Fabienne Comte

Claire Lacour

Data driven

HAL is

Introduction

Let us consider the following model:

(1) Y j = X j + ε j j = 1, . . . , n

where (X j ) 1≤j≤n and (ε j ) 1≤j≤n are independent sequences of i.i.d. variables. We denote by f the density of X j and by f ε the density of ε j . The aim is to estimate f when only Y 1 , . . . , Y n are observed. In the classical convolution model, f ε is assumed to be known, and this is often considered as an important drawback of this simple model. In many contexts however, preliminary calibration measures can be obtained in the absence of any signal X. This can be done each time a physical machine takes measures of a signal; when no signal is in input, only the noise is measured. In that case, the knowledge of f ε can be replaced by the observations of ε -1 , . . . , ε -M , a noise sample with distribution f ε , independent of (Y 1 , . . . , Y n ). Thus, a study can be conducted, in which we do not assume that f ε is known. Note that the availability of two distinct samples makes the problem identifiable.

On the one hand, there exists a huge literature concerning the estimation of f when f ε is known: see [START_REF] Carroll | Optimal rates of convergence for deconvolving a density[END_REF], [START_REF] Devroye | Consistent deconvolution in density estimation[END_REF], [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], [START_REF] Liu | A consistent nonparametric density estimator for the deconvolution problem[END_REF], [START_REF] Masry | Multivariate probability density deconvolution for stationary random processes[END_REF], [START_REF] Stefanski | Deconvoluting kernel density estimators[END_REF], [START_REF] Zhang | Fourier methods for estimating mixing densities and distributions[END_REF], [START_REF] Hesse | Data-driven deconvolution[END_REF], [START_REF] Cator | Deconvolution with arbitrarily smooth kernels[END_REF], [START_REF] Delaigle | Bootstrap bandwidth selection in kernel density estimation from a contaminated sample[END_REF] for kernel methods, [START_REF] Koo | Logspline deconvolution in Besov space[END_REF] for a spline method, [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF] and [START_REF] Fan | Wavelet deconvolution[END_REF] for wavelet strategies, , [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF] and [START_REF] Butucea | Sharp optimality in density deconvolution with dominating bias[END_REF] for studies of minimaxity of the rates, [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF], [START_REF] Comte | Finite sample penalization in adaptive density deconvolution[END_REF] for adaptive projection strategies. On the other hand, several authors have studied the exact problem which is considered in this paper, but only for particular type of smoothness for f ε or f or other type of risks. In this regard, we provide the first study of pointwise mean square risk (MSE). Moreover, we provide a general study of the mean integrated squared error (MISE) which substantially generalizes existing results. Then, to propose a model selection strategy, we use a minimum projection contrast expression of our collections of estimators: they depend on a bandwidth-type parameter for which we consider the difficult problem of automatic selection of this quantity. In other words, we explain how to select a relevant estimator in the collection: the study of an adaptive procedure in this context is essentially new.

Let us describe what has been done on the subject.

- [START_REF] Diggle | A Fourier approach to nonparametric deconvolution of a density estimate[END_REF] consider the same model and obtain result of the same type of our first Proposition (see Proposition 2). But then, they study the case of ordinary smooth noise and distribution function under M ≥ n.

-We may mention the work by [START_REF] Efromovich | Density estimation for the case of supersmooth measurement error[END_REF], but his context is slightly different, since he considers circular data. He assumes that the noise is supersmooth and the distribution function ordinary smooth. In this context, he proposes a data driven choice of M to make his estimator adaptive.

-Our work is more related to the paper of [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], since our estimator is rather equivalent to his and we borrow a useful Lemma from his work. He mainly considers the case of both ordinary smooth noise and distribution function. He does not perform any bandwidth selection, but he proves the minimax optimality of the bound he obtains in the case he considers. We shall of course refer to this lower bound.

- [START_REF] Meister | On the effect of misspecifying the error density in a deconvolution problem[END_REF] takes a rather different point of view, compared to our problem: he studies what happens when the function f * ε used for estimation is not the true one. For instance, he shows that it is safer to use an ordinary smooth noise characteristic function, if it is unknown.

-Lastly, [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF] recently studied the density deconvolution with unknown (but observed) noise and he is interested in the relation between M and n. Note that his estimator and his approach are very interesting and rather different from ours, his estimator depends on two bandwidth-type parameters, which, if relevantly chosen, lead to rate that are the same as in our work. But the data-driven selection of these bandwidths is not done.

Note that a similar question in the context of inverse problem is studied in [START_REF] Cavalier | Wavelet deconvolution with noisy eigen-values[END_REF].

Here is the plan of the present paper. In Section 2,we give the notations and define the estimator, first directly, and then as a projection-type estimator. We study in Section 3 both the pointwise and the integrated mean square risk (MSE and MISE) of one estimator, which allows to build general tables for the rates in both cases. Then, we study the link between M and n if one wants to preserve the rate found in the case where f * ε is known. Such a complete panorama is new in this setting. Next, we define and study in Section 4 an adaptive estimator by proposing a penalization device. A general integrated risk bound for the resulting estimator is given. The estimator is studied through simulation experiments, and its performances are compared with Neumann (1997)'s and Johannes (2007)'s ones. The influence of the size M of the noise sample is studied as well as the importance of some other theoretical constraints on the size of the collection of models. Most proofs are gathered in Section 6.

Estimation procedure

2.1. Notations. For z a complex number, z denotes its conjugate and |z| its modulus. For functions s, t : R → R belonging to L 1 ∩ L 2 (R), we denote by t the L 2 norm of t, that is t 2 = R |t(x)| 2 dx, and by s, t the scalar product: s, t = R s(x)t(x)dx. The Fourier transform t * of t is defined by

t * (u) = e -ixu t(x)dx
Note that, if t * also belongs to L 1 ∩ L 2 (R), then the function t is the inverse Fourier transform of t * and can be written t(x) = 1/(2π) e ixu t * (u)du. Finally, the convolution product is defined by (t * s)(x) = t(xy)s(y)dy.

2.2.

Basic definition of the estimator. It follows easily from Model (1) and independence assumptions that, if f Y denotes the common density of the Y j 's, then

f Y = f * f ε and thus f * Y = f * f * ε .
Therefore, under the classical assumption:

(A1) ∀x ∈ R, f * ε (x) = 0, the equality f * = f * Y /f * ε yields an estimator of f * by considering the following estimate of f * Y : f * Y (u) = 1 n n j=1 e -iuY j . Indeed, if f * ε is known, we can use the following estimate of f * : f * Y /f * ε .
Then, we should use inverse Fourier transform to get an estimate of f . As 1/f * ε is in general not integrable (think of a Gaussian density for instance), this inverse Fourier transform does not exist, and a cutoff is used. The final estimator for known f ε can thus be written:

|u|≤πm e iux f * Y (u)/f * ε (u)du.
This estimator is classical in the sense that it corresponds both to a kernel estimator built with the sinc kernel (see [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF]) or to a projection type estimator as in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF], as will be showed below. Now, f * ε is unknown and we have to estimate it. Therefore, we use the preliminary sample and we define

f * ε (x) = 1 M M j=1 e -ixε -j
the natural estimator of f * ε . Next, we introduce as in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] the truncated estimator:

(2)

1 f * ε (x) = 1 {| f * ε (x)|≥M -1/2 } f * ε (x) =    1 f * ε (x) if | f * ε (x)| ≥ M -1/2 0 otherwise.
Then we can consider

(3) fm (x) = 1 2π πm -πm e ixu f * Y (u) f * ε (u) du.
Note that this estimator is such that ( fm -πm,πm] . This formula is the only needed for the study of the MSE and the MISE, but it is not convenient to present the strategy which is required to select the parameter m. Indeed, m plays a bandwidth-type role, and has to be relevantly selected to lead to an adequate bias variance compromise. And the definition of this procedure must be done in terms that leads to MISE bounds on the final estimator.

) * = ( f * Y / f * ε )1 [
* , j in Z, ϕ m,j (x) = √ mϕ(mx -j). As ϕ * (x) = 1 [-π,π] (x),
we have, as a key property of the functions ϕ m,j , that ϕ * m,j (x) = e -ixj/m 1 [-πm,πm] (x)/ √ m. Note that {ϕ m,j } j∈Z is an orthonormal basis of the space of integrable functions having a Fourier transform with compact support included into [-πm, πm]. Note that m can be chosen in other sets than N * , and thinner grids may be useful in practice.

In the sequel, we use the following notation:

S m = Span{ϕ m,j } j∈Z .
We know (see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]) that the orthogonal projection of a function g in

(L 1 ∩ L 2 )(R) on S m , denoted by g m , is such that g * m = g * 1 [-πm,πm] ,
i.e. with Fourier inverse formula:

(4) g m (x) = 1 2π πm -πm
e ixu g * (u)du.

This explains why the order of bias terms is the same for the two expressions of the estimator.

2.3.2. Estimation of f for the classical deconvolution problem. We want to estimate f , the density of the X j in model (1). When f ε is known, we can estimate f by minimizing a contrast built as follows. A standard contrast in density estimation is

1 n j [ t 2 -2t(X j )].
It is not possible to use this contrast in the convolution model because we do not observe X 1 , . . . , X n . Only the noisy data Y j are available. The solution is given by exploiting the following lemma.

Lemma 1. For any function t, let v t be the inverse Fourier transform of t * /f * ε (-.), i.e.

v t (x) = 1 2π e ixu t * (u) f * ε (-u) du.
Then, for all 1 ≤ j ≤ n,

(1) E[v t (Y j )|X j ] = t(X j ) (2) E[v t (Y j )] = E[t(X j )]
The second assertion in Lemma 1 is an obvious consequence of the first one and leads us to consider the following contrast:

γ 0 n (t) = 1 n n j=1 [ t 2 -2v t (Y j )] with v * t (u) = t * (u) f * ε (-u)
.

Indeed, since t(X j ) and v t (Y j ) have the same expectation, it is natural to replace the unknown quantity t(X j ) in the contrast by v t (Y j ). We can observe that

Eγ 0 n (t) = (1/n) n j=1 [ t 2 -2E[v t (Y j )]] = (1/n) n j=1 [ t 2 -2E[t(X j )]] = t 2 -2 tf = t -f 2 -f 2 .
This contrast is used in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] to define a collection of estimators on each space S m and then a penalty is given to select an adequate space S m .

2.3.3. Estimation of f if the distribution of ε is unknown. Now, f * ε is unknown and we replace it by the estimator (2). We shall study in the following the new contrast

(5) γ n (t) = 1 n n j=1 [ t 2 -2ṽ t (Y j )] with ṽ * t (u) = t * (u) f * ε (-u) .
We define our estimators by minimizing this contrast on the projection spaces S m : It is sufficient to differentiate the contrast (5) to obtain formula (7). Actually, we should define fm = |l|≤Kn âml ϕ m,l because we can estimate only a finite number of coefficients. If K n is suitably chosen, it does not change the rate of convergence since the additional terms can be made negligible. For the sake of simplicity, we let the sum over Z. For an example of detailed truncation see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]. The notation fm is the same because the estimators coincide. Indeed, starting with (7), we have the following equalities:

fm (x) = l∈Z 1 n n j=1 ṽϕ m,l (Y j )ϕ m,l (x) = 1 2π l∈Z f * Y (-u) ϕ * m,l (u) fε (-u) du ϕ m,l (x) = 1 2π l∈Z f * Y f * ε , ϕ * m,l ϕ m,l (x) = 1 2π l∈Z f * Y f * ε * (-.), ϕ m,l ϕ m,l (x). (8)
This is the expression of the orthogonal projection on S m of (1/2π)( f * Y / f * ε ) * (-.). Using (4) and ( 8) yields (3). In practice, the coincidence is not exact because the sums over Z are truncated.

