i | 2 + E N i=1 |u i | 2 -|u - i | 2 = -E N i=1 A i + E N i=1 |u - i + ζ i | 2 -|u - i | 2 = -E N i=1 A i + 2E N i=1 u - i , ζ i + E N i=1 |ζ i | 2 .
Since Eζ i ≡ 0 (Property (iii) of the kicks), and u - i and ζ i are independent, then E u - i , ζ i = 0. Therefore, by (8), we have

E u - N +1 2 ≥ -2νE N +1 1 u(s) 2 1 + 0 + N I 0 .
On the other hand, by Corollary 3.2 (p = 2) there is a constant C 1 such that

E u - N +1 2 ≤ C 1 . Consequently 1 N N +1 1 E u(s) 2 1 ≥ N I 0 -C 1 2N ν -1 .
Choosing the smallest possible integer N verifying

N ≥ max 1, C 1 + 1 I 0 ,
we get the lemma's assertion.

We have reached our first goal: estimating from below the expected value of (12). Thus, we have a time-averaged lower estimate of the H 1 norm, which enables us to obtain similar estimates of H m norms for m ≥ 2. Lemma 4.2. We have

1 N N +1 1 E u(s) 2 m ≥ C(m)ν -(2m-1) , m ≥ 1,
where N is the same as in Lemma 4.1.

Proof. This statement is already proved in the previous lemma for m = 1, so we may assume that m ≥ 2. By Lemma 1.1 and Hölder's inequality we have

E u(s) 2 1 2m-1 ≤ C (m)E u(s) 2 m E |u(s)| 2 1,1 2m-2 . ( 13
)
Since by Corollary 3.1

E |u(s)| 2 1,1 ≤ K, t ∈ [1, N + 1],
where K > 0 is a constant, then, integrating (13) in time, we get

1 N N +1 1 E u(s) 2 m ≥ N +1 1 [E( u(s) 2 1 )] (2m-1) N C (m)K 2m-2
.

By Hölder's inequality,

N +1 1 [E( u(s) 2 1 )] (2m-1) ≥ N +1 1 E u(s) 2 1 (2m-1) N 2-2m ,
and then

1 N N +1 1 E u(s) 2 m ≥ N +1 1 E u(s) 2 1 (2m-1) N 2-2m N C (m)K 2m-2 = 1 N N +1 1 E u(s) 2 1 (2m-1) C (m)K 2m-2 .
Now the assertion follows from Lemma 4.1.

Since we impose no conditions on u 0 , we can consider a different positive integer "starting time". We may also consider a different averaging time interval of length T ≥ N . Finally, we obtain a general result for a non-integer starting time t ≥ 1 by considering the maximal interval [m 1 , m 2 ] ⊂ [t, t + T ] such that m 1 and m 2 are positive integers.

Theorem 4.1. We have

1 T t+T t E u(s) 2 m ≥ C(m) 4 ν -(2m-1) , t ≥ 1, T ≥ N + 1, m ≥ 1,
where N and C(m) are the same as in Lemma 4.2.

Upper estimates of H m norms

To estimate from above a Sobolev norm u m , m ≥ 1, of a solution u for (6), we differentiate between the kicks the quantity u(t) 2 m . Denote by B(u) the nonlinearity 2f (u)u x , and by L the operator -∂ xx . Integrating by parts, we get

d dt u 2 m = 2 u (m) , u (m) t = -2ν u 2 m+1 -L m u, B(u) . ( 14
)
We will need a standard estimate for the nonlinearity L m u, B(u) .

Lemma 5.1. For a zero mean value smooth function w such that |w| ∞ ≤ M , we have

| L m w, B(w) | ≤ C w m w m+1 , m ≥ 1, with C satisfying C ≤ C m (1 + M ) n , ( 15
)
where C m , as well as the natural number n = n(m), depend only on m.

Proof. Let C denote various positive constants satisfying an estimate of the type (15). Then we have

| L m w, B(w) | =2 w (2m) , (f (w)) (1) =2 w (m+1) , (f (w)) (m) ≤C m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m S 1 w (m+1) w (a 1 ) . . . w (a k ) f (k) (w) ≤C |f | C m [-M,M ] m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m S 1
|w (a 1 ) . . . . . . w (a k ) w (m+1) |.

By (3), |f | C m [-M,M
] satisfies an estimate of the type (15). By Hölder's inequality, we obtain that

| L m w, B(w) | ≤C w m+1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m w (a 1 ) 2m/a 1 . . . . . . w (a k ) 2m/a k .
Finally, the Gagliardo-Nirenberg inequality yields

| L m w, B(w) | ≤C w m+1 m k=1 1≤a 1 ≤•••≤a k ≤m a 1 +•••+a k =m ( w a 1 /m m |w| (m-a 1 )/m ∞ ) . . . ( w a k /m m |w| (m-a k )/m ∞ ) ≤C |w| m-1 ∞ w m w m+1 ≤C w m w m+1 ,
which proves the lemma's assertion.

Theorem 5.1. For any natural numbers m, n we have

E( sup t∈[k,k+1) u(t) n m ) ≤ C(m, n)ν -(2m-1)n/2 , k ≥ 2.
Proof. Fix k ≥ 2 and m ≥ 1. In this proof, Θ denotes various positive random constants which depend on m, such that all their moments are finite, and C denotes various positive deterministic constants, depending only on m.

We begin by noting that Corollary 3.1 and Property (ii) of the kicks imply the inequalities

|u(t)| 1,1 , ζ k m ≤ Θ, t ∈ [k -1, k + 1). ( 16
)
We claim that when u 2 m is too large, it decreases at least as fast as a solution of the differential equation y + (2m -1)y 2m/(2m-1) = 0, i.e. as t -(2m-1) . More precisely, we want to prove that for t ∈ [k -1, k + 1) we have

u(t) 2 m ≥ Θ 1 ν -(2m-1) =⇒ d dt u(t) 2 m ≤ -(2m -1) u(t) 4m/(2m-1) m , (17)
where Θ 1 is a random positive constant, chosen later. Random constants Θ below do not depend on Θ 1 . Indeed, assume that

u(t) 2 m ≥ Θ 1 ν -(2m-1) . (18)
We begin by observing that by Lemma 1.1 we have

u m ≤ C u (2m-1)/(2m+1) m+1 |u| 2/(2m+1) 1,1
, and hence

u m+1 ≥ C |u| -2/(2m-1)
1,1

u (2m+1)/(2m-1) m ≥ Θ -1 u (2m+1)/(2m-1) m (19)
(we used (16)). Now, ( 14), (16), and Lemma 5.1 imply that

d dt u 2 m ≤ -2ν u 2 m+1 + Θ u m u m+1 =(-2ν u 2/(2m+1) m+1 + Θ u m u -(2m-1)/(2m+1) m+1 ) u 4m/(2m+1) m+1 . (20)
Combining ( 20) and ( 19), we get

d dt u 2 m ≤(-2ν u 2/(2m+1) m+1 + Θ) u 4m/(2m+1) m+1 .
Therefore, by ( 19) and (18) we have

d dt u 2 m ≤ -νΘ -1 u 2/(2m-1) m + Θ u 4m/(2m+1) m+1 ≤ -Θ -1 Θ 1/(2m-1) 1 + Θ u 4m/(2m+1)

m+1

. Now we choose Θ 1 in such a way that the quantity in the parentheses is negative. Under this assumption, we get from (19) that

d dt u 2 m ≤ -Θ -1 Θ 1/(2m-1) 1 + Θ Θ -1 u 4m/(2m-1) m .
This relation implies (17) if we choose for Θ 1 a sufficiently big random constant with all moments finite. Now we claim that

u - k 2 m ≤ Θ 2 ν -(2m-1) , (21)
where

Θ 2 = max(Θ 1 , 1) has finite moments. Indeed, if u(t) 2 m ≤ Θ 1 ν -(2m-1) for some t ∈ [k -1, k), then (17) ensures that u(t) 2
m remains under this threshold up to t = k -. Otherwise, we consider the function 1) , y(t) increases at least as fast as t. Indeed,

y(t) = u(t) -2/(2m-1) m , t ∈ [k -1, k) . By (17), since u(t) 2 m > Θ 1 ν -(2m-
d dt y(t) = - 1 2m -1 u(t) 2 m -2m/(2m-1) d dt u(t) 2 m ≥ 1 2m -1 u(t) -4m/(2m-1) m (2m -1) u(t) 4m/(2m-1) m ≥ 1.
Therefore y(k -) 2 m ≥ 1. Since ν ≤ 1, then in this case we also have (21).

In exactly the same way, using (16), we obtain that for t ∈ [k, k + 1),

u(t) 2 m ≤ max(Θ 2 ν -(2m-1) , u(k) 2 m ) ≤ max Θ 2 , Θ + Θ 2 2 ν -(2m-1) ≤ Θ + Θ 2 2 ν -(2m-1) . Therefore u(t) 2 m ν 2m-1 is uniformly bounded by Θ + √ Θ 2 2 for t ∈ [k, k + 1)
. Since all moments of this random variable are finite, the lemma's assertion is proved.

Proof. Using Properties (i) and (iii) of the kicks (u - k and ζ k being independent), we get

E u + k 2 = E u - k 2 + 2E u - k , ζ k + E |ζ k | 2 = E u - k 2 + E |ζ k | 2 ≥ I 0 .
On the other hand, by Theorem 5.1 we have Corollary 6.1. We have

E u(t) 2 1 ≤ C ν -1 , t ∈ (k, k + 1). Since d dt |u(t)| 2 = -2ν u(t)
E|u(s)| 2 ≥ E u + k 2 -2(s -k)C ≥ I 0 -2C d ≥ I 0 2 .
k+1 k E|u(s)| 2 p 1/2 ≥ C, k ≥ 2, p ∈ [1, +∞],
where C does not depend on p.

Proof. It suffices to prove the inequality for p = 1. Using Hölder's inequality and integrating in time and in ensemble, and then using the Cauchy-Schwarz inequality, we get

k+1 k E|u| 2 1 ≥ k+1 k E|u| 4 |u| -2 ∞ ≥ k+1 k E|u| 2 2 k+1 k E|u| 2 ∞ -1
. Lemma 6.2 and Corollary 3.2 (p = +∞) complete the proof.

