Christophe Chesneau

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

Keywords: Deconvolution, Quadratic functionals, Adaptive curve estimation, Wavelets, Global thresholding

We observe a stochastic process where a convolution product of an unknown function and a known function is corrupted by Gaussian noise. We wish to estimate the squared L 2 -norm of the unknown function from the observations. To reach this goal, we develop adaptive estimators based on wavelet and thresholding. We prove that they achieve (near) optimal rates of convergence under the mean squared error over a wide range of smoothness classes.

Motivation

We observe the stochastic process {Y (t); t ∈ [0, 1]} where dY (t) = 1 0 f (t -u)g(u)du dt + n -1/2 dW (t), t ∈ [0, 1],

(1) {W (t); t ∈ [0, 1]} is a (non-observed) standard Brownian motion, f is an unknown one-periodic function such that 1 0 f 2 (t)dt < ∞, and g is a known one-periodic function such that 1 0 g 2 (t)dt < ∞. The goal is to estimate f , or a quantity depending on f , from {Y (t); t ∈ [0, 1]}. The convolution model (1) illustrates the action of a linear time-invariant system on an input signal f when the data are corrupted with additional noise. See, for instance, [START_REF] Bertero | Introduction to Inverse Problems in Imaging, institute of physics[END_REF] and [START_REF] Neelamani | Forward: Fourier-wavelet regularized deconvolution for ill-conditioned systems[END_REF]. This is a standard inverse problem in the field of function estimation. For related results on (1), we refer to [START_REF] Cavalier | Sharp adaptation for inverse problems with random noise[END_REF], Johnstone, Kerkyacharian, [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF] and [START_REF] Cavalier | Nonparametric statistical inverse problems[END_REF]. Extensions of (1) Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, Campus II, Science 3, 14032 Caen, France. E-mail: chesneau@math.unicaen.fr can be found in [START_REF] Willer | Deconvolution in white noise with a random blurring effect[END_REF], [START_REF] Cavalier | Wavelet deconvolution with noisy eigenvalues[END_REF] and [START_REF] Pensky | Functional deconvolution in a periodic setting[END_REF].

In the literature, the main effort was spent on producing adaptive wavelet estimators of the function f . See, for instance, [START_REF] Cavalier | Sharp adaptation for inverse problems with random noise[END_REF], [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF] and [START_REF] Chesneau | Wavelet estimation via block thresholding: A minimax study under the L p risk[END_REF]. In this paper, we focus our attention on a different problem: the estimation of the squared L 2 -norm of f defined by

f 2 2 = 1 0 f 2 (t)dt.
The problem of estimating such a value is closely connected to the construction of confidence balls in nonparametric function estimation. It has already been investigated for a wide variety of models (density, regression, Gaussian model in white noise, density derivatives, . . . ). See, for instance, [START_REF] Bickel | Estimating integrated squared density derivatives: Sharp best order of convergence estimates[END_REF], [START_REF] Donoho | Minimax quadratic estimation of a quadratic functional[END_REF], [START_REF] Kerkyacharian | Estimating nonquadratic functionals of a density using haar wavelets[END_REF], [START_REF] Efromovich | On optimal adaptive estimation of a quadratic functional[END_REF], [START_REF] Gayraud | Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law[END_REF], Johnstone (2001a,b), [START_REF] Laurent | Adaptive estimation of a quadratic functional of a density by model selection[END_REF], Cai andLow (2005, 2006) and [START_REF] Rivoirard | The maxiset point of view for estimating integrated quadratic functionals[END_REF].

To the best of our knowledge, the estimation of f 2 2 from a convolution problem has been firstly studied in [START_REF] Butucea | Goodness-of-fit testing and quadratic functional estimation from indirect observations[END_REF]. The model considered is different to (1) and can be described as follows: i.i.d. random variables X i , i = 1, . . . , n, having unknown density f are observed with additive i.i.d. noise, independent of the X i 's, with known probability density g. To estimate f 2 2 , a kernel estimator is developed. It enjoys good statistical properties under various assumptions on g. However, it is not adaptive.

In this paper, we consider a simpler convolution model, ( 1), but we focus on the adaptive estimation of f 2 2 . To reach this goal, we use wavelet estimators. As in [START_REF] Butucea | Goodness-of-fit testing and quadratic functional estimation from indirect observations[END_REF], Theorem 4), we distinguish two cases according to the nature of g: the ordinary smooth case where the Fourier coefficients of g decrease in a polynomial fashion, and the supersmooth case where they decrease in an exponential fashion.

For the ordinary smooth case, we develop an adaptive wavelet estimator Q n based on the global thresholding. The idea is the following: we decompose f 2 2 by using an appropriate wavelet basis, we estimate the associated coefficients via natural estimators, then, at each level of the wavelets, we keep all of these estimators if, and only if, the corresponding l 2 norm is greater than a fixed threshold. This technic has been initially developed by [START_REF] Gayraud | Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law[END_REF] for the estimation of f 2 2 in the standard Gaussian white noise model. For the supersmooth case, Q n is not adapted. We develop another adaptive wavelet estimator Q * n . We evaluate their theoretical performances via the asymptotic minimax approach under the mean squared error over Besov balls. If Q n denotes an estimator of f 2 2 and B s 2,∞ (M ) denotes the Besov balls, we aim to determine the smallest rates of convergence ϕ n such that

lim n→∞ ϕ -1 n sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 < ∞.
For the ordinary smooth case, we prove that Q n achieves (near) optimal rates of convergence. For the supersmooth case, we prove that Q * n achieves (exactly) optimal rates of convergence. The upper bounds follow from a suitable decomposition of the risk and some technical inequalities (concentration, moment, . . . ). The lower bounds are applications of a specific theorem established by [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF].