To complete the estimation procedure, we choose the best estimator among the collection ( fm ) m∈Mn where M n ⊂ {1, . . . , n} is the set of all considered indexes. To do this, we select the model which minimizes the following penalized criterion: where pen is a penalty term to be specified later. Finally, we consider f m as estimator of the density.

3. Bound on the L 2 risk 3.1. Notations. Let us recall first the following key lemma, proved in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] for p = 1:

Lemma 2. Let p ≥ 1 be an integer and

R(x) = 1 f * ε (x) - 1 f * ε (x)
.

Then there exists a positive constant C p such that

E|[R(x)| 2p ] ≤ C p 1 |f * ε (x)| 2p ∧ M -p |f * ε (x)| 4p
. The extension from p = 1 to any integer p is straightforward and therefore the proof is omitted. Moreover, we introduce the notations (10)

∆(m) = 1 2π πm -πm |f * ε (u)| -2 du and ∆ 0 (m) = 1 2π πm -πm |f * ε (u)| -1 du 2 and (11) ∆ f (m) = 1 2π πm -πm |f * (u)| 2 |f * ε (u)| 2 du and ∆ 0 f (m) = 1 2π πm -πm |f * (u)| |f * ε (u)| du 2 .
As we shall see, these quantities are involved in the bounds on the variance of our estimators.

3.2. Pointwise mean square risk. First, we study quickly the so-called MSE, the pointwise mean square error of the estimator. Let us denote by f m the orthogonal projection of f on S m . Then we have the following decomposition:

E[( fm (x) -f (x)) 2 ] ≤ 2(f m (x) -f (x)) 2 + 2E[( fm (x) -f m (x)) 2 ] ≤ 2(f m (x) -f (x)) 2 + 4Var 1 2π πm -πm e ixu f * Y (u) f * ε (-u) du +4E 1 2π πm -πm e ixu f * Y (u)R(u)du 2 (12)
The first (squared bias term) and second (variance term) terms of the right-hand-side of (12) are the usual ones, and are also found when f * ε is known; they are studied in [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF]. The last one is studied by analogous methods and Lemma 2, and is specific to the present context. We find that the following risk bound holds:

Proposition 1. Consider model ( 1) under (A1), then fm defined by (6) satisfies:

E[( fm (x) -f (x)) 2 ] ≤ 2 1 2π |t|≥πm |f * (t)|dt 2 + C n inf( |f * Y |∆(m), ∆ 0 (m)) + C ∆ 0 f (m) M It follows that, as f belongs to L 2 (R), if |f * Y |
< +∞ and M ≥ n, then the risk bound obtained in Proposition 1 is the same as the one we get when

f * ε is known. Indeed, ∆ 0 f (m) ≤ f 2 ∆(m) and ∆ 0 f (m) ≤ ∆ 0 (m).

This is summarized by the Corollary:

Corollary 1. Consider model ( 1) under (A1). Assume moreover that |f * Y | < +∞ and M ≥ n. Then fm defined by (6) satisfies:

(13) E[( fm (x) -f (x)) 2 ] ≤ 2 1 2π |t|≥πm |f * (t)|dt 2 + K n inf(∆(m), ∆ 0 (m)),
where K is a constant depending on |f * Y | and f . 3.3. Pointwise rates under regularity conditions. Assumption (A1) is generally strengthened by a parametric description of the rate of decrease of f * ε written as follows:

(A2) There exist s ≥ 0, b > 0, γ ∈ R (γ > 0 if s = 0) and k 0 , k 1 > 0 such that k 0 (x 2 + 1) -γ/2 exp(-b|x| s ) ≤ |f * ε (x)| ≤ k 1 (x 2 + 1) -γ/2
exp(-b|x| s ) Moreover, the distribution function f to estimate generally belongs to the following type of smoothness spaces:

A δ,r,a (l) = {f density on R and |f * (x)| 2 (x 2 + 1) δ exp(2a|x| r )dx ≤ l} (14) with r ≥ 0, a > 0, δ ∈ R and δ > 1/2 if r = 0, l > 0.
When r > 0, the function f is known as supersmooth, and as ordinary smooth otherwise. In the same way, the noise distribution is called ordinary smooth if s = 0 and supersmooth otherwise. The spaces of ordinary smooth functions correspond to classic Sobolev classes, while supersmooth functions are infinitely differentiable. It includes for example normal (r = 2) and Cauchy (r = 1) densities. We take the convention (a, r)

= (0, 0) if a = 0 or r = 0 and (b, s) = (0, 0) if b = 0 or s = 0. Remark. If f * ε satisfies (A2) and f belongs to A δ,r,a (l) as defined in (14), then |f * Y | 2 = |f * ε f * | 2 ≤ k 2 1 l (x 2 + 1) -(γ+δ) exp(-2b|x| s -2a|x| r )dx. Thus definition (14) implies that |f * Y | < +∞.
The optimality (minimaxity) of the rates resulting from (13) for known f * ε , when f * ε satisfies (A2) and f belongs to A δ,r,a (l), has been studied in [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF] and [START_REF] Butucea | Sharp optimality in density deconvolution with dominating bias[END_REF].

More generally, we can see that, if f * ε satisfies (A2) and if f ∈ A δ,r,a (l):

∆ 0 f (m) ≤ l 2πk 2 0 πm -πm (x 2 + 1) γ-δ exp (2b|x| s -2a|x| r ) dx.
Now, we can combine the rates related to M with standard pointwise rates (see Table 1 in [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF] or Table 2 in Butucea andComte (2007)) andwe obtain the Table 2 here.

s = 0 s > 0 r = 0 n -2δ-1 2δ+2γ + M -[1∧( 2δ-1 2γ )] (log M ) u (log n) -2δ-1 s + (log M ) -2δ-1 s u = 1 δ=γ+1/2 r > 0 (log n) 2γ+1 r n + 1 M see the discussion below. Table 1. Rates of convergence for the MSE if f * ε satisfies (A2) and f ∈ A δ,r,a (l).
We discuss the case r > 0, s > 0 for the integrated risk only, and thus omit this part of the study here. The principle would be the same, with slightly different orders. See also [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF].

3.4. Bound on the MISE. We shall study in more detail the integrated mean square risk, which is slightly simpler. Indeed now, by Pythagoras theorem, we have

(15) f -fm 2 = f -f m 2 + f m -fm 2 .
Moreover, writing fmf m according to (4) and ( 3) and applying the Parseval formula, we obtain

f m -fm 2 = 1 2π πm -πm f * Y (u) f * ε (u) - f * Y (u) f * ε (u) 2 du.
It follows that ( 16)

f m -fm 2 ≤ 1 π πm -πm | f * Y (u)| 2 |R(u)| 2 du + 1 π πm -πm | f * Y (u) -f * Y (u)| 2 |f * ε (u)| 2 du.
The last term of the right-hand-side of ( 16) is the usual term that is found when f * ε is known, and the first one is specific to the present framework.

We can prove the following result:

Proposition 2. Consider model ( 1) under (A1), then fm defined by (6) satisfies:

(17) E( fm -f 2 ) ≤ f m -f 2 + C ∆(m) n + C ∆ f (m) M with C and C numerical constants.
The first two terms in the right-hand-side of ( 17) are the usual terms when f * ε is known (see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]) and correspond to the bias and the variance term. The last term

∆ f (m)/M is due to the estimation of f * ε . Remark. As |f * (x)| ≤ 1, we have ∆ f (m) ≤ ∆(m). It follows that for any M ≥ n, then E fm -f 2 ≤ f m -f 2 +C∆(m)
/n and we recover the usual risk bound for deconvolution estimation when f * ε is known. Therefore, in all cases, the condition M ≥ n ensures that the rate of the estimator is the same as when f * ε was known. 3.5. Discussion about the resulting rates. In this section, we assume that f * ε satisfies Assumption (A2), with parameters γ, b, s and that the unknown function f belongs to a smoothness class A δ,r,a (l) given by ( 14). It is then possible to evaluate orders for the different terms involved in the bound (17).

Since

f * m = f * 1 [-πm,πm]
, the biais term can be bounded in the following way

f -f m 2 = 1 2π ([-πm,πm]) c |f * (u)| 2 du ≤ l 2π ((πm) 2 + 1) -δ e -2a(πm) r
The other terms are evaluated in the following lemma proved in Section 6.

Lemma 3. If f * ε satisfies Assumption (A2) then (1) ∆(m) (πm) 2γ+1-s e 2b(πm) s , (2) ∆ f (m) (πm) (1+2γ-s)∧2(γ-δ) + e 2b(πm) s 1 {s>r} + (πm) 2(γ-δ) + e 2(b-a)(πm) s 1 {r=s,b≥a} +1 {r>s}∪{r=s,b<a} .
Now distinguishing the different cases, we can state the following propositions.

Proposition 3. Assume that (A2) holds and that f ∈ A δ,r,a (l) given by ( 14). If s = 0 (ordinary smooth noise) and r = 0 (ordinary smooth function f ), then

E fm -f 2 ≤ C 0 m -2δ + C m 2γ+1 n + C m 2(γ-δ) +
M where C 0 , C and C are constants which do not depend on M nor n.

It is known from [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], that the optimal minimax rate when

f * ε is known is n -2δ 2γ+2δ+1 . It is preserved with unknown f * ε as soon as M ≥ n 2(γ∨δ)
2γ+2δ+1 . This bound is tighter than

M ≥ n. Now, choose m 0 = Int[n 1 2γ+2δ+1 ∧M 1 2(γ∨δ) ]
where Int[.] denotes the integer part. We obtain

E fm 0 -f 2 = O n - 2δ 2γ+2δ+1 + M -(1∧(δ/γ)) .
This is the lower bound proved by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], and thus the rate of our estimator is the optimal rate. Proposition 4. Assume that (A2) holds and that f ∈ A δ,r,a (l) given by ( 14). If s > 0 (supersmooth noise) and r = 0 (ordinary smooth function f ), then

E fm -f 2 ≤ C 0 m -2δ + C m 2γ+1-s e 2b(πm) s n + C m (1+2γ-s)∧2(γ-δ) + e 2b(πm) s M ,
where C 0 , C and C are constants which do not depend on M nor n.