Since the W 1,1 norm dominates the L ∞ norm, we get Corollary 6.2. We have

k+1 k E|u(s)| 2 1,1 (t) 1/2 ≥ C, k ≥ 2.
The cases m ≥ 2 and m = 1, p ≥ 2 follow from Lemma 4.

1 T t+T t E |u(s)| 2 m,p 1/2 ≥ C(m, p)ν -γ , t ≥ 1, T ≥ N + 1,
where N is the same as in Lemma 4.1. Now it remains to deal with the case m = 1 and p ∈ (1, 2). Lemma 6.4. For p ∈ (1, 2) we have 1

T t+T t E |u(s)| 2 1,p 1/2 ≥ C(p)ν -γ , t ≥ 2, T ≥ N + 1,
where N is the same as in Lemma 4.1. Note that here, γ = 1 -1/p.

Proof. In the proof of this lemma, C (p) denotes various positive constants depending only on p. By Hölder's inequality in space we have u(s) 2 1 ≤ |u(s)| p 1,p |u(s)| (2-p) 1,∞ . Therefore, using Hölder's inequality in time and in ensemble, as well as Lemma 6.1, we get 1 , n ≥ 2 follow from the lemmas above and Hölder's inequality.

T t+T t E u(s) 2 1 ≤ 1 T t+T t E |u(s)| 2 1,∞ (2-p)/2 • 1 T t+T t E |u(s)| 2 1,p p/2 ≤ C (p)ν (p-2) 1 T t+T t E |u(s)| 2 1,p p/2 . Furthermore, Lemma 4.1 implies that 1 T t+T t E |u(s)| 2 1,p ≥ C (p) ν (2-p) 1 T t+T t E u(s) 2 1 2/p ≥ C (p) ν (2-p) ν -1 2/p ≥ C (p)ν -(2p-

Conclusion

Putting together the estimates that we have obtained, we formulate our main result.

E sup t∈[k,k+1) |u(t)| n m,p 1/n ≤ C(m, p, n)ν -γ , n ≥ 1, k ≥ 2. ( 22
)
Moreover, there is an integer N ≥ 1 such that, for m ≥ 0 and p ∈ [1, +∞], we have

1 T t+T t E|u(s)| n m,p 1/n ≥ C(m, p)ν -γ , n ≥ 2, t ≥ 2, T ≥ N . ( 23
)
In both inequalities

γ = max 0, m - 1 p .
For a solution u of (6), we have obtained asymptotic estimates for expectations of a large class of Sobolev norms. The power of ν is clearly optimal except for m ≥ 2 and p = 1, since it coincides for upper and lower estimates: we are in a quasi-stationary regime. Let us stress again that the upper bound t = 2 for the time needed for a quasistationary regime to be established has no dependence on u 0 . The condition t ≥ T 0 for some time T 0 ≥ 1 is necessary: we need damping if u 0 is large and injection of energy at a kick point if u 0 is small. Now put ûk = a k (u) + ib k (u) (see ( 5)). For t ≥ 2 and T big enough (see Theorem 7.1), consider the averaged quantities

F s,θ = 1 T t+T t k∈I(s,θ) E|û k | 2 (τ ) k∈I(s,θ) 1
, s, θ > 0, where I(s, θ) = [ν -s+θ , ν -s-θ ). In the same way as in [2, formulas (1.6)-(1.8)], the inequalities (22-23) yield

F s,θ ≤ Cν 2s (24) F s,θ ≤ C(m)ν 2+2m(s-1-θ) , m > 0, s > 1 + θ (25) F 1,θ > Cν 2+2θ (26)
for ν ≤ ν(θ) with some ν(θ) > 0. These results have some consequences for the energy spectrum of u. Indeed, relation (25) implies that the energy of the k-th Fourier mode,

E k = 1 2T t+T t
E|û k | 2 , averaged around k = l, where l ν -1 , decays faster than any negative degree of l. On the other hand, by ( 24) and (26), the energy E k , averaged around k = ν -1 , behaves as k -2 . That is, the interval k ∈ (ν -1 , +∞) is the dissipation range, where the energy E k decays fast.

As the force η is smooth in x, then the energy is injected at frequencies k ∼ 1. The estimate (24) readily implies that the energy E = E k of a solution u is supported, when ν → 0, by any interval (0, ν -γ ), γ > 0. That is, the energy range of the solution u is the interval (0, ν 0 ] (see [START_REF] Frisch | Turbulence: The legacy of A.N. Kolmogorov[END_REF]).

The complement to the energy and dissipation ranges is the inertial range (ν 0 , ν -1 ). At k ∼ ν -1 we have E k ∼ k -2 . It is plausible that in this range E k decays algebraically; possibly E k ∼ k -2 . The study of the energy spectrum of solutions u in the inertial range is one of the objectives of our future research.

We recall that the behavior of the energy spectrum E k of turbulent fluid of the form "some negative degree of k in the inertial range, followed by fast decay in the dissipation range" is suggested by the Kolmogorov theory of turbulence (see [START_REF] Frisch | Turbulence: The legacy of A.N. Kolmogorov[END_REF]). Our results (following those of A.Biryuk in [START_REF] Biryuk | Spectral Properties of Solutions of the Burgers Equation with Small Dissipation[END_REF]) show that for the "burgulence" (described by the Burgers equation, see [START_REF] Bec | Burgers turbulence[END_REF]) the dissipation range is (ν -1 , +∞) and suggest that the power-law in the inertial range is

E k ∼ k -2 .
We also see that for ν → 0 + , solutions u display intermittency-type behavior (see [START_REF] Frisch | Turbulence: The legacy of A.N. Kolmogorov[END_REF]Chapter 8]). Indeed, in the quasi-stationary regime, up to averaging in time and in ensemble, max x∈S 1 u x ∼ 1, whereas S 1 u 2

x ∼ ν -1 . Thus, typically u has large negative gradients on a small subset of S 1 , and small positive gradients on a large subset of S 1 .

In a future paper, we will look at the same problem with the kick force replaced by a spatially smooth white noise in time (see [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF] for a possible definition). This problem is, heuristically, the limit case of the kick-forced problem with more and more frequent appropriately scaled kicks.

Lemma 1 . 1 .

 11 For a smooth zero mean value function w on S 1 , |w| β,r ≤ C |w| θ m,p |w| 1-θ q ,

  lemma's assertion. Now we can study the case m = 0 and p ∈ [1, +∞].

Lemma 6 . 3 .

 63 1 and Lemma 1.1 by interpolation in the same way as Lemma 4.2, for p > 1. The case p = +∞ follows from the case p = 1, since |u| m,1 ≥ |u| m-1,∞ , and γ(m, 1) = γ(m -1, +∞). If either m ≥ 2 and p ∈ [1, +∞], or m = 1 and p ∈ [2, +∞], then

Theorem 7 . 1 .

 71 For m ∈ {0, 1} and p ∈ [1, +∞], or for m ≥ 2 and p ∈ (1, +∞], we have

  2)/p .

	Remark 6.1. Upper estimates for
	1 T	t	t+T	m,p E|u(s)| n	1/n

Acknowledgements

First of all, I would like to thank my advisor S.Kuksin for formulation of the problem and guidance of my research. I would also like to thank A.Biryuk and K.Khanin for fruitful discussions. Finally, I am grateful to the faculty and staff at CMLS (Ecole Polytechnique) for their constant support during my PhD studies.

Estimates of other Sobolev norms.

The results in the three previous sections enable us to find upper and lower estimates for a large class of Sobolev norms. Unfortunately, while lower estimates extend to the whole Sobolev scale for m ≥ 0 and p ∈ [1, +∞], there is a gap, corresponding to the case m ≥ 2 and p = 1, for upper estimates.

Here and later on, Unfortunately, it cannot be applied for m ≥ 2 and p = 1, because Lemma 1.1 only allows us to estimate a W n,1 norm from above by other W n,1 norms.

Finally, the case m = 0 follows from Corollary 3.2.

The first norm that we estimate from below is the L 2 norm. Lemma 6.2. We have


Claude Manté
email: claude.mante@mio.osupytheas.fr

Iterated Bernstein operators for distribution function and density estimation: balancing between the number of iterations and the polynomial degree

Keywords: Density Estimation, Bernstein operator, roots of operators, regular histogram, shape restriction, Gnedenko test

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Although S. Bernstein simultaneously introduced both the polynomials and the operator that bear his name in his famous constructive proof of the Stone-Weierstrass theorem [START_REF] Bernstein | Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités[END_REF], both of these objects naturally split up with time. While there is a large interest in the Bernstein operator in the literature on Approximation Theory, (see for instance [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]; [START_REF] Sevy | Convergence of iterated boolean sums of simultaneous approximants[END_REF][START_REF] Sevy | Lagrange and least-squares polynomials as limits of linear combinations of iterates of Bernstein and Durrmeyer polynomials[END_REF]; [START_REF] Sahai | An iterative algorithm for improved approximation by Bernstein's operator using statistical perspective[END_REF]), researchers from other disciplines essentially focus on Bernstein polynomials. For instance, the attractive properties of this approximation prompted statisticians to apply it to Density Estimation [START_REF] Vitale | A Bernstein Polynomial Approach to Density Function Estimation[END_REF][START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF][START_REF] Bouezmarni | Bernstein estimator for unbounded density function[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc , , 2012a,b),b), Regression (A. Marco and J.J. Martinez , 2010;[START_REF] Curtis | A variable selection approach to monotonic regression with Bernstein polynomials[END_REF][START_REF] Wang | Shape restricted nonparametric regression with Bernstein polynomials[END_REF] or Bayesian Inference [START_REF] Petrone | Random Bernstein Polynomials[END_REF]. However, most of these authors paid little attention to the Bernstein operator itself.