The paper is organized as follows. In Section 2, we present wavelets and Besov balls. Section 3 clarifies the assumptions made on the Fourier coefficients of g and introduces some intermediate estimators. The main estimators of the study are defined in Section 4. Section 5 is devoted to the results. Perspectives and open questions are set in Section 6. The proofs of the results are postponed in Section 7.

Wavelets and Besov balls

Wavelets

We consider an orthonormal wavelet basis generated by dilations and translations of a "father" Meyer-type wavelet φ and a "mother" Meyer-type wavelet ψ. The features of such wavelets are that the Fourier transforms of φ and ψ have bounded support. More precisely, we have supp (

F (φ)) ⊂ [-4π3 -1 , 4π3 -1 ] and supp (F (ψ)) ⊂ [-8π3 -1 , -2π3 -1 ] ∪ [2π3 -1 , 8π3 -1 ], where, for any function h ∈ L 1 ([0, 1]), F (h) denotes the Fourier transform of h defined by F (h)(l) = 1 0 h(x)e -2iπlx dx. Moreover, for any l ∈ [-2π, -π] ∪ [π, 2π], there exists a con- stant c > 0 such that |F (ψ)(l)| ≥ c.
For further details about Meyer-type wavelets, see [START_REF] Walter | Wavelets and other orthogonal systems in applications[END_REF], and [START_REF] Zayed | Characterization of analytic functions in term of their wavelet coefficients[END_REF].

For the purposes of this paper, we use the periodised wavelet bases on the unit interval. For any x ∈ [0, 1], any integer j and any k ∈ {0, . . . , 2 j -1}, let φ j,k (x) = 2 j/2 φ(2 j x -k) and ψ j,k (x) = 2 j/2 ψ(2 j x -k) be the elements of the wavelet basis, and

φ per j,k (x) = l∈Z φ j,k (x -l), ψ per j,k (x) = l∈Z ψ j,k (x -l),
their periodised versions. There exists an integer τ such that the collection ζ defined by ζ = {φ per τ,k , k = 0, . . . , 2 τ -1; ψ per j,k , j = τ, . . . , ∞, k = 0, . . . , 2 j -1} constitutes an orthonormal basis of L 2 per ([0, 1]), the set of square integrable one-periodic functions on [0,1]. In what follows, the superscript "per" will be suppressed from the notations for convenience.

Let m be an integer such that m ≥ τ . A function f ∈ L 2 per ([0, 1]) can be expanded into a wavelet series as

f (x) = 2 m -1 k=0 α m,k φ m,k (x) + ∞ j=m 2 j -1 k=0 β j,k ψ j,k (x), x ∈ [0, 1],
where α m,k = 1 0 f (t)φ m,k (t)dt and β j,k = 1 0 f (t)ψ j,k (t)dt. For further details about wavelet bases on the unit interval, we refer to [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF].

Besov balls

Let M ∈ (0, ∞) and s ∈ (0, ∞). We say that a one-periodic function f belongs to the Besov balls B s 2,∞ (M ) if, and only if, there exists a constant M * > 0 such that ||f || 2 2 ≤ M * and the associated wavelet coefficients satisfy

sup j≥τ 2 2js 2 j -1 k=0 β 2 j,k ≤ M * .
These sets contain all the Besov balls B s p,∞ (M ) with p ≥ 2. See, for instance, [START_REF] Meyer | Wavelets and Operators[END_REF].

3 Preliminary study 3.1 Assumptions on g ; ordinary smooth and supersmooth cases

• Assumption (A g ): ordinary smooth case. We suppose that there exist three constants, c > 0, C > 0 and δ > 2 -1 , such that, for any l ∈ (-∞, -1] ∪ [1, ∞), the Fourier coefficient of g, i.e. F (g)(l), satisfies

c|l| -δ ≤ |F (g)(l)| ≤ C|l| -δ . (2)
For example, the square integrable one-periodic function g defined by g(x) = m∈Z e -|x+m| , x ∈ [0, 1], satisfies (A g ). Indeed, for any l ∈ Z, we have F (g)(l) = 2 1 + 4π 2 l 2 -1 . Hence, for any l ∈ (-∞, -1] ∪ [1, ∞), F (g)(l) satisfies (2) with c = 2(1 + 4π 2 ) -1 , C = (2π 2 ) -1 and δ = 2.

• Assumption (A * g ): supersmooth case. We suppose that there exist four constants, c > 0, C > 0, a > 0 and b > 0, such that, for any l ∈ (-∞, -1] ∪ [1, ∞), the Fourier coefficient of g, i.e. F (g)(l), satisfies

ce -a|l| b ≤ |F (g)(l)| ≤ Ce -a|l| b . (3)
For example, the square integrable one-periodic function g defined by g(x) = m∈Z e -(x+m) 2 /2 , x ∈ [0, 1], satisfies (A * g ). Indeed, for any l ∈ Z, we have

F (g)(l) = √ 2πe -2π 2 l 2 . Hence, for any l ∈ (-∞, -1] ∪ [1, ∞), F (g)(l) satisfies (3) with c = C = √ 2π, a = 2π 2 and b = 2.
Further details and examples about these cases can be found in [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF] and [START_REF] Fan | Wavelet deconvolution[END_REF].