For known f * ε , [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] proves that the optimal rate is of order (log n) -2δ s . It is preserved here with unknown f * ε as soon as

M ≥ n(log n) -s+[2(δ∧γ)+1-s] + s . Choose m 0 = Int[(1/π) 1 2b log[n(log n) -2δ+2γ+1 s ∧ M (log M ) -2δ+s+(1+2γ-s)∧2(γ-δ) + s ] 1/s ].
This yields

E fm 0 -f 2 = O (log n) -2δ s + (log M ) -2δ s .
Proposition 5. Assume that (A2) holds and that f ∈ A δ,r,a (l) given by ( 14). If s = 0 (ordinary smooth noise) and r > 0 (supersmooth function f ), then

E fm -f 2 ≤ C 0 m -2δ e -2a(πm) r + C m 2γ+1 n + C M ,
where C 0 , C and C are constants which do not depend on M nor n.

The optimal rate in this case is studied by [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF] when f * ε is known and is of order

(log n) 2γ+1 r /n. It is preserved even when f * ε is estimated, if the sample size for estimating it, M , is such that M ≥ n(log n) -2γ+1 r . Let us choose now m 0 = Int[(1/π) 1 2a log[n(log n) r-2δ-2γ-1 r ∧ M (log M ) -2δ r ] 1/r ].
We get

E fm 0 -f 2 = O (log n) 2γ+1 r n + 1 M .
This is summarized in Table 2.

s = 0 s > 0 r = 0 n -2δ 2δ+2γ+1 + M -[1∧( δ γ )] (log n) -2δ s + (log M ) -2δ s r > 0 (log n) 2γ+1 r n + 1 M see the discussion below.
Table 2. Rates of convergence for the MISE.

The last case, when both functions are supersmooth, is much more tedious, in particular if one wants to evaluate the rates. These are implicitly given in [START_REF] Butucea | Sharp optimality in density deconvolution with dominating bias[END_REF], who also study optimality; explicit formulae are available in [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF], see Theorem 3.1 therein. Proposition 6. Assume that (A2) holds and that f ∈ A δ,r,a (l) given by ( 14). If s > 0 (supersmooth noise) and r > 0 (supersmooth function f ), then

E fm -f 2 ≤ C 0 m -2δ e -2a(πm) r + C m 2γ+1-s e 2b(πm) s n + C ∆ f (m) M ,
where C 0 , C and C are constants which do not depend on M nor n. [START_REF] Lacour | Rates of convergence for nonparametric deconvolution[END_REF] that

Case r = s. We define ξ = [2bδ -a(2γ + 1 -s)]/[(a + b)s] and ω = [2(b -a)δ -2a(γ -δ) + ]/[bs] if b ≥ a, ω = 0 if b < a. It follows from Theorem 3.1 in
E fm 0 -f 2 = O n -a a+b (log n) -ξ + M -a a∨b (log M ) -ω , for πm 0 = Int[ log(n)-(α/s) log log(n) 2a+2b 1/s ∧ log(M )-(β/s) log log(M ) 2(a∨b)
1/s ] where α = 2δ + 2γ + 1s and β = 2δ + 2(γδ) + 1 b≥a . Case r < s. We define k = (s/r -1) -1 -1, where . is the ceiling function (i.e. x is the smallest integer larger than or equal to x). There exist coefficients b i recursively defined (see Theorem 3.1 in Lacour ( 2006)) and a choice m 0 such that

E fm 0 -f 2 = O (log n) -2δ/s exp[ k i=0 b i (log n) (i+1)r/s-i ] +(log M ) -2δ/s exp[ k i=0 b i (log M ) (i+1)r/s-i ]
Case r > s. We define k = (r/s -1) -1 -1. There exist coefficients d i recursively defined (see Theorem 3.1 in Lacour ( 2006)) and a choice m 0 such that

E fm 0 -f 2 = O (log n) (1+2γ-s)/r n exp[- k i=0 d i (log n) (i+1)s/r-i ] + 1 M
3.6. Lower bounds for the additional problem of estimating f ε . As mentioned above, [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] studied only one particular case from the lower bound point of view. But his proofs (for the additionnal problem of estimating f ε ) can be checked to be suitable in other cases. The following proposition establishes the optimality of our estimator with respect ot both risks in the cases where f is smoother than f ε and r ≤ 1.

Proposition 7. Let F γ,b,s = {f ε density such that there exist k 0 , k 1 > 0 such that

k 0 ≤ |f * ε (x)|(x 2 + 1) γ/2 exp(b|x| s ) ≤ k 1 } If r = s = 0 and γ < δ -1/2, or if 0 ≤ s < r ≤ 1 then inf f sup f ∈A δ,a,r (l),fε∈F γ,b,s E f -f 2 2 ≥ CM -1 inf f sup f ∈A δ,a,r (l),fε∈F γ,b,s E| f (x) -f (x)| 2 ≥ CM -1
Proof of Proposition 7. The proof of the lower bound (for the additional problem of estimating f ε ) given by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] can be used for the study of the pointwise risk. Indeed it suffices to use the same hypothesis functions f X,N,1 and f X,N,2 shifted at point x and to compute the distance |f X,N,1 (0)f X,N,2 (0)| 2 . These functions have been adjusted to deal with the integrated risk but in the case where the two risks have the same order, they can suit. Thus, if r = s = 0 and γ < δ -1/2, we obtain a lower bound CM -1 which proves the optimality of our estimator in this case.

In addition, this result can be generalized to a supersmooth noise distribution if s ≤ 1. Indeed, such densities verify the property (3.1) used by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF]. In the same way, an extension to supersmooth functions f can be done provided that r ≤ 1. Thus, if 0 ≤ s < r ≤ 1, the rate of convergence M -1 of our estimator is optimal for both the integrated and the pointwise risk.

Adaptation

The above study shows that the choice of m is both crucial and difficult. Thus, we provide a data driven strategy to perform automatically this choice. We assume that we are in case M ≥ n, so that our aim is to preserve here the rate corresponding to the case where f * ε is known. We consider thus the estimator f m defined by ( 9) where we have to define the penalty pen(.). We will work under Assumption (A2) and the following one, concerning the collection of models M n = {1, 2, . . . , m n }:

(A3) ∃α ∈]0, 1[, β ∈]0, 1/2[, (πm n ) 2γ e 2b(πmn) s M 1-α and n M exp(n 1/2-β ). The first Inequality in (A3) together with (A2) implies that ∀x ∈ [-πm n , πm n ], |f * ε (m n )| -2 M 1-α . If we choose M = n, we can see that Assumption (A3) is ensured if pen(.) is bounded over M n in the case b = s = 0 (that is if πm n ≤ n 1/(2γ+1) ) and if πm n ≤ (log(n)/(2b+1)) 1/s for b > 0, s > 0.
Clearly, it is difficult to choose M n and thus m n so that (A3) is fulfilled. This is a problem in the practical procedure which requires an explicit upper bound m n of M n . [START_REF] Diggle | A Fourier approach to nonparametric deconvolution of a density estimate[END_REF] suggest that regression methods applied to log(| f * ε |) may deliver some estimates of the parameter γ in the ordinary smooth case. An estimator of s is proposed in a semi-parametric framework in a recent work of [START_REF] Butucea | Adaptivity in convolution models with partially known noise distribution[END_REF]: if it is known that f ε is supersmooth, this strategy may be used to estimate s. Lastly, one can think of taking m n of order

(| f * ε | 2 ) -1 ( √ M )
, where the exponent -1 here denotes the reciprocal function.

We can provide another set of assumptions ensuring (A3). Assume that γ ∈ [γ, γ], s ∈ [s, s] and b ∈ [b, b] with s > 0, b > 0 whenever s > 0, b > 0. And consider also the following assumption: M exp(n 1/4 ) and Case (b, s) = (0, 0): (ordinary smooth noise):

(A4)      M n = m, (πm) 2γ+1 ≤ n M ≥ n 2γ+1 2γ+1 or    M n = m, (πm) ≤ √ n M ≥ n γ+1/2 . Case b > 0, s > 0: (supersmooth noise) (A4)          M n = m, (πm) 2γ+1-s e 2b(πm) s ≤ n M ≥ exp 2 2b (2b) s (log n) s/s
Then (A4) may require greater values of M , but also ensures (A3).

The simulation experiments of Section 5 are very useful here, to study the influence of the value of M and the importance of the set M n . It illustrates that we obtain very good results in practice, even when arbitrary limitations are set on m n or M .

We can prove the preliminary result given in Theorem 1. Note that the proof of this theorem is not standard in the present setting for the following reason. Usually, the contrast decomposition gives two types of terms:

• supremum of centered empirical processes (which in the independent bounded case are controlled thanks to Talagrand's type results): these terms impose the form of the penalty function.

• residual terms which are of negligible orders and which are not centered. Here, the residual terms which appear, even when controlled by using Lemma 2, are not negligible and have a weight in the penalty function. This makes the proof quite difficult.

Theorem 1. Assume that assumptions (A2) and (A3) are fulfilled and consider the estimator f m defined by ( 6) and ( 9) with

pen(m) = K 0 (πm) [s-(1-s) + /2] + ∆(m) n .
Then there exists C > 0 such that

E f m -f 2 ≤ 4 inf m∈Mn { f m -f 2 + pen(m)} + C n
where f m is the orthogonal projection of f on S m .

The presence of |f * ε | and s in the penalty is not admissible as it is an unknown quantity. That is why we state the following theorem.

Theorem 2. Assume that assumptions (A2) and (A3) are fulfilled and that s ≤ s for some given upper value s. Consider the estimator f = f m defined by ( 6) and

(18) m = arg min m∈Mn {γ n ( fm ) + pen(m)} with pen(m) = K 1 (πm) [s-(1-s) + /2] + πm -πm | f * ε | -2 n .
Then there exists C > 0 such that

E f -f 2 ≤ 4 inf m∈Mn { f m -f 2 + E pen(m)} + C n
Note that the restriction s ≤ 2 is very classical, so that s = 2 is generally suitable. Concluding Remarks. As we can prove that E pen(m) (πm) [s-(1-s) + /2] + ∆(m)/n, it follows from Theorem 2 that f automatically reaches the same rate as when f ε is known if s = s or if s = 0 (and thus s = 0 and this is known). For a discussion about the optimality of these rates (which holds in most cases), see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF].

In particular, in the case of ordinary smooth errors, the procedure is data driven and reaches the optimal rate, provided that M is taken large enough (Assumption (A4) case 1.

Simulations

Let us describe the estimation procedure. As noticed in (7), for each m, the estimator fm of f can be written To compute the coefficients âm,l , we use the Inverse Fast Fourier Transform. Indeed,

using ϕ * m,l (u) = e -ilu/m 1 [-πm,πm] / √ m, âm,l = 1 2π πm -πm 1 √ m e -ilu/m f * Y (-u) f * ε (-u) du = √ m 2 (-1) l 2 0 e ilπx f * Y f * ε (πm(x -1))dx
Then, for l = 0, ..., N -1, denoting h m (x) = ( f * Y / f * ε )(πm(x-1)), âm,l can be approximated by

√ m(-1) l 1 N N -1 k=0 e ilπ 2k N h m ( 2k N ) = √ m(-1) l (IFFT(H)) l
where H is the vector

(h m (0), h m (2/N ), . . . , h m (2(N -1)/N ). For l < 0, it is sufficient to replace h m (x) by h m (-x) = h m (x), i.e.
H by H. Following [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF], we choose K n = N -1 = 2 8 -1: indeed, a larger K n does not significantly improve the results. Thus, to compute f , we use the following steps:

• For each m ∈ M n and for each l, compute âm,l using function f * Y / f * ε and IFFT as described above • For each m compute γ n ( fm ) + pen(m) =l |â m,l | 2 + pen(m).