Nevertheless, an operator is attached to a pair of vector spaces, and not to particular bases of these spaces. We highlight in Section 3 that Bernstein polynomials consist in a natural output basis for the eponym operator, while the natural input basis is a Lagrange polynomial basis (see also [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF], Section 5). In addition, we must take into account the pair of bases associated with the eigendecompositon of the Bernstein operator, given by [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]. Bearing in mind a generalization of the Sevy convergence acceleration method [START_REF] Sevy | Convergence of iterated boolean sums of simultaneous approximants[END_REF][START_REF] Sevy | Lagrange and least-squares polynomials as limits of linear combinations of iterates of Bernstein and Durrmeyer polynomials[END_REF], we further investigate in Section 3 the matrix representation of powers of the Bernstein operator with respect to these bases. This enables us to define first, in Section 4, fractional Bernstein operators and second, in Section 5, fractional Sevy approximation sequences. This constitutes the basis for refining results obtained by [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF], where both the distribution function and the density approximation were obtained using Sevy's iteration scheme. Now, roughly speaking, density estimation or approximation by Bernstein polynomials [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF][START_REF] Bouezmarni | Bernstein estimator for unbounded density function[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc , , 2012a,b;,b;[START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] consists in fitting a Bernstein polynomial of some order m on a distribution function, and in differentiating it. More precisely, these authors estimate the distribution function (d.f.) F associated with a random variable X from m values of the empirical distribution function (e.d.f.) F N obtained from a N-sample of X :

F N,m (x) := m ∑ k=0 F N ( k m ) w m,k (x) ,
where w n,j (x) := ( n j )

x j (1 -x) n-j . The choice of an optimal number of bins m * is always a critical step. In the density estimation setting, most authors recommend either choosing m * = ν(N ), where N is the sample size and ν is some function stemming from asymptotic results [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc , , 2012a)), or else obtaining m * from cross-validation methods [START_REF] Bouezmarni | Bernstein estimator for unbounded density function[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF].

In Section 6 we propose another method, starting with the [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] upper value m 0 := N/ ln (N ). It consists in selecting m * ≤ m 0 in order that the same optimal m * should be obtained with a high probability from different N-samples (stability), and that the "coarsened" distribution functions associated with these m * bins should be close to the classical empirical distribution function F N (fidelity). The method is tested on real data in Section 7.

Notation

We will work in the Banach space C [0, 1] of continuous functions on [0, 1], equipped with the Chebyshev norm ∥f ∥ := max x∈ [0,1] |f (x)|. P n denotes the subspace of C [0, 1] consisting of polynomials of degree k ≤ n, and P n denotes the complement of P 1 in P n i.e. the vector space of polynomials of degree 1 < k ≤ n.

Consider an operator U : C [0, 1] → C [0, 1]; for n ≥ 2 (fixed), its restriction to P n (i.e. the operator U | Pn :

P n → C[0, 1] such that ∀ f ∈ P n , U | Pn (f ) = U (f )
) will be denoted

• U , and its restriction to P n will be denoted U . For the sake of simplicity, the restrictions of the identity operator to these subspaces will be denoted 1, instead of • 1 or 1.

In the finite dimensional setting, we will use the matrix p-norm (or

ℓ p - norm) ∥U ∥ p := sup v̸ =0 ∥U (v)∥ p ∥v∥ p
where ∥v∥ p is the usual vector ℓ p -norm. Notice that ∥U ∥ 1 and ∥U ∥ ∞ are the greatest sum of the absolute values of the matrix elements along columns and rows, respectively, while ∥U ∥ 2 is the spectral norm [START_REF] Farouki | On the stability of transformations between power and Bernstein polynomials forms[END_REF]. In this setting, M at (U ; L n , W n ) will denote the matrix representation of the operator U with respect to the bases L n and

W n .
Finally, the expression Y L = X denotes that both of the random variables X and Y obey the same probability law. The integer value of some real number x will be denoted ⌊x⌋.

Expression of powers of the Bernstein operator into different bases

The Bernstein operator [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF][START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF][START_REF] Sevy | Lagrange and least-squares polynomials as limits of linear combinations of iterates of Bernstein and Durrmeyer polynomials[END_REF] by:

B n : C [0, 1] → C [0, 1] is defined
B n [f ] (x) := n ∑ j=0 w n,j (x)f ( j n ), with w n,j (x) := ( n j ) x j (1 -x) n-j . Of course, its image R (B n ) is included in P n .
In this section, we will focus on the matrix representation of powers of B n with respect to three bases of P n : Lagrange and Bernstein bases, and the eigenfunctions of B n .

Expression of powers of B n relative to Lagrange and Bernstein bases

First, let us consider the Lagrange interpolation operator

L n : C [0, 1] → C [0, 1], defined by L n [f ] (x) := n ∑ j=0 ℓ n,j (x) f ( j n ),
where ℓ n,j (x) :

= n ∏ k=0 k̸ =j n x -k j -k
is the j th Lagrange polynomial in the equally spaced case. Clearly, R (L n ) = P n and, since L n is idempotent and the Lebesgue constant

∥L n ∥ = max ∥f ∦ =0 ∥Ln[f ]∥ ∥f ∥ ∼ 2 n
e n log(n) (see [START_REF] Mills | The Lebesgue constant for Lagrange interpolation on equidistant nodes[END_REF]) is bounded for any finite n, L n is the projection onto P n . Con-

sequently, any f ∈ C [0, 1] is the direct sum of two components: L n [f ] and the " Lagrange residual" (f -L n [f ]). Lemma 1. ∀k ≥ 1, B k n = • B k n • L n , where • B k n := ( • B n
) k denotes the power of order k of the restricted operator.

Proof. Because L n is interpolatory, we can write:

B n : C [0, 1] Ln -→ P n • Bn -→ P n .
In other words,

B n = • B n • L n ; furthermore, since L n is the projection onto P n , ∀k ≥ 1, L n • • B k n = • B k n
Consider now a polynomial P ∈ P n ; we have on the one hand

L n [P ] (x) = n ∑ j=0 ℓ n,j (x) P ( j n
) and on the other hand

• B n [P ] (x) = n ∑ j=0 w n,j (x)P ( j n
) . Thus, with respect to the bases

L n := {ℓ n,j (x) , 0 ≤ j ≤ n} and W n := {w n,j (x) , 0 ≤ j ≤ n} the matrix of • B n is the identity matrix: M at ( • B n ; L n , W n ) = I n . Let us
denote LW [n] the transformation matrix associated with the bases L n and W n , whose j th column consists in the coordinates of w n,j in the basis L n .

Lemma 2. The matrix of

( • B k n ; L n , W n ) = LW k-1 [n] .
Proof. One can easily verify that LW

[n] i,j = w n,j ( i n ) ; consequently, M at ( • B n ; W n , W n ) = LW [n]
. Thus, the iterated operator of order k can be represented by the diagram:

B k n : C [0, 1] Ln -→ (P n , L n ) In -→ (P n , W n ) LW k-1 [n] -→ (P n , W n )
and M at

( • B k n ; L n , W n ) = LW k-1 [n]

Expression of powers of B n relative to the eigenfunctions of B n

At present, the focus is on the eigenstructure of B n , which was completely elucidated by [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]. They demonstrated the following theorem.

Theorem 1. The Bernstein operator can be represented in the diagonal form

B n [f ] = n ∑ j=0 λ [n] j π [n] j µ [n] j (f ) , ( 1
)
where f ∈ C [0, 1], λ [n]
j and π

[n] j are the eigenvalues and eigenfunctions of B n , and µ

[n] j are the dual functionals to π

[n] j .

The eigenvalues are given by λ

[n] j = n! (n-j)! n j , while π [n]
j is a polynomial of degree j , which can be calculated with a recurrence formula given in [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]. As for the µ

[n] j , they constitute a basis for the dual space

P * n C [0, 1] * , such that ⟨ µ [n] j , π [n] k ⟩ = δ j,k ∀j, k.
Corollary 1. Using the classical notation u ⊗ v * (w) := u ⟨v * , w⟩ [START_REF] Bowen | Introduction to Vectors and Tensors Volume 1: Linear and Multilinear Algebra[END_REF], we can rewrite Eq. ( 1) in an alternative form:

B n [f ] = n ∑ j=0 λ [n] j π [n] j ⊗ π * [n] j (L n [f ]) . ( 2
)
Proof: see the appendix.

Thus, B n and the diagonal matrix associated with the λ

[n] j , we can now write the diagram:

B k n : C [0, 1] Ln -→ (P n , L n ) LΠ [n] -→ ( P n , Π [n] ) Λ k [n] -→ ( P n , Π [n] ) ΠW [n] -→ (P n , W n ) (3)
where LΠ [n] and ΠW [n] are the transformation matrices associated with these bases.

Fractional Bernstein operators

We propose in this section, for any integer K ≥ 2, a definition of the K th "root" of the operator

G n := (1 -B n ), denoted G 1/K
n ; this will enable us to generalize iterated boolean sums of operators studied by [START_REF] Sevy | Convergence of iterated boolean sums of simultaneous approximants[END_REF].

For a fixed continuous function f , consider the decomposition:

(1 -B n ) [f ] = (f -L n [f ]) + (L n [f ] -B n [f ]). While it is straightforward to write that (L n [f ] -B n [f ]) = ( 1 - • B n ) [L n [f ]],
we do not have much information

about the residual f -L n [f ].
In fact, the only thing that can be said is

that ∥f -L n [f ]∥ ≤ (1 + ∥L n ∥) ∥f ∥ ∼ 2 n
e n log(n) ∥f ∥ [START_REF] Laurent | Approximation et optimisation[END_REF][START_REF] Mills | The Lebesgue constant for Lagrange interpolation on equidistant nodes[END_REF]. This does not matter here, because the objective is to compute expressions like

( 1 -(1 -B n ) I
)

[f ] (see Section 5), and we have the following result (see appendix for proof).