Preliminary to the estimation of f 2 2

Thanks to the orthonormality of the wavelet basis, for any integer m ≥ τ , the unknown f 2 2 can be decomposed as

f 2 2 = 2 m -1 k=0 α 2 m,k + ∞ j=m 2 j -1 k=0 β 2 j,k .
Thus, the first step to estimate f 2 2 consists in estimating the unknown coefficients (α 2 m,k ) k and (β 2 j,k ) j,k . In what follows, we investigate the estimation of (β 2 j,k ) j,k . First of all, we aim to write the model (1) in the Fourier domain. For any l ∈ Z, we have

F 1 0 f (. -u)g(u)du (l) = F (f g) (l) = F (f )(l)F (g)(l),
where denotes the standard convolution product on the unit interval. Therefore, it we set y l = 1 0 e -2πilt dY (t), f l = F (f )(l), g l = F (g)(l), and e l = 1 0 e -2πilt dW (t), it follows from (1) that

y l = f l g l + n -1/2 e l .
(4)

The Parseval theorem and the translation theorem of the Fourier transform give

l∈Z y l g -1 l F (ψ j,k )(l) = l∈Z f l F (ψ j,k )(l) + n -1/2 l∈Z g -1 l F (ψ j,k )(l)e l = β j,k + n -1/2 l∈Z g -1 l e 2iπlk/2 j F (ψ j,0 )(l)e l . Since n -1/2 l∈Z g -1 l e 2iπlk/2 j F (ψ j,0 )(l)e l ∼ N 0, n -1 l∈Z |g l | -2 |F (ψ j,0 )(l)| 2 , we have β j,k = l∈Z y l g -1 l F (ψ j,k )(l) ∼ N β j,k , n -1 l∈Z |g l | -2 |F (ψ j,0 )(l)| 2 .
Therefore, the random variable θ j,k defined by

θ j,k = β 2 j,k -n -1 l∈Z |g l | -2 |F (ψ j,0 )(l)| 2
is an unbiased (and a natural) estimator of β 2 j,k . We are now in the position to describe the main estimators of the study.

Estimators

According to the ordinary smooth case and the supersmooth case, we describe two different estimators of Q(f ) = f 2 2 . We use the notations introduced in section 3.2.

Estimator: the ordinary smooth case

Suppose that (A g ) is satisfied. Let j 0 and j 1 be two integers such that

2 -1 n(log n) -1 1/(4δ+1) < 2 j0 ≤ n(log n) -1 1/(4δ+1) and 2 -1 n 1/(2δ+1/2) < 2 j1 ≤ n 1/(2δ+1/2) .
For any real number a, set (a) + = max(a, 0). Let κ be a positive real number. We define the thresholding estimator Q n by

Q n = 2 j 0 -1 k=0 α 2 j0,k -n -1 η 2 j0 + j1 j=j0   2 j -1 k=0 ( β 2 j,k -n -1 σ 2 j ) -κn -1 σ 2 j j2 j 1/2   + , (5)
where

α j0,k = l∈Dj 0 y l g -1 l F (φ j0,k )(l), β j,k = l∈Cj y l g -1 l F (ψ j,k )(l), (6)
η 2 j0 = l∈Dj 0 |g l | -2 F 2 (φ j0,0 )(l), σ 2 j = l∈Cj |g l | -2 |F (ψ j,0 )(l)| 2 .
Here, for any k ∈ {0, . . . , 2 j -1}, D j0 denotes the support of F (φ j0,k )(l) and

C j the support of F (ψ j,k )(l).
Let us briefly explain the construction of Q n . Firstly, we estimate the approximation term,

2 j 0 -1 k=0 α 2 j0,k
, by

2 j 0 -1 k=0 α 2 j0,k -n -1 η 2 j0
. Secondly, we estimate the "detail term", j1 j=j0 2 j -1 k=0 β 2 j,k , via the global thresholding rule described as follows. At each level j ∈ {j 0 , . . . , j 1 }, we keep

2 j -1 k=0 ( β 2 j,k - n -1 σ 2 j ) -κn -1 σ 2 j j2 j 1/2
if, and only if, we have

2 j -1 k=0 ( β 2 j,k -n -1 σ 2 j ) ≥ κn -1 σ 2 j j2 j 1/2 .
Then we sum all the selected terms. The idea is to only estimate the unknown coefficients β 2 j,k which characterized completely f 2 2 . As mentioned in Section 1, the global thresholding technic has been initially developed by [START_REF] Gayraud | Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law[END_REF] for the estimation of the squared L 2 -norm of f in the standard Gaussian white noise model.

Estimator: the supersmooth case

Suppose that (A * g ) is satisfied. In this case, the estimator Q n defined by ( 5) is not appropriate. One can prove that, in our minimax framework, it achieves a "suboptimal" bound. For this reason, we propose another estimator. It is described as follows. Let j * be an integer such that

2 -1 (16π3 -1 a) -1 (log n) 1/b < 2 j * ≤ (16π3 -1 a) -1/b (log n) 1/b .
We define the linear (or projection) estimator Q * n by

Q * n = 2 j * -1 k=0 α 2 j * ,k -n -1 η 2 j * , (7)
where α j * ,k and η j * are defined by ( 6) with j * instead of j 0 . The idea of Q * n is to estimate the approximation term

2 j * -1 k=0 α 2 j * ,k of f 2
2 (until the level j * ), and to neglect the detail term.