• Select the m which minimizes γ n ( fm ) + pen(m).

• Compute f = |l|≤Kn â m,l ϕ m,l .
Clearly, the use of FFT makes the procedure very fast. The penalty is chosen according to Theorem 2 with s = 2. Indeed densities with s > 2 are difficult to express in a closed form whereas usual densities all verify 0 ≤ s ≤ 2. The constant K 1 is chosen equal to 1/2 after intensive simulation experiments. However other values of K 1 can suit and, empirically, the procedure seems rather robust with respect to the choice of this constant. Thus, in all the examples below, we take:

pen(m) = 1 2n (πm) 2 πm -πm | f * ε | -2 .
Let us first compare our estimator to the one of [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF]. He denotes by f 0 (x) = e -|x| /2 and he considers two examples : -example 1: [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], n = 200 and M = 10 and the L 2 risk is computed with 100 random samples. As in [START_REF] Comte | Finite sample penalization in adaptive density deconvolution[END_REF], we consider that m can be fractional. More precisely, we take here

f = f 0 * f 0 * f 0 * f 0 and f ε = f 0 * f 0 -example 2: f = f 0 * f 0 and f ε = f 0 * f 0 * f 0 * f 0 We set, as in
M n = m = k 4π , k ∈ N * , k ≤ √ n
for the estimation with unknown noise. Actually, the bound on m, which is crucial in theory, turns out to be of little importance for practical purposes. The procedure chooses the appropriate model (often very small) even if the maximal model is very large. That is why in all our experiments we choose to keep a maximal m with order √ n. We also compute the estimator with known noise, replacing fε by f ε in the procedure. In this case, the choice of the maximal model has more impact, so we choose

M n = {m = k/(4π), k ∈ N * , k ≤ n 1/4 }.
Moreover, we take here pen(m) = 4∆(m)/n = (2/nπ) πm -πm (1 + x 2 ) 2d dx with d = 2 in example 1 and d = 4 in example 2. The integrated L 2 risks for 100 replications are given in Table 3 and show our improvement of the results of [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF].

ex 1 ex 2 f ε known 0.00257 0.01904 f ε unknown 0.00828 0.06592 ex 1 ex 2 f ε known 0.00225 0.01641 f ε unknown 0.00619 0.03327 Table 3. MISE for the estimators of Neumann (1997) (left) and for the penalized estimator (right).

In these examples, the signal and the noise are ordinary smooth (r = s = 0): this induces the rates of convergence n -15 24 + M -1 and n -7 24 + M -7 16 for examples 1 and 2 respectively. An example of estimation for supersmooth functions is given in [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF]. In his example 5.1, he considers a standard Gaussian noise and X ∼ N (5, 9). Again we use

M n = {m = k/(4π), k ∈ N * , k ≤ √ n}.
The penalty for a known noise is pen(m) = (πm) 3 1 0 exp{(πmx) 2 }dx/(2n). As [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF] presents only boxplots and for the sake of comparison, we give the third quartile for the L 2 risk in Table 4. In this case r = 2, δ = 1/2 and s = 2,γ = 0 and the rate of convergence is n -9 10 (log n) -1/2 + M -1 . The improvement brought by our method is striking. n = 100 n = 250 n = 500 f ε known 2.0 0.9 0.6 M = 100 2.0 1.0 0.7 M = 250 1.9 1.0 0.6 M = 500 1.9 0.9 0.6 Now we compute estimators for different signal densities and different noises. For the sake of simplicity (and since the chosen model are here larger), we take now

n =
M n = m = k 2π , k ∈ N * , k ≤ √ n
(for both known and unknown noise). Following [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] we study the following densities on the interval I:

(i) Laplace distribution: f (x) = e - √ 2|x| / √ 2, I = [-5, 5] (regularities δ = 2, r = 0) (ii) Mixed Gamma distribution: X = W/ √ 5.
48 with W ∼ 0.4Γ(5, 1) + 0.6Γ(13, 1), We consider two different noises with same variance 1/10:

I = [-1.5, 26] (regularities δ = 5, r = 0) (iii) Cauchy distribution: f (x) = (π(1 + x 2 )) -1 , I = [-10, 10] (regularities δ = 0, r = 1) (iv) Standard Gaussian distribution, I = [-4, 4] (regularities δ = 1/2, r = 2) n = 100 n = 250 n = 500 n = 1000
Laplace noise: In this case, the density of ε i is given by

f ε (x) = λ 2 e -λ|x| ; f * ε (x) = λ 2 λ 2 + x 2 ; λ = 2 √ 5.
The smoothness parameters are γ = 2 and b = s = 0. In the case when f ε is known, we use pen(m) = 4(πm + (2/(3λ 2 ))(πm) 3 + (1/(5λ 4 ))(πm) 5 )/n. Gaussian noise: In this case, the density of ε i is given by

f ε (x) = 1 λ √ 2π e -x 2 2λ 2 ; f * ε (x) = e -λ 2 x 2 2 ; λ = 1 √ 10 .
So γ = 0, b = λ 2 /2 and s = 2. In the case when f ε is known, we use pen(m) = 0.5(πm) 3 1 0 e (λπmx) 2 dx/n. The results are given in Table 5 and are very comparable to those of [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]. We notice that the estimation of the characteristic function of the noise does not spoil so much the procedure. It even happens that the estimation with unknown noise works better. We can also observe that, as expected, the risk decreases when M increases. The cases where the risk is larger for M = n correspond to a stabilization of the decrease and are due to the variance of the results. Figure 1 illustrates these results for two cases: a mixed Gamma density estimated through Laplace noise and a Laplace density estimated through Gaussian noise. Figure 2 shows the decrease of the integrated risk in these two cases. These curves confirm the theoretical result since the rate of convergence for a fixed n is M -1 in the first case and (log M ) -2 in the second case.

Proofs

For two sequences u n,M and v n,M , we denote u n,M v n,M if there exists a positive constant C such that u n,M ≤ Cv n,M . 6.1. Proof of Lemma 1. It is sufficient to prove the first assertion. First we write that v t (Y j ) = (1/2π) e iY j u t * (u)/f * ε (-u)du so that

E[v t (Y j )|X j ] = 1 2π E[e iY j u |X j ] t * (u) f * ε (-u) du.
By using the independence between X j and ε j , we compute

E[e iY j u |X j ] = E[e iX j u e iε j u |X j ] = e iX j u E[e iε j u ] = e iX j u f * ε (-u). Then E[v t (Y j )|X j ] = 1 2π e iX j u f * ε (-u) t * (u) f * ε (-u) du = 1 2π e iX j u t * (u)du = t(X j ).
6.2. Proof of Proposition 1. We start from Inequality (12). It follows from [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF] that:

(19) Var 1 2π πm -πm e ixu f * Y (u) f * ε (-u) du ≤ 1 2πn inf( |f * Y |∆(m), ∆ 0 (m)),
and (also, to see this, use ( 4) and

(f -f m )(x) = (1/2π)(f * -f * m ) * (-x)), (20)
(f m (x) -f (x)) 2 ≤ 1 2π |t|≥πm |f * (t)|dt 2 .
For the remaining term in (12), we write first:

E 1 2π πm -πm e ixu f * Y (u)R(u)du 2 ≤ 2E 1 2π πm -πm e ixu ( f * Y (u) -f * Y (u))R(u)du 2 +2E 1 2π πm -πm e ixu f * Y (u)R(u)du 2 := 2T 1 + 2T 2 .
Then we find

T 1 = 1 4π 2 e ix(u-v) cov( f * Y (u), f * Y (v))E(R(u) R(v))dudv ≤ 1 4π 2 n |f * Y (u -v)| E(|R(u)| 2 )E(|R(v)| 2 )dudv 1 4π 2 n |f * Y (u -v)| 1 |f ε (u)f ε (v)| dudv
by using Lemma 2. This term is clearly bounded by ∆ 0 (m). Moreover writing it as

|f * Y (u -v)| |f ε (u)| |f * Y (u -v)| |f ε (v)| dudv
and using first the Schwarz Inequality, and second the Fubini Theorem yields the bound

|f * Y |∆(m). Therefore (21) E 1 2π πm -πm e ixu ( f * Y (u) -f * Y (u))R(u)du 2 1 2πn inf( |f * Y |∆(m), ∆ 0 (m)),
and thus it has the same order as the usual variance term. Lastly,

T 2 ≤ 1 4π 2 |u|,|v|≤πm |f * Y (u)f * Y (v)| E(|R(u)| 2 )E(|R(v)| 2 )dudv ≤ 1 4π 2 πm -πm |f * Y (u)| E(|R(u)| 2 )du 2 1 4π 2 M πm -πm |f * Y (u)| 1 |f * ε (u)| 2 du 2 = 1 4π 2 M πm -πm |f * (u)| |f * ε (u)| du 2 = ∆ 0 f (m) 2πM . ( 22
)
Inserting the bounds ( 19) to ( 22) in Inequality ( 12), we obtain the result of Proposition 1. 6.3. Proof of Proposition 2. We start from ( 16) and take the expectation:

E( f m -fm 2 ) ≤ 2 π πm -πm E(| f * Y (u) -f * Y (u)| 2 |R(u)| 2 )du + 2 π πm -πm |f * Y (u)| 2 E(|R(u)| 2 )du + 1 π πm -πm n -1 |f * ε (u)| 2 du. Applying Lemma 2 yields: E( f m -fm 2 ) ≤ 2 π πm -πm E(| f * Y (u) -f * Y (u)| 2 )E(|R(u)| 2 )du + 2 π πm -πm |f * (u)| 2 |f * ε (u)| 2 E|R(u)| 2 du + 2 ∆(m) n πm -πm n -1 |f * ε (u)| -2 du + πm -πm |f * (u)| 2 |f * ε (u)| 2 M -1 |f * ε (u)| 4 du + ∆(m) n 1 M πm -πm |f * (u)| 2 |f * ε (u)| 2 du + ∆(m) n (23)
By gathering ( 15) and ( 23), we obtain the result. 6.4. Proof of Lemma 3. The proof of the first result is omitted. It is obtained by distinguishing the cases s > 2γ+1 and s ≤ 2γ+1 and with standard evaluations of integrals.

For the second point, we first remark that ∆ f (m) ≤ ∆(m). Next, using Assumption (A2),

∆ f (m) ≤ k -2 0 2π πm -πm (x 2 + 1) γ e 2b|x| s |f * (x)| 2 dx ≤ k -2 0 2π l sup x∈[-πm,πm]
((x 2 + 1) γ-δ e 2(b|x| s -a|x| r ) )

Then, if s > r, ∆ f (m) ≤ k -2 0 2π l((πm) 2 + 1) (γ-δ) + e 2b(πm) s If r = s and b ≥ a, ∆ f (m) ≤ k -2 0 2π l((πm) 2 + 1) (γ-δ) + e 2(b-a)(πm) s
If r > s or r = s and a > b, ∆ f (m) is bounded by a constant.