Lemma 3. For any integer I,

( 1 -(1 -B n ) I ) = ( 1 - ( 1 - • B n ) I ) • L n .
Consequently, we can proceed as if f ∈ R (L n ), and we do not need to worry about the Lagrange residual. Now, thanks to Eq. (2), we have:

• G n • L n [f ] := ( 1 - • B n ) [L n [f ]] = n ∑ j=0 ( 1 -λ [n] j ) π [n] j ⊗ π * [n] j (L n [f ]) .
Thus,

• G n can be considered as a symmetrical bilinear application, and its matrix relative to some basis

E [n] of P n is M at ( • G n ; E [n] , E [n] ) = Q [n] Γ [n] Q t [n] , where Q [n] is orthogonal and Γ [n] is the diagonal matrix associated with the vector (0, 0, 1/n, (3n -2) /n 2 , • • • , 1 -n!/n n ).
Consider now the restriction B n of • B n to P n . Since B n reproduces only the linear polynomials (even quadratic polynomials are not reproduced by B n -see [START_REF] Walz | Asymptotic expansions for multivariate polynomial approximation[END_REF]), the operator G n := 1 -B n is injective. All its eigenvalues are positive, and the maximal one is 1 [START_REF] Impens | Stirling's series made easy[END_REF]).

-n! n n < 1- √ 2πn exp ( -n + 1 12n -1 360n 3 ) (see
Since the maximum eigenvalue of B n is 1 -1 n , it is possible to define a new operator G n (α) from the classical results below (valid in a much larger setting than ours).

Proposition 1. [START_REF] Kato | Perturbation theory for linear operators[END_REF], Ch. 9&10)

(1) Let T be an operator of finite trace (trace class) in a separable Hilbert space H, such that its spectral radius is smaller than 1. Then we may define the operator

log (1 + T ) := ∞ ∑ k=1 (-1) k-1 T k k ,
which also belongs to the trace class.

(2) Let T be a bounded operator defined on a Banach space. Consider the Taylor series:

∞ ∑ k=0 (-1) k u k T k k! .
It is absolutely convergent for any complex number u, and defines an operator denoted exp (-u T ).

Using the first part of the Proposition above, we can first define the operator log ( G n ) and afterwards, thanks to the second part, we can define for any α > 0 the operator we need:

Definition 1. G n (α) := exp ( α log ( G n )) . ( 4
)
The matrix representation of this new operator is simple; it is given in the following result (see the appendix for a proof).

Proposition 2. Suppose E [n] is some basis of P n . Then, M at α) is the diagonal matrix associated with the vector

( G n (α) ; E [n] , E [n] ) = Q [n] Γ [n] (α) Q [n] t , where Γ [n] (
(( 1 n ) α , ( 3n-2 n 2 ) α , • • • , ( 1 -n! n n ) α ) and Q [n] is orthogonal.

Interpolating Sevy sequences

In order to accelerate the convergence of Bernstein approximations, Sevy (1993, 1995) proposed to replace B n by the iterated operator

I I n := ( 1 -(1 -B n ) I ) . ( 5
)
This method was re-discovered by [START_REF] Sahai | An iterative algorithm for improved approximation by Bernstein's operator using statistical perspective[END_REF], who noticed that one can

write C 0 [0, 1] ∋ F = B n [F ]+E, where E ∈ C 0 [0, 1] is an unknown " Bernstein
residual" which can be approximated by

B n [E]. Then, B n [F ] + B n [E] = ( 1 -(1 -B n ) 2 ) [F ] is a better approximation of F than B n [F ]
, and so on...

Sevy proved the following result :

Theorem 2. [START_REF] Sevy | Lagrange and least-squares polynomials as limits of linear combinations of iterates of Bernstein and Durrmeyer polynomials[END_REF], see also [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]) For some fixed n ≥ 1 and any function F defined on [0,1],

I I n [F ] -L n [F ] -→ 0 I→∞ . ( 6
)
Thus, Sevy sequences build a bridge between Bernstein approximation (which has nice shape-preserving properties, but converges slowly) and Lagrange interpolation, which is notoriously a bad approximate, especially in the case of equispaced knots (de Boor (1978, Ch. 2); see also Laurent (1972, Ch. 5)). Both of these polynomials can have bad properties: the first one can be suspected of excessive smoothness (especially when the sample size is small or moderate), while the second one is generally "bumpy". Searching for a trade-off, [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF] proposed to run across the

whole segment t B n [F ] + (1 -t) L n [F ] , 0 ≤ t ≤ 1.
We will follow a different line, specific to density approximation, to work out another trade-off between both types of approximations.

Proposition 3. Let P ∈ P n = P 1 ⊕ P n , and consider the associated decomposition: P = P 1 + P . We have:

∀ k ≥ 1, I k n (P ) = P 1 + I k n ( P ) .
Proof: see the appendix.

Because of Lemma 3 and the proposition above,

∀ f ∈ C [0, 1] , I k n (f ) = L 1 [f ] + I k n (L n [f ] -L 1 [f ]), with L 1 [f ] (x) = x f (1) + (1 -x) f (0) ∀ x ∈ [0, 1]. Consequently
I j n;K [f ] := L 1 [f ] + ( 1 -G n (j/K) ) (L n [f ] -L 1 [f ]) , j ≥ 1.
This sequence interpolates Sevy's one, since

I j K n;K [f ] = I j n (f ).
Proposition 4. The matrix of the restricted fractional operator is: M at

( • I j n;K ; L n , W n ) = ΠW [n] • Λ (j/K) [n]
• LΠ [n] , where Λ

(j/K) [n]
is the diagonal matrix associated with the vector

( 1, 1, 1 - ( 1 n ) (j/K) , 1 - ( 3n-2 n 2 ) (j/K) , • • • , 1 - ( 1 -n! n n ) (j/K) ) .
Proof. Using the blocks structure associated with the decomposition P n = P 1 ⊕ P n , we can see that M at

( • I j n;K ; Π [n] , Π [n] ) = Λ (j/K) [n]
. Thus, the fractional operator can be represented by a diagram similar to (3):

I j n;K : C [0, 1] Ln -→ (P n , L n ) LΠ [n] -→ ( P n , Π [n] ) Λ (j/K) [n] -→ ( P n , Π [n] ) ΠW [n] -→ (P n , W n )

Numerical difficulties

Because of Lemmas 2 and 3, computing a classical Sevy sequence amounts to computing powers of the transformation matrix LW [n] . Since M at

( • B n ; W n , W n ) = LW [n]
, the condition number of this matrix in the ℓ 2 -norm is [START_REF] Farouki | On the stability of transformations between power and Bernstein polynomials forms[END_REF]: [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF][START_REF] Impens | Stirling's series made easy[END_REF]. Thus, LW [n] is ill-conditioned in the ℓ 2 -norm sense, and one must expect to encounter numerical problems when n is big enough. The situation is more complicated in the case of fractional sequences, since Proposition 4 shows that the matrix of the restricted operator depends on both of the transformation matrices LΠ [n] and ΠW [n] . To our knowledge, the transformations between Lagrange polynomials, Bernstein polynomials, and the Bernstein operator eigenfunctions system have not been studied yet. However, it is wellknown that the transformations between power and Bernstein bases [START_REF] Farouki | On the stability of transformations between power and Bernstein polynomials forms[END_REF][START_REF] Farouki | The Bernstein polynomial basis: a centennial retrospective[END_REF] or between Hermite and Bernstein bases [START_REF] Hermann | On the stability of polynomial transformations between Taylor, Bernstein and Hermite forms[END_REF] are ill-conditioned.

∥LW[n]∥ 2 LW -1 [n] 2 = λ [n] 0 λ [n] n = n n n! ≈ e n √ 2πn
The idea here is merely to control numerical errors in the computation of Furthermore, we must add to this difficulty the computational cost of the eigenfunctions which becomes prohibitive for n ≥ 22. To sum up, for practical reasons, it seems necessary to restrict ourselves to polynomials of degree lower than 21.

I j n;K [f ]. First, notice that M at ( • B n ; L n , W n ) = ΠW [n] • Λ [n] • LΠ [n] ; thus, since M at ( • B n ; L n , W n ) = I n , the matrix norms ΠW [n] • Λ [n] • LΠ [n] -I n 1

Application of fractional sequences to distribution function and density estimation

Suppose F is a differentiable d.f. associated with a random variable X supported by [0, 1] and that

S N := {X 1 , • • • X N } is a N -sample of X, giving
rise to the e.d.f. F N (x). After [START_REF] Vitale | A Bernstein Polynomial Approach to Density Function Estimation[END_REF], who considered Bernstein density estimators for the first time, [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] proposed an estimator F N,m of F , consisting in smoothing the random step function F N :

F N,m (x) := m ∑ k=0 F N ( k m ) w m,k (x) = B m [F N ]. (7)
It is noteworthy that this estimator also smoothes another step function F N,m obtained by sub-sampling F N , whose jump set is:

{ (0, 0) , ( 1 m , 1 N N ∑ i=1 I ( X i < 1 m ) ) , • • • , ( k m , 1 N N ∑ i=1 I ( X i < k m ) ) , • • • , (1, 1) } . ( 8
)
In fact the expression "B m [F N ]" is slightly improper (F N is not continuous) and should be replaced by "B m [φ N ]", where φ N should be some continuous function (piecewise linear, spline, etc.) interpolating the jump set (8), or should be obtained from a well-suited histogram (see for instance [START_REF] Birgé | How many bins should be put in a regular histogram?[END_REF]; [START_REF] Davies | A comparison of automatic histogram constructions[END_REF]; [START_REF] Lugosi | Consistency of data-driven histogram methods for density estimation and classification[END_REF], and also Sections 6.1 & 6.2) by numerical integration. For the sake of simplicity, we will drop this refinement of no practical importance, except in the following proposition.

Proposition 5. If F is a continuous d.f., the Bernstein operator is a contraction. More precisely,

∥B m [F ]∥ ≤ ( 1 - 1 2 m-1 ) ∥F ∥ .
Consequently, if φ N is a continuous estimate of F derived from F N and such that φ N (0) = 0 and φ N (1) = 1, we can write:

∥B m [φ N -F ]∥ ≤ ( 1 - 1 2 m-1 ) ∥φ N -F ∥ .
Proof: see the appendix.