Remark. The consideration of two different estimators for each case (ordinary smooth and supersmooth) is standard for the adaptive wavelet estimation of f from convolution model. See, for instance, [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF] and [START_REF] Fan | Wavelet deconvolution[END_REF]. However, to the best of our knowledge, this is new for the adaptive wavelet estimation of f 2 2 .

Important remark. The estimators Q n and Q * n do not require any a priori knowledge on f in their constructions ; they are adaptive.

Results

This section is divided into two parts. The first part is devoted to the adaptive estimation of ||f || 2 2 for the ordinary smooth case (A g ), and the second part, for the supersmooth case (A * g ).

Results: ordinary smooth case

Under (A g ), Theorem 1 below determines the rates of convergence achieved by Q n under the mean squared error over the Besov balls B s 2,∞ (M ). Theorem 1 Consider the convolution model defined by (1). Suppose that (A g ) is satisfied. Let Q n be the estimator defined by (5) with a large enough κ. Then there exists a constant C > 0 such that, for n large enough,

sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≤ Cϕ n ,
where

ϕ n = n -1 , when s > δ + 4 -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) , when s ∈ (0, δ + 4 -1 ], i.e., ϕ n = max n -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) .
The proof of Theorem 1 uses several auxiliary results. In particular, we establish a functional upper bound for the risk similar to Proposition 3 of [START_REF] Rivoirard | The maxiset point of view for estimating integrated quadratic functionals[END_REF]. It is obtained by proving two technical concentration inequalities satisfied by the estimators ( β j,k ) k=0,...,2 j -1 .

The result of Theorem 1 raises the following question: are the rates of convergence ϕ n the optimal ? Theorem 2 below provides the answer by determining the minimax lower bounds.

Theorem 2 Consider the convolution model defined by (1). Suppose that (A g ) is satisfied. Then there exists a constant c > 0 such that, for n large enough,

inf e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ cϕ * n ,
where inf e Qn denotes the infimum over all the possible estimators of f 2 2 , and

ϕ * n = n -1 , when s > δ + 4 -1 , n -4s/(2s+2δ+1/2) , when s ∈ (0, δ + 4 -1 ].
The proof of Theorem 2 is based on a general theorem established by [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF].

The results of Theorems 1 and 2 show that, under (A g ), Q n is near optimal under the mean squared error over the Besov balls B s 2,∞ (M ). "Near" is due to the case s ∈ (0, δ + 4 -1 ] where there is an extra logarithmic term.

Results: supersmooth case

Under (A * g ), Theorem 1 below determines the rates of convergence achieved by Q * n under the mean squared error over the Besov balls B s 2,∞ (M ). Theorem 3 Consider the convolution model defined by (1). Suppose that (A * g ) is satisfied. Let Q * n be the estimator defined by (7). Then there exists a constant C > 0 such that, for n large enough,

sup f ∈B s 2,∞ (M ) E Q * n -f 2 2 2 ≤ C(log n) -4s/b .
The proof of Theorem 3 is based on a good decomposition of the risk and technical inequalities (l 2 norm, moment, . . . ). As in Theorem 2, but under (A * g ), Theorem 4 below determines the minimax lower bounds of the model.

Theorem 4 Consider the convolution model defined by (1). Suppose that (A * g ) is satisfied. Then there exists a constant c > 0 such that, for n large enough,

inf e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ c(log n) -4s/b ,
where inf e Qn denotes the infimum over all the possible estimators of f 2 2 .

The proof of Theorem 4 is based on a general theorem proved by [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF]. Theorems 3 and 4 show that, under (A * g ), υ n = (log n) -4s/b is the minimax rate of convergence under the mean squared error over the Besov balls B s 2,∞ (M ). Moreover the adaptive estimator Q * n is optimal.

Conclusion and open questions

The following table summarizes the minimax results of the paper:

Case ordinary smooth supersmooth

Estimator

Q n Q * n Upper bound max n -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) (log n) -4s/b
Optimality optimal, when s > δ + 4 -1 , near optimal, when s ∈ (0, δ + 4 -1 ]. optimal

Our rates of convergence are similar to those determined in [START_REF] Butucea | Goodness-of-fit testing and quadratic functional estimation from indirect observations[END_REF], Theorem 4) which considers the density convolution, a non-adaptive estimator based on kernel, and the risk: R(h, f

) = E h -f 2 2 , h ∈ R.
In comparison to this result, we consider another convolution model, simpler to manipulate, but we provide a contribution to the adaptive estimation of f 2 2 .

Some perspectives and open questions are presented below:

-A possible extension of this work is to adapt our estimators to other convolution models. For instance, the density convolution model: i.i.d. random variables X i , i = 1, . . . , n, having unknown density f are observed with additive i.i.d. noise, independent of the X i 's, with known probability density g. Another interesting convolution model is the one studied in [START_REF] Pensky | Functional deconvolution in a periodic setting[END_REF]: one observes n random variables {Y i ; i = 1, . . . , n} where

Y i = 1 0 f (i/n -u)g(u)du + i , i = 1, . . . , n, { i ; i = 1, .
. . , n} are (nonobserved) i.i.d. Gaussian standard random variables, f is an unknown oneperiodic function such that 1 0 f 2 (t)dt < ∞, and g is a known one-periodic function such that 1 0 g 2 (t)dt < ∞. Despite the "equivalences" which exist between these models and (1), they are more difficult to manipulate. In particular, they can not be directly written as a sequence model of the form (4) with "f l × g l " and/or "e l " independent N (0, 1). For instance, this last point is an obstacle to the application of the Cirelson inequality used in the proof of Theorem 1 (see Lemma 3). Thus, the estimation of f 2 2 from these models via our estimators requires technical tools certainly not used in our study.