6.5. Proof of Theorem 1. We observe that for all t, t

γ n (t) -γ n (t ) = t -f 2 -t -f 2 -2ν n (t -t )
where

ν n (t) = (n) -1 j ṽt (Y j ) -t(x)f (x)dx .
Let us fix m ∈ M n and recall that f m is the orthogonal projection of f on S m . Since γ n ( f ) + pen( m) ≤ γ n (f m ) + pen(m), we have

f m -f 2 ≤ f m -f 2 + 2ν n ( f m -f m ) + pen(m) -pen( m) ≤ f m -f 2 + 2 f -f m sup t∈B(m, m) ν n (t) + pen(m) -pen( m)
where, for all m, m , B(m, m ) = {t ∈ S m + S m , t = 1}. Then, using inequality 2xy ≤ x 2 /4 + 4y 2 , (24)

f m -f 2 ≤ f m -f 2 + 1 4 f m -f m 2 + 4 sup t∈B(m, m) ν 2 n (t) + pen(m) -pen( m). But f m -f m 2 ≤ 2 f m -f 2 + 2 f -f m 2 so that, introducing a function p(., .) f m -f 2 ≤ 3 f m -f 2 + 8[ sup t∈B(m, m) ν 2 n (t) -p(m, m)] + 8p(m, m) + 2pen(m) -2pen( m).
If p is such that for all m, m , (25) 4p(m, m ) ≤ pen(m) + pen(m )

then (26) E f m -f 2 ≤ 3 f m -f 2 + 8E[ sup t∈B(m, m) ν 2 n (t) -p(m, m)] + 4pen(m).
With Lemma 1 in mind, ν n (t) can be split into two terms : ν n (t) = ν n,1 (t) + S n (t) with ( 27)

           ν n,1 (t) = 1 n n j=1 {v t (Y j ) -E[v t (Y j )]} S n (t) = 1 n n j=1 (ṽ t -v t )(Y j ), ,
For ν n,1 , we use the following proposition, proved in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]:

Proposition 8. Let p 1 (m, m ) = K(πm ) [s-(1-s) + /2] + ∆(m )
/n where ∆(m) is defined in (10) and m = max(m, m ) and K is a constant. Then, under assumptions of Theorem 1, there exists a positive constant C such that

(28) E 0 := E sup t∈B(m, m) ν 2 n,1 (t) -p 1 (m, m) + ≤ C n .
Note that Theorem 1 in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] is proved under the assumption that the penalty is bounded, but it is easy to check that it also holds under (A3) (and the assumption M exp(n 1/2-β ) is used here).

For S n we need additional decompositions. We write

S n (t) = 1 n n j=1 (ṽ t -v t )(Y j ) = 1 2π ( 1 n n j=1 e iuY j )t * (u)R(-u)du = 1 2π f * Y (u)t * (-u)R(u)du = 1 2π ( f * Y (u) -f * Y (u))t * (-u)R(u)du + 1 2π f * Y (u)t * (-u)R(u)du Now, let E(x) = {| f * ε (x)| ≥ 1/ √ M } and write R(x) = 1 E(x) f * ε (x) - 1 f * ε (x) = 1 E(x) 1 f * ε (x) - 1 f * ε (x) - 1 E(x) c f * ε (x) = (f * ε (x) -f * ε (x)) f * ε (x) 1 E(x) f * ε (x) - 1 E(x) c f * ε (x) = (f * ε (x) -f * ε (x)) f * ε (x) R(x) + (f * ε (x) -f * ε (x)) (f * ε (x)) 2 - 1 E(x) c f * ε (x)
.

Thus we have

S n (t) = R n,1 (t) + R n,2 (t) -R n,3 (t) -R n,4 (t)
where

                     R n,1 (t) = 1 2π ( f * Y (u) -f * Y (u))t * (-u)R(u)du, R n,2 (t) = 1 2π f * (u)t * (-u)(f * ε (u) -f * ε (u)))R(u)du, R n,3 (t) = 1 2π f * (u)t * (-u) f * ε (u) -f * ε (u) f * ε (u) du, R n,4 (t) = 1 2π f * (u)t * (-u)1 E(x) c du.
Now, we prove in the following subsections the result:

Proposition 9. We denote here by m * = m ∨ m. Under assumptions of Theorem 1, there exists a positive constant C such that

(29) E 1 := E sup t∈B(m, m) |R n,1 (t)| 2 -C 1 ∆(m * ) n ≤ C n . (30) E 2 := E sup t∈B(m, m) |R n,2 (t)| 2 -C 2 ∆ f (m * ) M ≤ C n .
(31)

E 3 := E sup t∈B(m, m) |R n,3 (t)| 2 -p 3 (m, m) + ≤ C n .
with p 3 (m, m ) = K (πm ) [s-(1-s) + /2] + ∆(m )/M , m = max(m, m ).

(32)

E 4 := E sup t∈B(m, m) |R n,4 (t)| 2 ≤ C n .
It follows that

E[ sup t∈B(m, m) ν 2 n (t) -p(m, m)] + ≤ 5(E 1 + E 2 + E 3 + E 4 + E 0 ) (33) as soon as 5 ∆(m") n + ∆ f (m") n + p 3 (m, m ) + p 1 (m, m ) ≤ p(m, m )
for all m, m in M n . Therefore, the choice p(m, m ) = K"(πm ) [s-(1-s) + /2] + ∆(m )/n for a numerical constant K" large enough, is suitable. The choice of pen(.) given in Theorem 1 ensures then that (25) holds true. Now, gathering (28), ( 29)-( 32) and ( 33) yields

E[ sup t∈B(m, m) ν 2 n (t) -p(m, m)] + ≤ C n ,
which, together with (26), ends the proof of Theorem 1.

6.6. Proof of Proposition 9.

6.6.1. Study of R n,1 (t) and proof of (29). We recall that m * is the maximum max(m, m) and we define Ω(x) the set Ω(

x) = Ω 1 (x) ∩ Ω 2 (x)
where

Ω 1 (x) = {| f * Y (x) -f * Y (x)| ≤ n α/8-1/2 M α/4 } and Ω 2 (x) = {|R(x)| ≤ M α/8-1/2 /|f * ε (x)| 2 )},
for α ∈ (0, 1) as defined in (A3).

For t in S m + S m = S m * , we can bound the term |R n,1 (t)| 2 in the following way

|R n,1 (t)| 2 ≤ 1 4π 2 πm * -πm * | f * Y -f * Y | 2 |R| 2 so that sup t∈B(m, m) |R n,1 (t)| 2 ≤ 1 4π 2 πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ω + 1 4π 2 πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ω c
On the one hand

πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ω ≤ πm * -πm * n α/4-1 M α/2 M α/4-1 |f * ε | -4 ≤ n α/4-1 M 3α/4-1 πm * -πm * M 1-α |f * ε (x)| -2 dx n α/4-1 M -α/4 ∆(m * ) ∆(m * ) n n M α/4 ∆(m * ) n
On the other hand

E( πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ω c ) ≤ πmn -πmn E 1/2 (| f * Y -f * Y | 4 )E 1/2 (|R| 4 )P 1/2 (Ω c ) πmn -πmn n -1 M -1 |f * ε (x)| -4 P 1/2 (Ω(x) c )dx n -1 M -α ∆(m n ) P 1/2 (Ω c ) ∞
But, using the Markov inequality,

P(Ω(x) c ) ≤ n -p(α/8-1/2) M -pα/4 E| f * Y -f * Y | p + M -2p(α/8-1/2) |f * ε (x)| 4p E|R| 2p ≤ n -pα/8 M -pα/4 + M -pα/4 M -pα/4 , then E( πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ω c ) n -1 M -α ∆(m n )M -pα/8 ) m n M 1-α M n M 1-α-pα/8 ≤ M 1+1/(2γ)-2α-αp/8 n where 1/(2γ) = 1/(2γ) if s = 0, γ > 0 and 1/(2γ) = 1 if γ = 0, s > 0.
We choose p large enough (p ≥ 8(1/(2γ) + 1 -2α)/α) so that M 1+1/(2γ)-2α-pα/8 = O(1). We obtain (29).

6.6.2. Study of R n,2 (t) and proof of (30). The following result obviously holds E

[|f * ε - f * ε | p ] M -p/2 . Moreover, let Ξ(x) = {|f * ε (x) -f * ε (x)| ≤ M -ω and |R(x)| ≤ M -ω /|f * ε (x)| 2 }, where 0 < ω < 1/2. We can bound the term |R n,2 (t)| 2 in the following way sup t∈Sm+S m |R n,2 (t)| 2 ≤ 1 4π 2 πm * -πm * |f * | 2 |f * ε -f * ε | 2 |R| 2 1 Ξ + 1 4π 2 πm * -πm * |f * | 2 |f * ε -f * ε | 2 |R| 2 1 Ξ c
On the one hand

πm * -πm * |f * | 2 |f * ε -f * ε | 2 |R| 2 1 Ξ ≤ πm * -πm * |f * | 2 M -4ω |f * ε | -4 ≤ πm * -πm * |f * | 2 M -4ω |f * ε | -2 M 1-α ∆ f (m * ) M (M 2-4ω-α ) ∆ f (m * ) M ,
as soon as ω satisfies (2α)/4 ≤ ω < 1/2, e.g. we can take 0 < ω = (2α)/4 < 1/2. On the other hand

E( πm * -πm * |f * | 2 |f * ε -f * ε | 2 |R| 2 1 Ξ c ) ≤ πmn -πmn |f * | 2 E 1/4 (|f * ε -f * ε | 8 )E 1/4 (|R| 8 )P 1/2 (Ξ c ) πmn -πmn M -2 |f * (x)| 2 |f * ε (x)| -4 P 1/2 (Ξ(x) c )dx ∆ f (m n )M -1-α P 1/2 (Ξ c ) ∞ m n M -α P 1/2 (Ξ c ) ∞ M 1/(2γ)-α P 1/2 (Ξ c ) ∞ .
Then, using the Markov inequality,

P(Ξ(x) c ) ≤ M 2pω E|f * ε -f * ε | 2p + M 2ωp |f * ε (x)| 4p E|R| 2p M p(2ω-1) . Thus E( πm * -πm * | f * Y -f * Y | 2 |R| 2 1 Ξ c ) M 1/(2γ)-α-p(1-2ω) n 1 M for p ≥ (1/(2γ) -α)/(1 -2ω) = 2(1/(2γ) -α)/α.
This yields (30). 6.6.3. Study of R n,3 (t) and proof of (31). We can write

R n,3 (t) = 1 M M k=1 [F t (ε -k ) -E(F t (ε -k ))] with F t (u) = 1 2π f * (x) f * ε (x) t * (-x)e -ixu dx.
Moreover,

E sup t∈B(m, m) |R n,3 (t)| 2 -p 3 (m, m) + ≤ m ∈Mn E sup t∈B(m,m ) |R n,3 (t)| 2 -p 3 (m, m )
+ which replaces the supremum on a random unit ball ( m is random) by suprema on deterministic unit balls. Then we use the following Lemma Lemma 4. Let T 1 , . . . , T M be independent random variables and ν M (r) = (1/M ) M j=1 [r(T j )-E(r(T j )], for r belonging to a countable class R of measurable functions. Then, for > 0,

(34) E[sup r∈R |ν M (r)| 2 -(1 + 2 )H 2 ] + ≤ C v M e -K 1 M H 2 v + B 2 M 2 C 2 ( ) e -K 2 C( ) √ M H B with K 1 = 1/6, K 2 = 1/(21 √ 2), C( ) = √ 1 + -1 and C a universal constant and where sup r∈R r ∞ ≤ B, E sup r∈R |ν M (r)| ≤ H, sup r∈R 1 M M j=1
Var(r(T j ) ≤ v.