As a corollary, B m [F N ] inherits all the good asymptotic properties of the e.d.f. in the Chebyshev norm (i.e. in the Kolmogorov-Smirnov metric) [START_REF] Servien | Estimation de la fonction de répartition: revue bibliographique[END_REF][START_REF] Leblanc | Chung-Smirnov property for Bernstein estimators of distribution functions[END_REF][START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] because for large enough samples, F N can always be closely approached by continuous functions.

Now, what about

I I m;K [F N ] -F ?
The situation is much more intricate than in the classical Bernstein operator case, since I I m;K is not a positive operator. Let us denote ∆ N := F N -F , and consider the sampled val-

ues δ m;N := { ∆ N (0), ∆ N ( 1 m ) • • • , ∆ N ( m-1 m ) , ∆ N (1)
} . We will also denote

H [m;K] (δ m;N ) as the coordinates of I 1 m;K [∆ N ] in the equispaced Lagrange basis.
Proposition 6. We can write:

I I m;K [F N ] -F ≤ H I [m;K] ∞ ∥δ m;N ∥ ∞ ∥L m ∥ + I I m;K [F ] -F .
In addition:

lim I→∞ I I m;K [F ] -F ≤ (1 + ∥L m ∥) inf P ∈Pn ∥P -F ∥ ,
where

∥L m ∥ ≈ 2 m
e m log(m) denotes the Lebesgue constant [START_REF] Mills | The Lebesgue constant for Lagrange interpolation on equidistant nodes[END_REF].

Proof: see the appendix. Remark 1. To compute H I [m;K] (except if I = K), we need to compute LΠ -1 [m] (see the proof of Proposition 6). Since LΠ [m] is ill-conditioned (like any change of polynomial basis), H I [m;K] ∞ increases with m, just like ∥L m ∥,
while ∥δ m;N ∥ ∞ clearly depends on the structure of F , and can be optimized (see Sections 6.1 & 6.2). On the other hand, a small value of m controls the possibly explosive behaviour of the undesirable (and essentially unknown) term I I m;K [F ] -F when I is big. Thus, Proposition 6 shows that the choice of (m, I) must result from a delicate tuning of these parameters.

The choice of the number of bins m < N in formula 7 was previously discussed in [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF]; [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc ( , 2012a)). [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] proved the almost sure convergence of (7) when F is continuous, and gave conditions under which its rate of stochastic convergence can be determined, as well as the rate of convergence of the associated density estimator

f N,m (x) := d dx F N,m (x)
when F is differentiable with derivative f := dF dx . More precisely, they proved that f N,m almost surely converges towards f , under the condition m = o (N/ ln (N )). Furthermore, they inferred from simulations that the upper value m = N/ ln (N ) is indeed acceptable. But notice that N ≥ 100 ⇒ m > 21. Thus, the numerical issues brought up is Section 5.1 will arise even with moderate sample size. Consequently, it is necessary to determine a number of bins m ≤ 21 such that the associated partition of [0, 1] is well-suited for F .

A similar problem was tackled by [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] but, instead of an e.d.f., the data consisted of a discretized distribution function {F (x j ) , 0 ≤ j ≤ N } sampled on an imposed mesh 0 < x 0 ≤ x 1 < . . . , < x N -1 ≤ x N < 1. The method proposed by [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] consisted firstly in determining a submesh of size n ≤ N well-suited for Bernstein approximation and, secondly, in optimizing the number of iterations in formula 5, under the constraint that the associated density approximation f n

(I * )
is bona fide according to [START_REF] Gajek | On improving density estimators which are not bona fide functions[END_REF], i.e. belongs to both the closed convex cone of positive functions F + and the closed convex set F 1 of functions integrating to one. A number of discretized distribution functions (e.g. grain size curves) were processed in that way [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF]Manté and Stora , 2012). Sometimes we found I * = 1; in such cases, the usual approximation cannot be improved by using Sevy's iteration scheme, because f n

(2) / ∈ F + ∩ F 1 , while f n (1) ∈ F + ∩ F 1 .
But we can indeed get finer trajectories by slowing down Sevy's acceleration method! We just have to supersede integers by rational numbers in formula (5), that is to say to use a K-fractional sequence (see Definition 2), whose resolution increases with K. This will be done in the next section.

But for the moment, the objective is to determine what number of bins is best-suited for a given data set. Since the upper value m = N/ ln (N ) is often too big to use in practice, we propose to lower it according to the structure of F N (x). Since 1/m can be considered as a kind of bandwidth [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF], lowering m could cause oversmoothing, but one can expect that fractional iterations will offset this phenomenon. So, let us start with m 0 := N/ ln (N ), and consider the sequence of uniform meshes

{U m : 1 ≤ m ≤ m 0 } such that U m := { i m , 0 ≤ i ≤ m } .
We propose here a method to select m * ≤ min [m 0 , 21] such that U m * is well-suited for F N . By "well-suited", we mean that the same m * should be obtained with a high probability from different samples of size N of X (stability), and that the step functions F N,m * and F N should be close to each other (fidelity).

A stability/fidelity test

We first propose a criterion based on half-sampling [START_REF] Stephens | On the half-sample method for goodness-of-fit[END_REF] and on a classical two-sample test. Suppose N = 2M (if N is odd, get rid of an observation). From S N , we randomly draw (without replacement) a sample of size M of X, the learning sample S L M , and obtain the test sample S T M := S N ⊖ S L M . The subsamples S L M and S T M are independent, and the associated e.d.f.s will be denoted F L M (x) and F T M (x). Even if the hypothesis (H 0 ) := F L = F T is actually true, due to sampling fluctuations (or to a descendant of Maxwell's demon), the e.d.f. of the subsamples S L M and S T M can be quite different, especially in the case of small samples. For instance, consider a N -sample of the uniform distribution: the probability of drawing a learning M -sample of numbers lower than 0.5 and a test M -sample of numbers greater than 0.5 is not null (with M = 5, it is 0.0625 and with M = 10, it is about 0.0020). This indeed depends upon the power of the test, and [START_REF] Stephens | On the half-sample method for goodness-of-fit[END_REF] observed that the power of the half-sample goodness-of-fit test is uneven.

Consequently, we suggest to discard ill-suited subsamples such that the

Kolmogorov-Smirnov random distance D KS ( F L M , F T M ) := sup x∈[0,1] F L M (x) -F T M (
x) is excessive: in such a case, finding from the learning sample a mesh wellsuited for the test sample is hopeless! Consider two samples of same size M of the same distribution, and the distribution-free statistics D KS (F 1 , F 2 ) associated with the Kolmogorov-Smirnov homogeneity test. Gnedenko and Korolyuk (1951) obtained the exact distribution of this statistics; this probability measure D M is defined by: [START_REF] Der Megreditchian | Un test non paramétrique unilatéral de rupture d'homogénéité de "K" échantillons[END_REF] or Gnedenko and Korolyuk (1951)). In our case, because (H 0 ) is true, we can write:

P (D M ≥ x) = 1 when x ≤ 1/M , P (D M ≥ x) = 0 when x ≥ 1 and P (D M ≥ x) = 1 - ⌊1/x⌋ ∑ i=-⌊1/x⌋ (-1) i   2M M -i ⌊M x⌋     2M M   , when x ∈]1/M, 1[ (see Der
D KS ( F L M , F T M ) L = D M .
Suppose now we randomly draw a pair of subsamples S L M and S T M from the data, and let

d := D KS ( F L M , F T M )
be the computed distance between the associated e.d.f.s. If the p-value P (D M ≥ d) is big enough (≥ 0.95, say) the pair (L, T ) is "good" since (H 0 ) may be accepted with little risk.

In this case we will use S L M to build a sequence

{ F L M,m : 1 ≤ m ≤ m 0 } of
"coarsened" e.d.f.s, each F L M,m being described by its jump set given by ( 8). If this isn't the case (i.e. if the pair (L, T ) is "bad"), we draw another pair of subsamples, until (H 0 ) is acceptable.

Suppose now that (H 0 ) is accepted. It is noteworthy that the m th coarsening process introduced above actually consists in replacing each

X L i ∈ [ k m , k+1
m [ by the value k m . In other words, this is a nonlinear transformation

C m : [0, 1] → U m such that x ∈ [ k m , k+1 m [⇒ C m (x) = k m .
Consequently, we consider that the e.d.f. F L M,m has been obtained from a sample of size M of the induced probability distribution, C m * X. Thus, computing D KS

( F L M,m , F L M )
should enable us to decide whether or not the hypothesis (H m ) :

C m * X L = X
is acceptable, i.e. whether or not the coarsening significantly alters the data.

But, since both these e.d.f.s are based on the same learning sample, testing

this hypothesis from D KS ( F L M,m , F L M ) is impossible. On the other hand, us- ing D KS ( F L M,m , F T M )
is straightforward, since S L M and S T M are independent: to accept or reject (H m ), we just have to test whether or not the computed

distance D KS ( F L M,m , F T M ) is an unlikely observation of D M .
We could compute all the distances [START_REF] Beer | Upper semicontinuous functions and the Stone approximation theorem[END_REF].

{ D KS ( F L M,m , F T M ) , 1 ≤ m ≤ m 0 } from if the theoretical d.f. F is continuous, the propositions ∥F K -F ∥ -→ 0 K→∞ and d H (F K , F ) -→ 0 K→∞ are equivalent
To sum up, we will select m * ≤ 21 such that both (H m * ) is acceptable and d H (F N,m * , F N ) is visually small (see the lower panels of Figures 2 &5).

Numerical illustrations

The method is tested hereunder on two data sets which can be found in the classical book of [START_REF] Silverman | Density estimation for Statistics and Data Analysis[END_REF].

The suicide Data

This data set is a classical benchmark in Density Estimation [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF](Leblanc , , 2012a;;[START_REF] Silverman | Density estimation for Statistics and Data Analysis[END_REF][START_REF] Eilers | Flexible smoothing with B-splines and penalties (with discussion)[END_REF], which consists of the duration (in days) of psychiatric treatment for 86 patients used in a study of suicide risks. These durations range between 1 and 737; consequently they must be rescaled to the unit interval with a transformation ψ a,b (x) := x-a b-a . Following [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc ( , 2012a)), we chose a = 0 and b = 800.