-Our minimax results can be extended to the Besov balls B s p,∞ (M ) with p ≥ 2. However, the case p < 2 is beyond the scope of the paper. As underlined in [START_REF] Cai | Optimal adaptive estimation of a quadratic functional[END_REF] for the standard Gaussian sequence model (page 2300, lines 18-19): "The sparse case where p < 2 presents some major new difficulties which requires a novel approach for the construction of adaptive procedures". In response to these difficulties, [START_REF] Cai | Optimal adaptive estimation of a quadratic functional[END_REF] have developed a sophisticated adaptive estimator which uses both block thresholding and term-by-term thresholding. Taking the minimax approach under the mean squared error over B s p,∞ (M ), it is (near) optimal for p ≥ 1. An open question is: for the model (1), more complex than the standard Gaussian sequence model due to the presence of g, how can we adapt this estimator in order to keep these optimal properties over B s p,∞ (M ) for any p ≥ 1 ?

-And, finally: can we construct only one adaptive estimator Q n which are (near) optimal for both ordinary smooth case and supersmooth case ?

All these aspects need further investigation that we leave for a future work.

Proofs

In this section, c and C denote positive constants which can take different values for each mathematical term. They are independent of f and n.

Proof of Theorem 1. Proposition 1 below presents a "functional" upper bound for the mean squared risk of Q n without any assumption on the smoothness of f . Proposition 1 Let Q n be the estimator defined by (5) with a large enough κ. There exist two constants C 1 > 0 and C 2 > 0 such that

E Q n -f 2 2 2 ≤ C 1 (A + B + C + D), where A = n -2 2 (1+4δ)j0 , B = n -1 f 2 2 , C =   ∞ j=j1+1 2 j -1 k=0 β 2 j,k   2 and D =   j1 j=j0 min   2 j -1 k=0 β 2 j,k , C 2 κ2 2δj n -1 (j2 j ) 1/2     2 .
Therefore, to prove Theorem 1, it is enough to investigate the upper bounds for the terms A, B, C and D defined in Proposition 1 above.

The upper bound for A. Thanks to the definition of j 0 , we have

A = n -2 2 (1+4δ)j0 ≤ n -1 ≤ max n -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) = ϕ n .
The upper bound for B. Since f ∈ B s 2,∞ (M ), we have f 2 2 ≤ M * . Therefore

B = n -1 f 2 2 ≤ C max n -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) = Cϕ n .
The upper bound for C. Since f ∈ B s 2,∞ (M ), we have

C =   ∞ j=j1+1 2 j -1 k=0 β 2 j,k   2 ≤ C   ∞ j=j1+1 2 -2js   2 ≤ C2 -4j1s ≤ Cn -4s/(2δ+1/2) ≤ C max n -1 , n(log n) -1/2 -4s/(2s+2δ+1/2) = Cϕ n .
The upper bound for D. We distinguish the case s > δ + 4 -1 and the case s ∈ (0, δ + 4 -1 ].

-The case s > δ + 4 -1 . Since f ∈ B s 2,∞ (M ), the definition of j 0 yields

D =   j1 j=j0 min   2 j -1 k=0 β 2 j,k , C 2 κ2 2δj n -1 (j2 j ) 1/2     2 ≤ C   j1 j=j0 2 j -1 k=0 β 2 j,k   2 ≤ C   j1 j=j0 2 -2js   2 ≤ C2 -4j0s ≤ C n(log n) -1 -4s/(4δ+1) ≤ Cn -1 .
-The case s ∈ (0, δ + 4 -1 ]. Let j 2 be an integer such that

2 -1 n(log n) -1/2 1/(2s+2δ+1/2) < 2 j2 ≤ n(log n) -1/2 1/(2s+2δ+1/2) .
Notice that j 0 ≤ j 2 ≤ j 1 . We have the following decomposition

D = (L + M) 2 ,
where

L = j2 j=j0 min   2 j -1 k=0 β 2 j,k , C 2 κ2 2δj n -1 (j2 j ) 1/2   and M = j1 j=j2+1 min   2 j -1 k=0 β 2 j,k , C 2 κ2 2δj n -1 (j2 j ) 1/2   .
Let us study the upper bounds for L and M in turn.

It follows from the definition of j 2 that

L ≤ Cn -1 j2 j=j0 j 1/2 2 (2δ+1/2)j ≤ Cn -1 (log n) 1/2 2 (2δ+1/2)j2 ≤ C n(log n) -1/2 -2s/(2s+2δ+1/2) . Since f ∈ B s 2,∞ (M ), the definition of j 2 yields M ≤ j1 j=j2+1 2 j -1 k=0 β 2 j,k ≤ C j1 j=j2+1 2 -2js ≤ C2 -2j2s ≤ C n(log n) -1/2 -2s/(2s+2δ+1/2) .
Putting these two inequalities together, we obtain

D = (L + M) 2 ≤ C n(log n) -1/2 -4s/(2s+2δ+1/2) .
It follows from Proposition 1 and the obtained upper bounds for A, B, C and D, that

E Q n -f 2 2 2 ≤ C 1 (A + B + C + D) ≤ Cϕ n .
Theorem 1 is proved.