Inequality ( 34) is a straightforward consequence of the [START_REF] Talagrand | New concentration inequalities in product spaces[END_REF] inequality given [START_REF] Birgé | From model selection to adaptive estimation[END_REF]. Moreover, standard density arguments allow to apply it to the unit ball of a finite dimensional linear space.

Let us determine B, H and v is our problem.

For t ∈ S m + S m = S m , F t 2 ∞ ≤ 1 4π 2 πm -πm |f * (x)| 2 |f * ε (x)| 2 dx πm -πm |t * (-x)| 2 dx. Then sup t∈B(m,m ) F t 2 ∞ ≤ ∆ f (m ) and we set B = ∆(m ).
If t belongs to B(m, m ), it can be written t = l∈Z a m ,l ϕ m ,l and

|R n,3 (t)| 2 ≤ l∈Z a 2 m ,l l∈Z | 1 M M j=1 [F ϕ m ,l (ε -j ) -E(F ϕ m ,l (ε -j ))]| 2
As the ε -j are i.i.d., Then we insert these quantities in the Inequality given in Lemma 4. For the first term of the right-hand-side bound, we obtain

m ∈Mn v M exp(-K 1 M H 2 v ) ≤ C M
with adequate choices of . The study of this term is the same as in the proof of Theorem 1 in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] and is omitted here. The second term of the right-hand-side bound is less than (up to multiplicative constants):

m ∈Mn ∆(m ) M 2 exp(-C √ M ) ≤ M 1/γ-1-α exp(-C √ M ) 1 M .
Inserting the value of and applying Lemma 4 leads to

P(| f * ε (x)| < M -1/2 ) ≤ P(| f * ε (x) -f * ε (x)| > |f * ε (x)| -M -1/2 ) ≤ P(| f * ε (x) -f * ε (x)| > M -1/2 ) ≤ κ exp(-κM |f * ε (x)| 2 ) = O(M -p |f * ε (x)| -2p
), for all p ≥ 1. Then

E sup t∈B(m, m) |R n,4 (t)| 2 πmn -πmn |f * (u)| 2 M -p |f * ε (u)| -2p du ∆ f (m n )M -p M (1-α)(p-1)
Then as 2γ) , it is sufficient to take p ≥ 2+1/(2αγ) > 0 to obtain

∆ f (m n ) ≤ ∆(m n ) ≤ m n M ≤ M 1+1/(
E sup t∈B(m, m) |R n,4 (t)| 2 1 M 1 n .
Therefore, (32) holds. Then

E( f -f 2 1 Λ ) ≤ 4 inf m∈Mn { f m -f 2 + E pen(m)} + C n
We still have to prove that

E( f -f 2 1 Λ c ) ≤ C n
First we compute, using formula (3) ,

f 2 = 1 2π | f * m| 2 = 1 2π π m -π m | f * Y | 2 | f * ε | 2 ≤ 1 2π π m -π m | f * ε | -2 But | f * ε (x)| -2 = | f * ε (x)| -2 1 {| f * ε (x)|≥M -1/2 } ≤ M . Then f 2 ≤ M m ≤ M m n ≤ M 1/(2γ)+1
and thus E( ff 2 1 Λ c ) M 1/(2γ)+1 P (Λ c ). Now, using Markov and Jensen inequalities, .

P(Λ c
Finally E( ff 2 1 Λ c ) M 1+1/(2γ) P (Λ c ) M -αp+1+1/(2γ) m 1+s(p-1) n

. If s = 0, then γ > 0 and the bound become M -αp+1+1/γ , so that p ≥ (2 + 1/γ)/α implies E( ff 2 1 Λ c ) M -1 . If s > 0, then under (A3), m n (log(M )) 1/s , so that the previous inequality holds if p > 2/α.

2. 3 .

 3 Definion of the adaptive estimator. 2.3.1. Projection spaces. Let us consider the function ϕ(x) = sin(πx)/(πx) and, for m in N

Figure 1 .

 1 Figure 1. True function f (bold line) and estimators for n = 500. Left: mixed Gamma density with Laplace noise. Right : Laplace density with Gaussian noise

Figure 2 .

 2 Figure 2. MISE against M for n = 100 in two cases

  H = ∆(m )/M as f 1 = 1. Lastly, standard methods give v = C min(∆(m ), f ε ∆ 2 (m )) with ∆ 2 (m) = 1 f * ε 4

6. 7 .

 7 Proof of Theorem 2. We use the following setΛ = {∀m ∈ M n , |R| 2 + 2 ∆(m), we can write on Λ, ∆(m) ≤ ∆(m)/2 + 2 ∆(m) and then ∆(m)1 Λ ≤ 4 ∆(m)1 Λ Reasoning as in the proof of Theorem 1, if p is such that for all m, m , 4p(m, m )1 Λ ≤ pen(m)1 Λ + pen(m )1 Λ then ff 2 1 Λ ≤ 3 f mf 2 + 8[ sup t∈B(m, m) ν 2 n (t)p(m, m)]1 Λ + 4 pen(m)1 Λ .It follows from the proof of Theorem 1 that8E[ sup t∈B(m, m) ν 2 n (t)p(m, m)] + ≤ C/n with p(m, m ) = K(πm ) [s-(1-s) + /2] + ∆(m )/n. Thus, choosing pen(m) = 16K(πm) [s-(1-s) + /2] + ∆(m)/n, on Λ, 4p(m, m ) = 4K(πm ) [s-(1-s) + /2] + ∆(m )/n ≤ 4K(πm) [s-(1-s) + /2] + ∆(m)/n + 4K(πm ) [s-(1-s) + /2] + ∆(m )/n ≤ 16K(πm) [s-(1-s) + /2] + ∆(m)/n + 16K(πm ) [s-(1-s) + /2] + ∆(m )/n ≤ pen(m) + pen(m ).

  + 1) 2γp exp(4pb|x| s )dx (πm) 4γp+1-s e 4pb(πm) s ) 2γ e 2b(πm) s ] p [(πm) 2γ+1-s e 2b(πm) s ] p m 1-s-p+sp M (1-α)p ∆(m) p m 1-s-p+sp Hence P(Λ c ) M -ap m∈Mn m -s+sp M -αp (m n ) 1+s(p-1)

Table 4 .

 4 Third quartile of the MISE ×100 for the estimators of Johannes (2007) (left) and for the penalized estimator (right).

		100 n = 250 n = 500
	f ε known	0.34	0.20	0.08
	M = 100	0.29	0.15	0.09
	M = 250	0.27	0.13	0.08
	M = 500	0.23	0.11	0.08

Table 5 .

 5 MISE

	H H H H H H f f ε	Lap. Gauss. Lap. Gauss. Lap. Gauss. Lap. Gauss.
	Laplace	f ε known 2.185 2.250 1.261 1.168 0.836 0.924 0.583 0.633 M = √ n 4.868 4.791 2.811 2.845 1.777 1.781 1.153 1.109
		M = n	5.107 5.114 2.892 2.876 1.757 1.748 1.090 1.110
	Mixed Gamma M = f ε known 1.001 0.945 0.603 0.554 0.278 0.274 0.177 0.202 √ n 0.971 1.025 0.751 0.777 0.454 0.472 0.232 0.230
		M = n	1.037 1.039 0.745 0.765 0.467 0.500 0.222 0.218
	Cauchy	f ε known 1.072 0.979 0.468 0.475 0.341 0.251 0.243 0.137 M = √ n 1.276 1.343 0.791 0.802 0.400 0.398 0.189 0.194
		M = n	1.266 1.362 0.762 0.782 0.363 0.364 0.172 0.172
	Gaussian f ε known 0.810 0.589 0.771 0.287 0.500 0.191 0.373 0.134 M = √ n 1.045 1.114 0.397 0.346 0.241 0.181 0.139 0.170
		M = n	0.904 0.986 0.252 0.256 0.150 0.182 0.100 0.094

E( f -f 2 ) × 100 averaged over 100 samples

  |R n,3 (t)| 2p 3 (m, m ) We use that, as |f * ε (x)| -2 ≤ M 1-α , it holds that |f * ε (x)| ≥ 2/ √ M .Thus, proceeding as in[START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], we apply Bernstein Inequality and we get

		E		sup
		m ∈Mn	t∈B(m,m )
	and thus
	E	sup t∈B(m, m)	|R n,4 (t)| 2 ≤	1 2π	πmn -πmn	|f

+

≤

C M which implies (31).

6.6.4. Study of R n,4 (t). It is easy to see that

sup t∈B(m, m) |R n,4 (t)| 2 ≤ 1 2π πm * -πm * |f * (u)| 2 1 E c du, * (u)| 2 P(E c )du. Now, P(E c ) = P(| f * ε (x)| < 1/ √ M ).


F Plouraboué

H-C Chang

Attraction between two similar particles in an electrolyte: effects of Stern layer absorption

When Debye length is comparable or larger than the distance between two identical particles the overlapping between particles doublelayers can play an important role in their interactions. This paper presents a theoretical analysis of the interaction between two identical particles with overlapped double-layers. We particularly focus on the effect of a Stern electrostatic condition from linearization of the adsorption isotherm near the isoelectric (neutrality) point in order to capture how polyvalent ion condensation affects and reverses the surface charge. The stationary potential problem is solved within the framework of an asymptotic lubrication approach for a mean-field Poisson-Boltzmann model. Both spherical and cylindrical particles are analyzed. The results are finally discussed in the context Debye-Hückel (D-H) limit and beyond it.

Introduction

Recent AFM studies have shown that, in the presence of a polyvalent counterion, two similarly charged or identical surfaces can develop a short-range attractive force at a distance comparable to the Debye screening length λ [Zohar et al.(2006), Besterman et al.(2004)]. This observation is most likely related to earlier reports on attraction between identical colloids in an electrolyte, although the role of poly-valency is not as well established for colloids [Han and Grier(1999)]. It is also related to the condensation of likecharged molecules like DNAs [Gelbart et al.(2000)]. Such attraction has raised some debate during the past ten years since for two identical objects is necessarily repulsive according to the classical Poisson-Boltzmann (PB) mean-field theory [Neu(1999)]. For this reason, theories for like-charge attraction phenomena have sought mechanisms beyond the classical mean-field description to include spatial correlation of charge fluctuations [Lubatsky and Safran(2008), Netz and Orland(2000), Lau and Pincus(2002), Lau(2008)]. In a previous contribution [Plouraboué and Chang(2008)] we realized that including a Stern layer for the mean-field boundary condition is compatible with previous field-theoretical analysis of the role of fluctuations on mean-field description [Lau and Pincus(2002), Lau(2008)]. In this framework we obtained an implicit analytic solution for the mean-field potential and compute the attraction between two planar surfaces. In this contribution we derive an asymptotic computation of the potential between two spherical or cylindrical identical particles. As opposed to the situation where the particle distance is large compared to the Debye length for which the DLVO approximation holds, we focus here on the possible non-linear interaction between particles double-layers.