Notice that the integer closest to 86/ ln (86) is m 0 = 19, which was also the data-driven optimal choice found by Leblanc (2012a). Plots of the criteria proposed in the previous section are shown in Figure 2 )

: 1 ≤ i ≤ 50 } were lower than 0.80. The reader can see that this is also a satisfactory value from the fidelity point of view (lower panel of Figure 2).

Next, putting together the methodology of [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] and Definition 2, we first fix K, which determines the resolution of the discrete trajectory:

{ P m * -1 ∋ f m * (K+j) := dI K+j m * ;K [F N,m * ](x) dx , 0 ≤ j } associated with the K th "root" of the restricted operator G m * (1/K) . This
trajectory consists in a sequence of polynomials, computed through Proposition 4. Remember that ∀ (m, K) ,

I K m;K [f ] = I 1 m (f ) = B m [f ]. Consequently,
such a trajectory which starts in

F + ∩ F 1 ( f m (K) = d F N,m (x) dx = f N,m (x) is al-
ways bona fide) and progressively get out of this closed convex set (in general,

dI ∞ m;K [F N,m * ](x) dx = dLm[F N,m * ](x) dx / ∈ F + ∩ F 1
). Thus, once m * has been determined, it is quite natural to search for the first

I * > K such that f m * (I * ) belongs to F + ∩ F 1 while f m * (I * +1)
doesn't. For that purpose, we proposed [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] to control the graph of f m * (i)

through two "stresses" : the positivity stress

π(i) := 1 ∫ 0 ( f m * (i) -f m * (i) ) (x)dx, ( 9
)
and the unit total mass stress

ν(i) := 1 ∫ 0 ( f m * (i) + f m * (i) ) (x)dx -2. ( 10
)
The approximation

f m * (i)
is bona fide if and only if both of these stresses are null.

Remark 3. As in [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF], all of the computations are made in the Bernstein basis.

We fixed K = 10, and plotted stresses ( 9) and (10) in Figure 3, together with the Kolmogorov distance (in percents)

K.D.(i) := 100 sup x∈[0,1] x ∫ 0 f m * (K+i) (t)dt -F N (x) .
Since in our case

∫ 1 0 f m * (K+i) (
I K+I * m * ;K [F N,m * ],
which is also a polynomial of degree 18.

On the lower panel, we plotted the three corresponding density estimators.

Please note that the exponential aspect of these three densities have been highlighted in previous studies [START_REF] Eilers | Flexible smoothing with B-splines and penalties (with discussion)[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc , , 2012a)). Nevertheless, kernel [START_REF] Silverman | Density estimation for Statistics and Data Analysis[END_REF] or spline estimators [START_REF] Eilers | Flexible smoothing with B-splines and penalties (with discussion)[END_REF], p. 99) behaved differently from ours near zero. Such differ- Approximation with m 18 and r 1.7 ences likely come from the fact that, contrary to most other methods, Bernstein estimators are well-behaved near boundaries [START_REF] Bouezmarni | Bernstein estimator for unbounded density function[END_REF]Leblanc , 2012b).

The Old faithful data

This data set consists of 107 eruption lengths of the Old Faithful geyser, situated in the Yellowstone National Park. These lengths range between 1.67 and 4.93 minutes. Thus, we embedded these data in the interval [1.5, 5] and rescaled them to the unit interval (following [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]). In this case, m 0 = 23 although Figure 5 shows that choosing m * = 18 is quite reasonable: (of degree 18) while our estimate is rather different: it is similar to estimates obtained by various authors with kernel [START_REF] Silverman | Density estimation for Statistics and Data Analysis[END_REF] p.17, S.T. [START_REF] Chiu | Bandwidth selection for kernel density estimation[END_REF]) p. 1897[START_REF] Sain | Comment of "Flexible smoothing with B-splines and penalties[END_REF]) or spline estimators [START_REF] Eilers | Flexible smoothing with B-splines and penalties (with discussion)[END_REF], pp. 99 and 118). The Bernstein estimator of [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF] was more "bumpy", probably because it belonged to P 65 , while ours lies in P 18 . Nevertheless it is worthwhile to take the comparison of these results a bit further.

Following the pioneering work of [START_REF] Vitale | A Bernstein Polynomial Approach to Density Function Estimation[END_REF], [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF] proved that f N,m (x) := d dx F N,m (x) is biased, and proposed instead the biased-corrected estimator 2 f N,m (x) -f N,m/2 (x). He was inspired by a paper by [START_REF] Politis | Bias-corrected nonparametric spectral estimation[END_REF] which was dedicated to spectral density estimation.

Roughly speaking, the method of Politis & Romano consisted in reducing the bias of the Bartlett spectral estimator f (ω) by computing instead 2 f (ω) -f (ω), where f (ω) was an over-smoothed Bartlett spectral estimator. Now, in the setting of density estimation and with the notations of [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF], if m has been well-chosen, f N,m is a pertinent estimator while f N,m/2 is necessarily an oversmoothed estimator. Consider now our

estimate f m * (2) = dI 2 m * [F N,m * ](x) dx
, and notice that (see formula 5):

I 2 m * [F N,m * ] = ( 1 -(1 -B m * ) 2 ) [F N,m * ] = ( 2 B m * -B 2 m * ) [F N,m * ].
It is well-known [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF] that, because of the eigenvalues of B n , iterated operators B k n act as filters, such that lim

k→∞ B k n [f ] = L 1 [f ]. Thus, f m * (2)
has the same structure as the biased-corrected estimator of [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF], except that the over-smoothed component B 2 m * [F N,m * ] is built differently. This might explain why for a lot of data sets studied the optimal fractional iteration number r * := K+I * K was close to 2. Approximation with m 18 and r 1.9

determines the best subspace while r * corresponds to the optimal number of iterations of an operator acting inside this subspace. Thus, simultaneous bivariate optimization is unnatural.

It is noteworthy that m 0 was independently proposed by [START_REF] Birgé | How many bins should be put in a regular histogram?[END_REF] as an upper number of classes in the setting of automatic histograms construction. Unfortunately, m 0 is generally too big for us (numerical issues), but we stress that big values of m are indeed linked to the sluggish convergence of Bernstein approximations. For instance, the Voronovsky theorem [START_REF] Davis | Interpolation and approximation[END_REF] proves that the rate of uniform convergence of the Bernstein approximation of a twice differentiable function is O (1/m), while the rate of convergence of the best polynomial approximation is much better: it is ε (m) ln (m) /m 2 , with lim m→∞ ε (m) = 0 [START_REF] Laurent | Approximation et optimisation[END_REF], p. 303). In the special case of distribution functions, we proved [START_REF] Manté | Application of iterated Bernstein operators to distribution function and density approximation[END_REF] that one can expect only O (1/ √ m) as a rate of convergence of the Bernstein approximation. However, it is possible to compensate for this drawback thanks to the acceleration of convergence method used here: similar estimates can be obtained either with a small number of iterations in a large space of polynomials or else with a large number of iterations in a smaller space (compare Figures 6 & 7; see also Remark 1 ).

Contrary to the method of [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF], the method of [START_REF] Birgé | How many bins should be put in a regular histogram?[END_REF] gives an optimal value m BR which is generally too small for our purpose. For instance, in the case of the suicide data, it gives the single optimum m BR = 6 ≪ m 0 = 19. In the case of the geyser data, one should choose m BR = 9 ≪ m 0 = 23 (this is the greater optimum of this criterion).

Thus, the associated density estimators are of very low degree (respectively and8). This can be offset by using a large iteration number: indeed, in the first case we could find a good approximation with r * = 5 while in the second case r * = 18 would be correct (see Figure 7). On the contrary, the classical Bernstein estimators, of the same degree, (gray curves on Figure 7) are extremely oversmoothed. This is displayed in more detail on Figure 8, where we plotted the difference between the raw e.d.f. F 107 and its Bernstein estimate B 9 [F 107,9 ], superimposed with F 107 -I 20 9;1 [F 107,9 ], also of degree 9. Thus, using K-fractional Sevy approximation sequence, it is possible to obtain satisfactory estimators, even with low-degree polynomials, when the density is smooth enough! Clearly, in the case of the Old Faithful data, degree 9 is not enough for regions of strong curvature (both the extremities of the curves), but it is enough for weakly curved regions (see Figure 8) when the number of iterations is sufficient.

The main issue is indeed the condition that m ≤ 21, which stems from the numerical problems raised in Section 5.1. These difficulties are due to the ill-conditioning of matrices involved in the expression: M at

( • B n ; L n , W n ) = ΠW [n] •Λ [n] •LΠ [n] .
At first sight, one cannot do anything against the curse of ill-conditioning... Nevertheless, we stressed in Section 5.1 the excessive computational cost of the polynomials calculated from the complicated recurrence formula of [START_REF] Cooper | The eigenstructure of the Bernstein operator[END_REF]. The observed loss of accuracy is probably due to the complexity of these calculations.

Please note that on the one hand M at

( • B n ; W n , W n ) = LW [n] and M at ( • B n ; L n , L n ) = LW [n] (see Lemma 2) while on the other hand, B n [f ] = n ∑ j=0 λ [n] j π [n] j ⊗ π * [n] j (L n [f ]
) . As a consequence, Π [n] can be calculated by polynomial interpolation of the eigenvectors of LW [n] . This is quicker and much more numerically stable, and makes it possible to go beyond the limit m = 21 (for further details, see (Manté , 2014)).

In spite of this exciting perspective, one should be warned that, no matter what, m is necessarily bounded by max [m 0 , M ] (where M is to be determined) because

• for large values of this upper value, the method proposed in Section 6.1

for determining m * would be excessively computationally expensive

• m * itself must be bounded, because of the ill-conditioning problems, and more generally because "it is impractical to employ polynomials with degrees running to hundreds or thousands in "real-world" problems" [START_REF] Farouki | The Bernstein polynomial basis: a centennial retrospective[END_REF].

Indeed, the proposed method is heuristic, constructive, and well-suited for moderate sample sizes in its current state.