Proof of Proposition 1. The proof of Proposition 1 is similar to the proof of Proposition 3 of [START_REF] Rivoirard | The maxiset point of view for estimating integrated quadratic functionals[END_REF]. If we analyze this proof, Proposition 1 follows from Lemmas 1 and 2 below.

Lemma 1 If (A g ) is satisfied, then here exist two constants c > 0 and C > 0 such that, for any integer j ≥ τ , we have

c2 2δj ≤ σ 2 j ≤ C2 2δj .
Moreover, we have η 2 j0 ≤ C2 2δj0 .

Proof of Lemma 1. Using the inequalities sup l∈Cj |F (ψ j,0 )(l)| 2 ≤ C2 -j , Card(C j ) ≤ C2 j and (A g ), we have

σ 2 j = l∈Cj |g l | -2 |F (ψ j,0 )(l)| 2 ≤ C2 -j l∈Cj |g l | -2 ≤ C sup l∈Cj |g l | -2 ≤ C2 2δj .
In the same way, we prove that η 2 j0 ≤ C2 2δj0 .

Recall that, for any l ∈ [-2π, -π]∪[π, 2π], there exists a constant c > 0 such that |F (ψ)(l)| ≥ c. Therefore, there exists a set H j such that inf l∈Hj |F (ψ j,0 )(l)| 2 ≥ c2 -j , Card(H j ) ≥ c2 j and, thanks to (A g ), inf l∈Hj g -2 l ≥ c2 2δj . Hence,

c2 2δj ≤ c inf l∈Hj g -2 l ≤ c2 -j l∈Hj |g l | -2 ≤ l∈Cj |g l | -2 |F (ψ j,0 )(l)| 2 = σ 2 j .
This ends the proof of Lemma 1.

Lemma 2 below presents two concentration inequalities satisfied by the estimator β j,k .

Lemma 2 Let β j,k be the estimator defined by ( 6). For λ large enough, there exists a constant C > 0 such that, for any integer j ≥ τ , we have

P    2 j -1 k=0 β j,k ( β j,k -β j,k ) ≥ λn -1/2 j 1/2 2 δj   2 j -1 k=0 β 2 j,k   1/2    ≤ 2 exp -Cλ 2 j (8) and P      2 j -1 k=0 ( β j,k -β j,k ) 2   1/2 ≥ λn -1/2 2 j/2 2 δj    ≤ 2 exp -Cλ 2 2 j . (9)
Proof of Lemma 2. The first part of the proof is devoted to the bound (8) and the second part, to the bound (9).

-Proof of the bound (8). Since the random variable (e l ) l are i.i.d. with e l ∼ N (0, 1), we have

2 j -1 k=0 β j,k ( β j,k -β j,k ) = n -1/2 l∈Cj e l g -1 l 2 j -1 k=0 β j,k F (ψ j,k )(l) ∼ N 0, θ 2 j ,
where

θ 2 j = n -1 l∈Cj |g l | -2   2 j -1 k=0 β j,k F (ψ j,k )(l)   2 . (A g ) yields sup l∈Cj |g l | -2 ≤ C2 2δj
. It follows from this, the Plancherel theorem and the orthonormality of the wavelet basis that

θ 2 j = n -1 l∈Cj |g l | -2   F   2 j -1 k=0 β j,k ψ j,k   (l)   2 ≤ Cn -1 2 2δj l∈Cj   F   2 j -1 k=0 β j,k ψ j,k   (l)   2 = Cn -1 2 2δj 1 0   2 j -1 k=0 β j,k ψ j,k (x)   2 dx = Cn -1 2 2δj 2 j -1 k=0 β 2 j,k .
Applying a standard Gaussian concentration inequality, we obtain

P    2 j -1 k=0 β j,k ( β j,k -β j,k ) ≥ λn -1/2 j 1/2 2 δj   2 j -1 k=0 β 2 j,k   1/2    ≤ 2 exp   -   λn -1/2 j 1/2 2 δj   2 j -1 k=0 β 2 j,k   1/2    2 /(2θ 2 j )    ≤ 2 exp -Cλ 2 j .
This ends the proof of the bound ( 8).

-Proof of the bound (9). To obtain the bound (9), we need the Cirelson inequality presented in Lemma 3 below.

Lemma 3 [START_REF] Cirelson | Norm of Gaussian sample functions[END_REF]) Let D be a subset of R. Let (η t ) t∈D be a centered Gaussian process. If E (sup t∈D η t ) ≤ N and sup t∈D V (η t ) ≤ V then, for any x > 0, we have

P sup t∈D η t ≥ x + N ≤ exp -x 2 /(2V ) . We have β j,k -β j,k = n -1/2 l∈Cj e l g -1 l F (ψ j,k )(l) ∼ N 0, n -1 σ 2 j , where σ 2 j = l∈Cj |g l | -2 |F (ψ j,0 )(l)| 2 . Set Ω = {a = (a k ) ∈ R; 2 j -1 k=0 a 2 k ≤ 1}.
For any a ∈ Ω, let Z(a) be the centered Gaussian process defined by

Z(a) = 2 j -1 k=0 a k ( β j,k -β j,k ) = n -1/2 l∈Cj e l g -1 l 2 j -1 k=0 a k F (ψ j,k )(l).
By an argument of duality, we have sup a∈Ω Z(a) =

2 j -1 k=0 ( β j,k -β j,k ) 2 1/2
. Now, let us determine the values of N and V which appeared in the Cirelson inequality (see Lemma 3).