2 Problem under study

Governing equation

We very briefly discuss here the stationary electro-kinetic problem that one has to solve for the electric potential φ ′ (Cf for example [Karniadakis et al.(2004)] for more details). We consider an electrolyte solution composed of Z-charged positive/negative ions. Boltzmann equilibrium associated with the concentration/potential leads to the non-linear mean-field PB relation :

∇ 2 φ ′ = 2 ZF C ∞ ǫ 0 ǫ p sinh ZF φ ′ R g T (1)
Where F is the Faraday constant, ǫ p the solution relative permittivity, ǫ 0 the dielectric permittivity of vacuum, R g the perfect gas constant, T the temperature and C ∞ a reference concentration in the far-field region. These parameters are usually used to define the Debye length

λ = ǫ 0 ǫ p R g T /F 2 C ∞ . a p + p - z' 2h m r' h'(r')
Figure 1: Slide view of the two indentical particles under study.

Particle shape and boundary conditions

Let us now discuss the surface shape h ′ (r ′ ) of the particles sketched on figure 1. This figure represent a section of either a cylindrical or a spherical particle. In the first case, the problem under study is translationally invariant along the direction perpendicular to the figure plane, aligned along the cylinder main axis. In the second case, the problem under study is rotationally invariant along the z ′ axis. In both case the particle shape in the section follows a circle. From elementary trigonometry identities one gets

(a -(h ′ -h m )) 2 + r ′2 = a 2 , (2) so that, h ′ (r ′ ) = h m + a - √ a 2 -r ′2 . (3)
This problem can be associated with different boundary conditions at the particles surfaces. We will mainly focus in this study on the influence of a Stern layer at the particles surfaces

∂ n φ ′ (±h ′ (r ′ )) = Kφ ′ (±h ′ (r ′ )) (4)
We consider an iso-electric point situation for which the reference potential is taken to be zero so that the right-hand side of ( 4) is a linear function of φ ′ . As already discussed in [Plouraboué and Chang(2008)], the constant K scales as the inverse of the Debye length λ from previous analysis [Lau and Pincus(2002), Lau(2008)].

We will compare the obtained results for Stern-Layer with other boundary conditions such as an applied surface field at each particles

-∂ n φ ′ (±h(r ′ )) = ∓E ′ p± (5)
Where E ′ p± stands for the Electric field prescribed either at the top p+ or the bottom p-particle. From using Gauss's theorem, one realizes that prescribing the field is equivalent to prescribe a surface charge at the particle surfaces. Another widely used boundary condition is to prescribe the electrical potential at the particle surface :

φ ′ (±h(r ′ )) = φ ′ p± . (6)
In the following we will compare these three boundary conditions in the D-H approximation.

Asymptotic formulation

Dimensionless formulation

Let us first discuss some physics associated with the problem for choosing an interesting dimensionless formulation. First, the distance between the particles involves one characteristic length which is the minimum distance h m . As discussed in the introduction non-trivial effects for particles interactions arise when this distance is of the same order or shorter than the Debye length λ. Furthermore, difficulties in quantifying this interaction are associated with the importance of non-linear effects in the double-layer. Numerical estimation can be used but necessitates an accurate description of rapid potential variations inside double layers which can be computationally challenging.

A possible way to get around this difficulty is to use the specific limit for which those double-layer effects matters, i.e when the distance h m is sufficiently small. From realizing that the Debye length λ generally lies between nanometer to micron scale, it can be seen that many interesting situations are associated with particle radius a larger than Debye length a ≫ λ. In the limit h m ≪ a and thus, an asymptotic "lubrication" analysis of the problem can be sough for. More specifically, in this limit, most of the potential variation holds along the transverse direction between two particles whose typical length is h m rather that in the longitudinal direction, roughly parallel to the particles surfaces, for which the potential variations holds along a typical length-scale √ ah m . This discussion suggests the following dimensionless formulation of transverse coordinates z, h and longitudinal ones r:

z ′ = h m z h ′ = h m h r ′ = √ h m ar (7)
Those coordinates associated with rapid variation of the potential inside a central region are "inner" coordinates and are used in section 2.3.2. Another choice could have been taken from simply considering the potential variations far from the confined region, for which the only relevant length-scale is the particle radius. In this case, 'outer' dimensionless coordinates can be defined with upper-case notations

z ′ = aZ r ′ = aR h ′ = aH (8)
that will be subsequently used in section 2.3.3.

Asymptotic expansion : inner region

Introducing the small parameter ǫ = h m /a, one can then re-write the shape equation ( 3)

h(r) = 1 ǫ + 1 - 1 ǫ (1 -ǫr 2 ) 1/2 ≃ 1 + r 2 2 + ǫ r 4 8 (9)
This behavior suggests the following asymptotic sequence for the shape :

h = h 0 + ǫh 1 h 0 (r) = 1 + r 2 2 h 1 (r) = r 4 8 (10)
The normal vector n to the particle surfaces can also be computed :

n ≃ -0 1 + √ ǫ y 0 + ǫ 0 1 y 2 2 + ... (11)
Using the usual dimensionless formulation for the potential φ = ZF φ ′ /R g T the normal derivative of the dimensionless potential on the upper particle reads :

∂ n φ = ∇φ.n ≃ -∂ z φ + ǫ(∂ y φy + ∂ z φ y 2 2 ) (12)
A similar result with opposite sign for the first term holds for the lower particle. Dimensionless PB problem (1) reads

∂ 2 z + ǫ∇ 2 // φ = 2 h m λ 2 sinh φ (13)
Where ∇ 2 // is the Laplacian contribution orthogonal to the (e z , e r ) section which is different for cylindrical or spherical particles. We will not need to specify it further since we are just going to compute the leading order contribution to the following asymptotic sequence in the inner region suggested by ( 10)

φ = φ 0 + ǫφ 1 . ( 14
)
Introducing this sequence in the governing equation ( 13) leads to the leading order

∂ 2 z φ 0 = β m sinh φ 0 , (15)
where β m = 2(h m /λ) 2 . From ( 12) the associated Boundary conditions, with prescribed electric fields (5) reads at the leading order :

∂ z φ 0 (±h 0 ) = ±E p± (16)
Where we have used dimensionless electric fields E p± = ZF E ′ p± /R g T h m . For prescribed potentials (6), the leading order boundary conditions reads :

φ 0 (±h 0 ) = φ p± (17)
And finally the Stern layer boundary condition reads at leading order

∂ z φ 0 (±h 0 ) = ∓µβ m φ 0 (±h 0 ) (18)
Where we have introduced a parameter µ which stands for dimensionless prefactor between the potential and its gradient at the particle surface boundary condition.

Outer region

Using dimensionless formulation, (8) PB problems (1) reads :

ǫ 2 ∇ 2 Φ = β m sinh Φ. ( 19
)
The inner problem suggests the following asymptotic sequence for the potential in the outer region :

Φ = Φ 0 + ǫΦ 1 (20)
Injecting this sequence in the outer governing equation ( 19) leads to the following leading order problem :

sinh Φ 0 = 0, ( 21
)
whose solution is Φ 0 = 0, so that the matching condition at leading order is just lim y→∞ φ 0 (0, y) = 0.

Results

Since the outer region solution is trivial we now focus on the inner region solution which is going to provide the interesting potential variations for particle interactions.

Debye-Hückel approximation

We examine here the linearized limit of small dimensionless potential φ ≪ 1. Let us first consider prescribed electrical fields at the particles surfaces. In this case the solution of the linearized limit of (15) with boundary conditions (16) reads

φ 0 = 1 2 E + cosh z sinh h 0 + E -sinh z cosh h 0 ( 22
)
where we have introduced notation

E + = E p+ + E p-and E -= E p+ - E p-.
The case of symmetrical particles corresponds to E -= 0, since the corresponding fields at each particle are identical and of opposite sign. We then recover in this case that ∂ z φ 0 (r, z = 0) = 0. Let us now write-down the solution associated with the boundary conditions ( 17)

φ 0 = 1 2 φ + cosh z cosh h 0 + φ -sinh z sinh h 0 ( 23
)
where we have introduced notation φ + = φ p+ + φ p-, and φ -= φ p+ -φ p-.

One can also see that in this case symmetrical particles are associated with φ -= 0 so that the resulting field will also fulfills ∂ z φ 0 (r, z = 0) = 0. Finally let us now discuss the D-H limit solution associated with boundary conditions (18) for which a family of non-symmetrical solutions can be found

φ 0 ∼ sinh z h 0 (24)
The amplitude coefficients of this eigenfunction cannot be specified since boundary conditions ( 18) are linearly varying with the potential. Furthermore it can be shown that only specific values of h 0 can match these boundary conditions.

Non-linear PB problem

It is now interesting to realize that the leading order problem (15) associated with boundary conditions ( 16),( 17) or ( 18) can be expressed in a single variable ζ = z/h 0 (r) independently of any other explicit Dependance on the longitudinal variable r. Hence, one can then map the leading order PB problem between two identical particle onto the same problem between two parallel planes. Hence, we recover here the solution previously studied in [Plouraboué and Chang(2008)] for Stern layer boundary conditions. Let us now recall here the main steps of the solution. A first integral of (15) using the variable change

ζ = z/h 0 (r) for φ 0 (ζ) is 1 2 (∂ ζ φ 0 ) 2 = β(r) (cosh φ 0 ) + d ( 25
)
where d depends on the value prescribed at the particle, and β(r) = 2(h m h 0 (r)/λ) 2 . Introducing notation d ′ = d/β, one finds that this constant depends upon the applied boundary condition. For prescribed electric field (16) one finds

d ′ (r) = d β(r) = 1 2 E 2 p+ -coshφ 0 (1) = 1 2 E 2 p--coshφ 0 (-1), (26)
whilst, in the case of prescribed potentials (17)

d ′ (r) = d β(r) = 1 2 [∂ ζ φ 0 (1)] 2 -cosh φ p+ = 1 2 [∂ ζ φ 0 (-1)] 2 -cosh φ p-. (27)
Finally in the case of Stern layer boundary conditions (18) this constant is

d ′ (r) = d β(r) = 1 2 [µφ 0 (1)] 2 -cosh φ 0 (1) = 1 2 [µφ 0 (-1)] 2 -cosh φ 0 (-1) (28) 0 1 2 β 0 2 4 6 φ(1) µ=2 µ=3 µ=4
Figure 2: Surface potential of an anti-symmetric solution φ(1) at ζ = 1 versus dimensionless parameter β. The dotted lines display an asymptotic value which can be computed (Cf [Plouraboué and Chang(2008)] for more details on this point).