Figures captions

Figure 1: plot of the logarithm of the norms

{ ΠW [n] • Λ [n] • LΠ [n] -I n ∞ , 1 ≤ n ≤ 35 } ,
as an indicator of loss of numerical accuracy in the computation of M at

( • B n ; L n , W n ) ,
due to transformation matrices.

{ D KS ( F L i 43,m , F T i 43 ) , 1 ≤ i ≤ 50; 1 ≤ m ≤ m 0 =
{ D KS ( F L i 53,m , F T i 53 ) , 1 ≤ i ≤ 50; 1 ≤ m ≤ m 0 = 23 } , assuming that each D KS ( F L 53,m , F T

Appendix: proofs of intermediate results

Proof of Corollary 1

Since B n [(f -L n [f ])] = 0, we can restrict ourselves to the case where

f ∈ P n = R (B n ). Since { π [n] 0 , • • • , π [n] n
} is a basis of this space, and { µ

[n] 0 , • • • , µ [n] n
} is a basis of P * n we can write, using the standard notation u ⊗ v * (w) := u ⟨v * , w⟩ [START_REF] Bowen | Introduction to Vectors and Tensors Volume 1: Linear and Multilinear Algebra[END_REF]:

B n [f ] = • B n • L n [f ] = n ∑ j=0 λ [n] j π [n] j ⟨ µ [n] j , L n [f ] ⟩ = n ∑ j=0 λ [n] j π [n] j ⊗ π * [n] j (L n [f ])

Proof of Lemma 3

Notice first that:

( 1 -(1 -B n ) I ) = I ∑ k=1 (-1) k-1   I k   B k n .
Then, thanks to Lemma 1, we can write:

I ∑ k=1 (-1) k-1   I k   B k n =   I ∑ k=1 (-1) k-1   I k   • B k n   •L n = ( 1 - ( 1 - • B n ) I ) •L n

Proof of Proposition 2

Remember that

• G n can be considered as a symmetrical bilinear application, whose matrix can be written M at

( • G n ; E [n] , E [n] ) = Q [n] Γ [n] Q t [n]
, where Q [n] is orthogonal and Γ [n] is the diagonal matrix associated with the vector (0, 0, 1/n, (3n -2)/n 2 , • • • , 1 -n!/n n ). Eliminating its two first ) ∥f -g∥

Proof of Proposition 6

Notice first that:

I I m;K [F N ] -F ≤ I I m;K [∆ N ] + I I m;K [F ] -F ,
and, because of Theorem 2, we have the asymptotical relation [START_REF] Laurent | Approximation et optimisation[END_REF] From another side, consider the vector:

δ m;N := { ∆ N (0), ∆ N ( 1 m ) • • • , ∆ N ( m -1 m ) , ∆ N (1)
} .

The • LΠ [m] • δ m;N (see Diagram (3)). In particular:

H K [m;K] (δ m;N ) = LΠ -1 [m] • Λ (1) [m] • LΠ [m] • δ m;N = W L [m] • δ m;N = B m [∆ N ] ,
and of course:

lim I→∞ I I m;K [∆ N ] = LΠ -1 [m] • Λ (
I I m;K [∆ N ] ≤ H I [m;K] ∞ ∥δ m;N ∥ ∞ ∥L m ∥

  , it is quite natural to propose the following definition of fractional sequences. Definition 2. Let K ≥ 2 be an integer, and f ∈ C [0, 1]. The K-fractional Sevy approximation sequence of f is defined by:

  Figure 1:

  Figure 2:

  Figure 3:

  Figure 4:

  Figure 5:

  Figure 6:

  Figure 7:

  Figure 8:

Figure 2 :

 2 Figure 2: (suicide data). Upper panel: box-plots of the p-values of

Figure 3 :

 3 Figure 3: (suicide data) Plot, with the step 1/10, of the stresses ν(i) and π(i) characterizing f 18 (i), against the Kolmogorov distance (in percents)

Figure 4 :

 4 Figure 4: (suicide data). Upper panel: plot of both the Bernstein estimators B 19 [F 86,19 ] of Babu et al. (2002) (dashed gray) and B 18 [F 86,18 ] (gray) superimposed to the proposed one, I 17 19;10 [F 86,19 ] (black), and to F 86 (dots) and the associated Gnedenko confidence bands with coverage probability 0.95 (red) and 0.999 (green). Lower panel: the density estimates, with the same graphic directives as in the upper panel.

Figure 5 :

 5 Figure 5: (geyser data). Upper panel: box-plots of the p-values of

53 )

 53 obeys D 53 . Lower panel: plot of the Hausdorff distances {d H (F 107,m , F 107 ) , 1 ≤ m ≤ 23}.

Figure 6 :

 6 Figure 6: (geyser data). Upper panel: plot of both the Bernstein estimators B 23 [F 107,23 ] of Babu et al. (2002) (dashed gray)and B 18 [F 107,18 ](gray) superimposed to the proposed one, I 19 18;10 [F 107,18 ] (black), and to F 107 (dots) and the associated Gnedenko confidence bands with coverage probability 0.95 (red) and 0.999 (green). Lower panel: the density estimates, with the same graphic directives as in the upper panel.

Figure 7 :

 7 Figure 7: (geyser data). Upper panel: plot of both the Bernstein estimators B 23 [F 107,23 ] of Babu et al. (2002) (dashed gray) and B 9 [F 107,9 ](gray) superimposed to the proposed one, I 17 9;1 [F 107,9 ] (black), and to F 107 (dots) and the associated Gnedenko confidence bands with coverage probability 0.95 (red) and 0.999 (green). Lower panel: the density estimates, with the same graphic directives as in the upper panel.

Figure 8 :

 8 Figure 8: (geyser data). Plot of the differences F 107 -B 9 [F 107,9 ] (gray) and F 107 -I 20 9;1 [F 107,9 ] (black) between the raw d.f. and iterated Bernstein estimates.

Proof of Proposition 5

 5 Let us first define the closed subsetC a,b := {f ∈ C [0, 1] | f (0) = a, f (1) = b}.If F is a d.f. supported by the unit interval, F ∈ C 0,1 while, if F N a e.d.f. associated with a N -sample, (F N -F ) ∈ C 0,0 .Both the inequalities claimed result from the fact that the restrictionB m | C a,b : C a,b → C a,b is a contraction. More precisely,[START_REF] Rus | Iterates of Bernstein operators, via contraction principle[END_REF] proved that for all f, g ∈ C a,b , ∥B m [f -g]∥ ≤ (

  [F ] -F ≤ (1 + ∥L m ∥) inf P ∈Pn ∥P -F ∥ .

  coordinates of I I m;K [∆ N ] in the Lagrange basis are given by H I [m;K] (δ m;N ) := LΠ -

  ∞) [m] • LΠ [m] • δ m;N = δ m;N = L m (∆ N ) .Thus, for each I ≥ K + 1, I I m;K [∆ N ] depends on the matrixH I [m;K] : δ m;N → δ I m;N , such that H I [m;K] (δ m;N ) =

Acknowledgements

The author is very grateful to the referees for their numerous and helpful comments and suggestions, and to Starrlight Augustine for greatly improving the English text.

some good pair (L, T ) and scan the corresponding list of p-values; acceptable values of m will be those for which the p-value is big enough (≥ 0.90, say). But since such random lists are highly fluctuating, it seems preferable to perform a reasonable number (e.g. 50) of good trials (such that (H 0 ) is acceptable), and to summarize the obtained 50 lists of p-values by the associated m 0 box-plots (see the upper panel of Figures 2 &5).

The reader can see in the upper panel of Figure 2, for instance, that pvalues corresponding to m < 7 are very small. Consequently, such a coarsening would deeply alter the histogram structure. On the contrary, for m > 15 most p-values are greater than 0.5 and we can conclude that such a coarsening is quite acceptable.

A complementary fidelity criterion

We just proposed a method for obtaining a list of acceptable numbers of bins in histograms, but there are generally several candidates. To select one of them, we proceed with the complete sample S N . This time, we compute the list of Hausdorff distances {d H (F N,m , F N ) , 1 ≤ m ≤ m 0 }, which quantify the similarity of successive coarsened distributions with the complete e.d.f. (see the lower panel of Figures 2 &5). Notice that these coarsened distributions are tightly associated with the classical estimator [START_REF] Vitale | A Bernstein Polynomial Approach to Density Function Estimation[END_REF][START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF][START_REF] Bouezmarni | Bernstein estimator for unbounded density function[END_REF][START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc , , 2012a,b) ,b) through Eq. ( 7). [START_REF] Beer | Upper semicontinuous functions and the Stone approximation theorem[END_REF] and [START_REF] Cuevas | On visual distances in density estimation : the Hausdorff choice[END_REF] : d H is a metric in the space of Upper Semi Continuous (USC) functions, and any d.f. is USC. Furthermore,

Remark 2. The choice of the Hausdorff distance is supported by the works of

Discussion

In this paper we propose an original method for estimating distribution functions and densities with Bernstein polynomials. On the one hand, we take advantage of results about the eigenstructure of the Bernstein operator to improve Sevy's convergence acceleration method. On the other hand, we work out an original adaptative method for choosing the number of bins m of a regular histogram. As [START_REF] Birgé | How many bins should be put in a regular histogram?[END_REF] noticed: this is an old and still open problem. In the setting of Bernstein estimation of distribution functions and densities, [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] proposed the upper value m 0 := N/ ln (N ) as an "acceptable" solution to this problem, even if one should theoretically choose m = o (N/ ln (N )). In theory, the number of bins should not be the same for fitting both the d.f. and the density. [START_REF] Leblanc | A Bias-reduced approach to density estimation using Bernstein polynomials[END_REF]Leblanc ( , 2012a) ) ) 2

) in the latter one.

Thus the density estimator should be built from a smoother d.f. estimator than the optimal d.f. estimator itself. A similar result was proved by [START_REF] Hjort | A note on kernel density estimators with optimal bandwidth[END_REF] regarding kernel density estimation. This is probably due to the fact that, roughly speaking, the differentiation operator is a high-pass filter whose action must generally be balanced by smoothing.