Value of N . The Cauchy-Schwarz inequality and Lemma 1 imply that

E sup a∈Ω Z(a) = E      2 j -1 k=0 ( β j,k -β j,k ) 2   1/2    ≤   2 j -1 k=0 E ( β j,k -β j,k ) 2   1/2 ≤ C   n -1 2 j -1 k=0 σ 2 j   1/2 ≤ Cn -1/2 2 j/2 2 δj . Hence N = Cn -1/2 2 j/2 2 δj .
Value of V . Using the fact that, for any l, l ∈ Z, we have E (e l e l ) =

1 0 e -2iπ(l-l )t dt = 1 {l=l } , (A g ), the Plancherel inequality and the orthonormality of the wavelet basis, we obtain

sup a∈Ω V(Z(a)) = sup a∈Ω   E   2 j -1 k=0 2 j -1 k =0 a k β j,k -β j,k )a k ( β j,k -β j,k )     = n -1 sup a∈Ω 2 j -1 k=0 2 j -1 k =0 a k a k l∈Cj l ∈Cj g l -1 F (ψ j,k )(l)(g l ) -1 F (ψ j,k )(l )E (e l e l ) = n -1 sup a∈Ω   2 j -1 k=0 2 j -1 k =0 a k a k l∈Cj |g l | -2 F (ψ j,k )(l)F (ψ j,k )(l)   ≤ Cn -1 2 2δj sup a∈Ω   2 j -1 k=0 2 j -1 k =0 a k a k l∈Cj F (ψ j,k )(l)F (ψ j,k )(l)   = Cn -1 2 2δj sup a∈Ω   2 j -1 k=0 2 j -1 k =0 a k a k 1 0 ψ j,k (x)ψ j,k (x)dx   = Cn -1 2 2δj sup a∈Ω   2 j -1 k=0 a 2 k   ≤ C2 2δj n -1 .
Hence V = C2 2δj n -1 . By taking λ large enough and x = 2 -1 λn -1/2 2 j/2 2 δj , the Cirelson inequality described in Lemma 3 yields

P      2 j -1 k=0 ( β j,k -β j,k ) 2   1/2 ≥ λn -1/2 2 j/2 2 δj    ≤ P sup a∈Ω Z(a) ≥ x + N ≤ exp -x 2 /(2V ) ≤ 2 exp -Cλ 2 2 j .
The bound ( 9) is proved. The proof of Lemma 2 is complete.

This ends the proof of Proposition 1.

Proof of Theorem 2. We distinguish the case s > δ + 4 -1 and the case s ∈ (0, δ + 4 -1 ].

The case s > δ + 4 -1 . We consider the two functions

f 0 (x) = 1, f 1 (x) = 1 + n -1/2 , x ∈ [0, 1].
Clearly, f 0 and f 1 belong to B s 2,∞ (M ). Moreover, we have

f 0 2 2 -f 1 2 2 = 2n -1/2 + n -1 ≥ n -1/2 = κ n
Now, in order to apply Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF], we aim to bound (by a constant C) the chi-square divergence χ 2 (P f1 , P f0 ) defined by

χ 2 (P f1 , P f0 ) = dP f1 dP f0 2 dP f0 -1, (10)
where P h denotes the probability distribution of {Y (t); t ∈ [0, 1]} indexed by the function h. Let be the standard convolution product on the unit interval. The Girsanov theorem yields

dP f1 dP f0 = exp(n 1 0 ((f 1 g)(t) -(f 0 g)(t)))dY (t) - 2 -1 n 1 0 (f 1 g) 2 (t) -(f 0 g) 2 (t) dt) = exp n 1/2 1 0 g(u)du 1 0 dY (t) -2 -1 n 1 0 g(u)du 2 (2n -1/2 + n -1 ) .
So, under P f0 , we have

dP f1 dP f0 = exp 1 0 g(u)du 1 0 dW (t) -2 -1 1 0 g(u)du 2 (2n 1/2 + 1) .
Using the equality exp 2

1 0 g(u)du 1 0 dW (t) dP f0 = exp 2 1 0 g(u)du 2 ,
the Cauchy-Schwarz inequality and the fact that

1 0 g 2 (t)dt < ∞, we have dP f1 dP f0 2 dP f0 = exp - 1 0 g(u)du 2 (2n 1/2 + 1) exp 2 1 0 g(u)du 1 0 dW (t) dP f0 = exp - 1 0 g(u)du 2 (2n 1/2 + 1) exp 2 1 0 g(u)du 2 ≤ exp 1 0 g(u)du 2 ≤ exp 1 0 g 2 (u)du ≤ C.
Therefore, there exists a constant C > 0 such that

χ 2 (P f1 , P f0 ) = dP f1 dP f0 2 dP f0 -1 ≤ C < ∞.
It follows from Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF] that

inf e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ cκ 2 n = cn -1 = cϕ * n . (11)
The case s ∈ (0, δ + 4 -1 ]. We consider the two functions

f 0 (x) = 0, f 1 (x) = γ j * 2 j * k * =0 w k * ψ j * ,k * (x), x ∈ [0, 1], ( 12
)
where j * is an integer to be chosen below, γ j * is a quantity to be chosen below, and w 0 , . . . , w 2 j * -1 are i.i.d. Rademacher random variables (i.e., for any u ∈ {0, . . . , 2 j * -1}, P(w u = 1) = P(w u = -1) = 2 -1 ).