Hence in each case the function d ′ (r) either depends on the potential solution at the particle surface or on its gradient. In the case of Stern layer boundary condition (18) the anti-symmetrical solution associated with parameter µ > 1 is always attractive [Plouraboué and Chang(2008)]. This is thus the solution onto which we will focus on. A symmetrical solution also exists for parameter µ < 1 but this case is not considered in this study. Let us now briefly recall here the main steps for finding a implicit solution to the PB non-linear problem. We use Boltzmann transformation

ψ 0 = e -φ 0 /2 (29) so that (25) reads 1 2 (∂ ζ φ 0 ) 2 = 2e φ 0 (∂ ζ ψ 0 ) 2 = 1 2 (ψ 2 0 + 1 ψ 2 0 ) + d ′ (30)
so that the problem on new variable ψ 0 becomes

∂ ζ ψ 0 = ± 1 2 ψ 4 0 + 2d ′ ψ 2 0 + 1 = ± 1 2 (ψ 2 -α -)(ψ 2 -α + ), ( 31
) 10 with, α ± = -d ′ ± d ′ 2 -1, (32)
which solution can be found formally equal to

± β(r) 2 ζ + c = - √ α -F (ψ 0 √ α + , α -) , (33)
up to a constant c to be specified. Evaluating (33) at the upper and lower particle boundary ζ = z/h 0 = ±1 leads to the following implicit condition for the potential value ψ 0 (±1)

β(r) = - √ α -[F (ψ 0 (1) √ α + , α -) -F (ψ 0 (-1) √ α + , α -)] , (34)
Now, collecting relation ( 29) and ( 34) with one of the boundary condition associated with the d ′ (r) value ( 26), (28) gives a system of two transcendental equations for φ 0 (±1) that can be solved numerically [Plouraboué and Chang(2008)] for each β(r). Figure 2 shows the result of this numerical computation. It is interesting to note that for value of β smaller than a critical value which depends on µ β < β c (µ) the resulting surface potential is zero, and thus the solution will be zero every-where-else in between the two particles (Cf [Plouraboué and Chang(2008)] for the expression of β < β c (µ)). In these regions, the local interaction will obviously be zero at leading order.

4 Computation of the force

Local pressure contribution

We now compute the force between the particle. From using Green's theorem it is possible to show that the particle/particle interaction can be computed from evaluating the Maxwell stress tensor contraction with the normal surface of any closed surface around one particle [Neu(1999)]. Since, at infinity, the matching condition gives vanishing field perturbations, any closed far-field surface around one of the two particles which intersects the mean-plane, has no contribution to the force. Hence, the only contribution on the force is the scalar product of the stress tensor on the normal to the z = 0 mean-plane.

Let us now compute the force from considering the asymptotic expansion of the Maxwell stress tensor. For dimensionless formulation σ ′ = σǫ p (RT /ZF ) 2 /λ 2 , this tensor reads

σ = 1 β(r) -1 2 (∂ ζ φ) 2 + ǫ 2 (∂ r φ) 2 √ ǫ∂ r φ∂ ζ φ √ ǫ∂ r φ∂ ζ φ -1 2 (∂ ζ φ) 2 -ǫ 2 (∂ r φ) 2 (35)
So that, one can write :

σ = 1 β(r) - 1 2 (∂ ζ φ) 2 I + √ ǫ∂ r φ∂ ζ φ 0 1 1 0 + ǫ(∂ r φ) 2 1 0 0 -1 + .. (36)
Using ( 11), one can now obtain that the contribution of the stress tensor to the mean-plane is :

e z .σ.e z = 1 β(r) - 1 2 (∂ z φ) 2 - ǫ 2 (∂ r φ) 2 + .. ( 37
)
Now using ( 14) one finds

e z .σ.e z = 1 β(r) - 1 2 (∂ z φ 0 ) 2 -ǫ 1 2 (∂ r φ 0 ) 2 + ∂ z φ 0 ∂ z φ 1 + .. ( 38
)
which is the contribution of the Maxwell stress. The dimensionless osmotic contribution p 0 associated with a far-field zero reference potential is

p 0 = cosh φ -1 = cosh φ 0 -1 + O(ǫ) (39)
Finally, to the leading order, one can find the total local pressure at ζ = 0

p(r) = - 1 2β(r) [∂ ζ φ 0 (r, 0)] 2 + cosh φ 0 (r, 0) -1 + O(ǫ) (40)
As previously indicated it is interesting to note that in the case of antisymmetrical solution for which φ 0 (r, 0) = 0 this pressure is always negative and thus attractive. For any symmetrical solution for which on the contrary, ∂ z φ 0 (r, 0) = 0 and φ 0 (r, 0) = 0, we observe that this pressure is positive, so that the interaction is repulsive, because the osmotic contribution is always positive. Finally it is interesting to note from (25) that this force is simply related to the constant

d ′ p(r) = -d ′ (r) -1 + O(ǫ) (41)
This show that solving for the potential at the particle surface φ 0 (r, ζ = ±1) is enough to compute the total force from using ( 26) or (28) to deduce constant d ′ for a given value of β(r), that is to say a given value of r. Let us now explicitly estimate this force for spherical or cylindrical particles.

Total force

The total force formulation is the integral of the local force over the horizontal plane z = 0. We define two distinct dimensionless force in the case of spherical or cylindrical particles. For spherical particles we scale the force to the square of the sphere radius F s = aλǫ p (RT /ZF ) 2 /λ 2 F ′ . For cylindrical particles, we rather consider the product of radius a to the cylinder length L :

F c = √ aλLǫ p (RT /ZF ) 2 /λ 2 F ′ .
The integration nevertheless differs between spherical or cylindrical particles. In the case of two spheres one finds :

F s = 2π h m λ ∞ 0 p(r)rdr. (42)
In the following we will also use the equivalent formulation

F s = 2π ∞ hm λ p(h 0 )dh 0 . (43)
In the case of two cylinders, the total force per unit length is :

F c = 2 h m λ ∞ 0 p(r)dr (44)
Let us now first evaluate the forces in the D-H limit.

Force in the Debye-Hückel approximation

• In the case of prescribed fields the evaluation of (40) in the φ 0 ≪ 1 limit, using solution (22) leads to

p(r) = 1 8 λ h m 2 E 2 + cosh 2 h 0 -E 2 -sinh 2 (h 0 ) cosh 2 (h 0 ) sinh 2 (h 0 ) , (45)
One can see that in the case of symmetrical boundary conditions E -= 0, this pressure is positive leading to repulsion. In the fully non-symmetrical case, then E + = 0, and this pressure is negative leading, as expected to attraction between the particles. For sphere one finds :

F s = λ h m π 4 (E 2 + I s1 -E 2 -I s2 ) (46)
where,

I s1 = ∞ 0 rdr sinh 2 (1+r 2 /2) = 2 e 2 +1 ≃ 0.23840 I s2 = ∞ 0 rdr cosh 2 (1+r 2 /2) = 2 e 2 -1 ≃ 0.31303 (47)
For cylinders one finds :

F c = λ h m 3/2 1 4 (E 2 + I c1 -E 2 -I c2 ) (48)
where,

I c1 = ∞ 0 dr sinh 2 (1+r 2 /2) ≃ 0.5895922 I c2 = ∞ 0 dr cosh 2 (1+r 2 /2) ≃ 0.40108449 (49)
• Let us now consider the case with prescribed potentials. From linearization of (40) in the φ 0 ≪ 1 limit, the solution (24) leads to the following local pressure

p = λ h m 2 1 8 φ 2 + sinh 2 (h 0 ) -φ 2 -cosh 2 (h 0 ) cosh 2 (h 0 )sinh 2 (h 0 ) , (50)
As expected, this pressure is again always repulsive for symmetrical boundary conditions φ -= 0, and might be attractive for fully anti-symmetric conditions φ + = 0. For two spheres one finds the total force

F s = λ h m π 4 (φ 2 + I s2 -φ 2 -I s1 ), (51)
Whilst for two cylinders

F c = λ h m 3/2 1 4 (φ 2 + I c2 -φ 2 -I c1 ). (52)
Hence, in the Debye-Hückel approximation, the force can only be attractive for prescribed non-symmetrical fields. We do not discuss here the D-H limit for the Stern layer boundary condition since the solution is only specified up to a multiplicative constant in this regime, so that the absolute value of the force is not define. One needs to go to the non-linear PB problem to find a definite answer to this question.

Force for Stern-layer boundary condition

In this case a numerical computation has been carried out from the solution found for the potential field at the surface, which permits to deduce the d ′ (r) from ( 28) and the local pressure from (41).

• The numerical integration is then performed in the spherical case from formulation (43) with a simple trapezoidal rule. The result obtained is plotted on figure 3a where one can observe a saturation of the Force when the gap is smaller than the critical ratio β c for which the local pressure tends to zero.

• A different behavior for the total force is found in the case of two cylinders for which the formulation (44) associated with an integration along variable r is chosen to obtain again a simple direct integration. The total force display a different behavior with localized minimum at h m /λ ≃ β c /2 as represented on figure 3b.

Discussion

Let us discuss here the main results that have been obtained. We found in the D-H limit that for any imposed electric field or potential at the par-ticle surface the only possible attractive regime exists for non-symmetrical boundary conditions as expected from previous works [Neu(1999)]. On the quantitative point of view for any imposed electric field or potential we found in the D-H limit that the (attractive or repulsive) force display a divergent behavior with the minimum distance h m which is λ/h m for a sphere or a (λ/h m ) 3/2 for a cylinder.

This results differ with the one obtained for Stern layer boundary conditions at the iso-electric point. For two spheres the force decreases up to a critical ratio of h m /λ which is related to the parameter β c below which the local pressure tends to zero. The total force then reaches a constant plateau for very small h m /λ values. This behavior is different for two cylinders for which a local minimum is first reached, followed by a markedly inflected divergence at very small h m /λ. Both behavior are very different from those obtained for prescribed electric field and potential.

In any case two important remarks have to be added to better grasp the validity range of the presented results in the Stern layer case.

First, it is interesting to note that our computation is not valid for very small value of h m /λ ≪ 1. It is important to realize that the most substantial part of potential variations is mostly concentrated in the thin region of width √ ah m whereas it is very small outside this region. Since there is no interaction for distances smaller than β c , for there is no local pressure, a critical in-plane distance r c is associated with the critical parameter β c such that β c /2 = h m /λ(1 + r 2 c /2) for any interaction to occur. If r c exceeds √ ah m our leading order estimate will not give an accurate answer to the resulting very small interaction that will be associated to the problem. This gives a lower bond for the ratio h m /λ which has to be larger than h m /λ > 2β c ǫ for our analysis to be valid.

For smaller value of the h m /λ one should then consider the influence of O(ǫ) corrections to the force which might change the final picture.

Conclusion

We compute the electro-osmotic interaction between two particles when the gap h m is smaller than Debye length λ. We have shown that in the confined regime for which h m ≪ a and h m < λ the problem can be mapped onto a one dimensional planar formulation in a reduced parameter z/h 0 (r) which encapsulate any radial shape of the particles. We analyzed the influence of a Stern layer boundary condition at the iso-electric point on the interaction and found distinct new and interesting behavior for the particle interaction. Further extension of this work to non iso-electric point situations could be considered in the future.

  Dimensionless Force computed versus the minimum-gap to Debye layer ratio for different value of the mixed boundary condition parameter µ. (a) Between two spheres, (b) between two cylinders.