Our two-step method takes both functions into account: m * ≤ m 0 -wellsuited for density estimation [START_REF] Babu | Application of Bernstein polynomials for smooth estimation of a distribution and density function[END_REF] is first tuned according to the structure of the e.d.f., and then r * is tuned according to the density which should be bona fide. These steps cannot be interchanged because m * eigenfunction (basis of Ker(

amounts to restrict this operator to P n .

As a consequence, we may write M at

where and Γ [n] is the diagonal matrix associated with the vector (1/n, (3n -2)/n 2 , • • • , 1 -n!/n n ). Then, we have first: log

where ∆ [n] is the diagonal matrix associated with the vector

Proceeding the same way with the exponential operator, we find:

t

Proof of Proposition 3

We demonstrate this proposition by induction. Firstly, note that I 1 n (P ) = B n (P ) = P 1 +B n ( P ) = P 1 +I 1 n ( P ) ; let us now suppose that for some k > 1, I k n (P ) = P 1 + I k n ( P ) . Then:


Vincent Luboz

Emmanuel Promayon

Yohan Payan

come

Measurements of the linear elastic properties of the face soft tissues using an aspiration device

Measurements of the linear elastic properties of the face soft tissues using an aspiration device Vincent Luboz, Emmanuel Promayon, Yohan Payan Abstract: Maxillo-facial surgeries have a direct impact on the face shape and appearance. The anatomical variations in the face soft tissues (skin thickness, skin or muscle material parameters…) make it hard to predict the face final shape accurately. Biomechanical modeling could help in this prediction by simulating the soft tissue displacement following the bone remodeling and therefore the patient's face visible outcome [START_REF] Chabanas | Comparison of linear and non-linear soft tissue models with postoperative CT scan in maxillofacial surgery[END_REF]. Measuring the variation of the skin properties across a range of subjects could help to build accurate patient-specific biomechanical simulations. In [START_REF] Barbarino | Experimental and numerical study on the mechanical behavior of the superficial layers of the face[END_REF], the skin initial Young's modulus was evaluated to 22.8 kPa using a Mooney Rivlin model and measurements on different locations on the face of one subject. In this study, a specifically designed aspiration device, called LASTIC (Light Aspiration device for in vivo Soft TIssue Characterization), is used to characterize the elastic modulus of the face soft tissues at four different locations (cheek, cheekbone, forehead, and lower lip) on 16 subjects. LASTIC is sterilizable and could therefore be used in the operating room. It is a two-compartment 33 mm x 34 mm metal cylinder using the aspiration technique to estimate the elastic modulus [START_REF] Schiavone | In Vivo Measurement of Human Brain Elasticity Using a Light Aspiration Device[END_REF]. The bottom compartment is an airtight chamber, opened at the bottom by a 12 mm diameter circular aperture lying on the soft tissues where a negative pressure can be applied to aspirate the tissues. The upper compartment holds a miniature digital camera which images the height of the deformation resulting from the aspiration. Measuring the deformation height corresponding to several steps of increasing negative pressures leads to an estimation of the behavior of the tissues using an inverse analysis based on a Neo Hookean model. To study the possible variations between subjects, LASTIC was used to estimate the elastic moduli of 16 healthy subjects, eight males and eight females, of different ages and body mass indexes (mean age = 29.2±6.3, mean BMI = 21.5±2.1). The initial Young's modulus of the cheek, cheekbone, forehead, and lower lip are respectively estimated to be in the range [31 kPa: 42kPa]. Significant intra-subject differences in tissue stiffnesses between the cheek and forehead and between the lower lip and forehead are highlighted by these estimations, and can be explained by the tissue thickness. They also show important inter-subject variabilities for some locations even when mean value stiffness show no statistical difference. This study stresses the importance of using a measurement device capable of evaluating the patient-specific tissue elastic moduli before and during an intervention in order to improve the accuracy of a surgery outcome.


Gabriel Antherieu
email: gabriel.antherieu@imag.fr

Yohan Payan

Denis Favier

Nathanaël Connesson

Pierre Mozer

An innovative NiTi based stent as an emergency treatment for acute urinary retention in case of benign prostatic hyperplasia

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

As life expectancy is increasing, population is getting older. Ageing diseases are thus becoming more and more numerous. Benign Prostatic Hyperplasia (BPH) is a pathology mostly striking elder men : 40% of men over 50 years old show evidences of BPH [START_REF] Mcnicholas | Benign prostatic hyperplasia[END_REF]. BPH consists in an anarchic proliferation of prostatic stromal and epithelial tissues [START_REF] Mcnicholas | Benign prostatic hyperplasia[END_REF], resulting in an increase of the prostate volume. Since the prostatic urethra is surrounded by the prostate, BPH may eventually obstruct it, leading to Acute Urinary Retention (AUR). AUR is one of the most problematic complications of BPH, and requires an emergency treatment. Today, the first line treatment of AUR consists in catheterization. However, this procedure comes with various drawbacks :

• it is uncomfortable, and often painful,

• it reduces the patient autonomy and social life (due to the catheter and drainage bag),

• it may lead to Nosocomial Urinary Tract Infection (NUTI). Urinary infection is a key issue, since catheterization is responsible for 80% of NUTI [START_REF] Plowman | An economic model to assess the cost and benefits of the routine use of silver alloy coated urinary catheters to reduce the risk of urinary tract infections in catheterized patients[END_REF]. It is worth noting that infection risk increases by 3% to 10% each passing day [START_REF] Alfandari | Prévention des infections urinaires nosocomiales : effets de lâinfection urinaire nosocomiale sur la durée de séjour, le coût et la mortalité[END_REF]. Patients presenting AUR due to BPH are often old, polypathologic, and thus more likely to develop complications due to NUTI. Beyond the medical concern, NUTI is also responsible for a dramatic increase of the treatment costs.

To circumference these issues, an innovative NiTi based stent has been developped as an alternative to catheterization.

Urethral stenting and shape memory alloys

First intra urethral stentings began around 1985, notably with the P rostakath R prosthesis [START_REF] Yachia | Stenting the urinary system[END_REF]. Various urethral stents have been developped ever since. Urethral stenting enables to treat AUR, with a minimized infection risk and discomfort. However, due to their complex and time-consuming implantation mechanism (for instance M emokath R stent requires sterile hot/cold water for its expansion/removal), existing conventional stents are inconvenient as an emergency treatment, and thus remain scarcely used.

The aim of the work introduced is this paper is to provide a suitable solution for the emergency treatment of AUR. In order to fulfil the clinical needs, this stent presents different innovative features :

• at least two distinct shape memories triggered successively by increasing its temperature : (1) an extension phase to release the urethra, and (2) a contraction phase to easily remove the stent, • a bio-compatible silicone coating. This coating prevents the stent, heated using Joule effect, from damaging surrounding tissues, • a global implantation procedure suitable for emergency treatment. Double shape memory effect is achieved using NiTi shape memory alloy (SMA). SMA show interesting properties, as a rough description [START_REF] Otsuka | Physical metallurgy of TiâNi-based shape memory alloys[END_REF] :

• Through thermal treatments, SMA is given a transition temperature (T trans ), associated to the "set shape" of the alloy. • For T < T trans , SMA is in martensitic state, and can be easily pseudo plastically deformed (SAM shows pseudo-plasticity around 1% strain). • For T > T trans , SMA austenitic state is triggered and presents super elasticity (up to 12%). • When heated above T trans , an initially deformed sample almost instantly recovers its set shape, if its shape is unconstrained : this is the shape memory effect.

Stent design

To achieve a two shape memory stent, a specific design was developed. (1) A NiTi wire, diameter 0.5mm, was given for shape memory an helical shape at diameter D wire , and a T trans = T wire (Fig 1,a).

(2) A NiTi tube, inner diameter=0.5mm and wall thickness=0.05mm, was given for shape memory an helical shape at diameter D tube > D wire , and a T trans = T tube < T wire (Fig 1,b). The wire was then inserted inside the tube.

The stent obtained this way shows three different behaviours and shapes, according to its temperature T :

• For T < T tube < T wire , both wire (1) and tube [START_REF] Mcnicholas | Benign prostatic hyperplasia[END_REF] are in martensitic state. The stent can be easily deformed (Fig 1,c). Notably, the stent can easily be rolled around a specific insertion catheter. • For T tube < T < T wire , the tube (2) shape memory is triggered while the wire (1) remains in martensitic state. The stent is thus expanded, and releases the urethra (Fig 1,d). The patient is then able to return home during HBP treatment, without discomfort or NUTI risk. • For T tube < T wire < T , the wire (1) shape memory is triggered. The stent collapses, allowing an easy removal (Fig 1 ,e). An alternative solution to obtain a double shape memory is to use a single wire, with two distinct set shapes associated to two distinct T trans . This can be achieved using local thermal treatment, such as laser treatment for example.

To ensure that the stent is suitable for emergency use, the activation (T > T tube ) will be made using Joule effect, a custom made insertion catheter enabling electrical contact between the stent and the power supply. In fact, currently available urethral stents are mostly activated using hot water. Consequently, the clinical staff has to heat sterilized water and then to inject it inside the stent to expand it. It is understandable that this type of activation prevents the stent from being used in emergency. Using Joule effect allows to activate the stent fastly (less than 10s), without any specific procedure. To ensure that the urethral tissues are not burnt during activation, the stent is coated with a biocompatible silicone layer. A preliminary axis-symmetric thermal simulation provided the duration of the release operation t activation = 8s, and the maximum temperature of the urethra T max urethra = 57˚C. This temperature can easily be reduced by increasing the thickness of the silicone coating, knowing that for the simulation the coating thickness was 0.25mm.

Conclusion

A innovative NiTi based stent has been developped. This stent is meant to be used as an emergency treatment for AUR due to BPH as an alternative to catheterization. Thanks to its double shape memory effect, the stent can be easily expanded, and removed, without hurting surrounding tissues. As a consequence, this stent will reduce the risk of NUTI and associated morbidity.

Acknowledgement

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme (Labex CAMI) under reference ANR-11-LABX-0004.

Figure 1 :

 1 Figure 1: The double shape memory stent : an helical wire (a) inserted into an helical tube (b) ; and its various configurations (c,d,e)