Clearly, f 0 belongs to B s 2,∞ (M ). Thanks to the orthogonality of the wavelet basis, for any integer j ≥ τ and any k ∈ {0, . . . , 2 j -1}, we have β j,k = 1 0 f 1 (x)ψ j,k (x)dx = γ j * w k for j = j * , and 0 otherwise. Thus, if γ 2 j * = M * 2 -j * (2s+1) , then

2 2j * s 2 j * -1 k=0 β 2 j * ,k = 2 2j * s γ 2 j * 2 j * -1 k=0 w 2 k = γ 2 j * 2 j * (2s+1) = M * .
Therefore, for such a choice of γ j * , we have f 1 ∈ B s 2,∞ (M ). Let j * be an integer such that

2 -1 n 1/(2s+2δ+1/2) ≤ 2 j * ≤ n 1/(2s+2δ+1/2) . (13)
With the previous value of γ j * , the orthonormality of the wavelet basis gives

f 0 2 2 -f 1 2 2 = f 1 2 2 = γ 2 j * 2 j * -1 k=0 w 2 k = γ 2 j * 2 j * ≥ C2 -2j * s ≥ Cn -2s/(2s+2δ+1/2) = κ n .
In what follows, we consider the function f 1 described by ( 19) with the integer j * defined by ( 13) and the quantity γ j * = M 1/2 * 2 -j * (s+1/2) . In order to apply Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF], we aim to bound the chi-square divergence χ 2 (P f1 , P f0 ) (see ( 10)) by a constant.

Let be the standard convolution product. Due to the definitions of f 0 , f 1 , the random variables w 0 , . . . , w 2 j * -1 , and the Girsanov theorem, we have

dP f1 dP f0 = 2 j * -1 k=0 [2 -1 exp nγ j * 1 0 (ψ j * ,k g)(t)dY (t) -2 -1 nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt + 2 -1 exp -nγ j * 1 0 (ψ j * ,k g)(t)dY (t) -2 -1 nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt ].
So, under P f0 , we have

dP f1 dP f0 = 2 j * -1 k=0 [2 -1 exp n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) -2 -1 nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt + 2 -1 exp -n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) -2 -1 nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt ] = 2 j * -1 k=0 2 -1 exp -2 -1 nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt × [exp n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) + exp -n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) ].
Hence,

dP f1 dP f0 2 dP f0 = 2 j * -1 k=0 2 -2 exp -nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt × [exp n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) + exp -n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) ] 2 dP f0 . Since exp 2n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) dP f0 = exp -2n 1/2 γ j * 1 0 (ψ j * ,k g)(t)dW (t) dP f0 = exp 2nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt ,
we have

dP f1 dP f0 2 dP f0 = 2 j * -1 k=0 2 -1 exp nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt + exp -nγ 2 j * 1 0 (ψ j * ,k g) 2 (t)dt .
The inequality 2 -1 (e x + e -x ) ≤ e x 2 /2 , x ∈ R, implies that

dP f1 dP f0 2 dP f0 ≤ 2 j * -1 k=0 exp 2 -1 n 2 γ 4 j * 1 0 (ψ j * ,k g) 2 (t)dt 2 . (14)
Using (A g ) and the Plancherel theorem, we obtain

1 0 (ψ j * ,k g) 2 (t)dt = l∈Z (F (ψ j * ,k g)) 2 (l) = l∈Cj * |F (ψ j * ,k ) (l)| 2 |g l | 2 ≤ C2 -2δj * l∈Cj * |F (ψ j * ,k ) (l)| 2 = C2 -2δj * 1 0 ψ 2 j * ,k (x)dx = C2 -2δj * .
We deduce that

dP f1 dP f0 2 dP f0 ≤ 2 j * -1 k=0 exp Cn 2 γ 4 j * 2 -4δj * = exp Cn 2 γ 4 j * 2 (1-4δ)j * .
Thanks to the definitions of j * and γ j * , we have

n 2 γ 4 j * 2 (1-4δ)j * ≤ Cn 2 2 -j * (4s+2) 2 (1-4δ)j * ≤ Cn 2 2 -j * (4s+1+4δ) ≤ C.
It follows the existence of a constant C > 0 such that

χ 2 (P f1 , P f0 ) = dP f1 dP f0 2 dP f0 -1 ≤ C < ∞.
Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF] yields

inf e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ cκ 2 n = cn -4s/(2s+2δ+1/2) = cϕ * n . (15)
Putting ( 11) and ( 15) together, we prove the existence of a constant c > 0 such that inf

e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ cϕ * n ,
where ϕ * n = max n -1 , n -4s/(2s+2δ+1/2) . Theorem 2 is proved.

Proof of Theorem 3. We need the following lemma.

Lemma 4 If (A * g ) is satisfied, then there exists a constant C > 0 such that, for any integer j ≥ τ , η 2 j ≤ Ce (8π3 -1 a)2 bj .

Proof of Lemma 4. Using the inequalities sup l∈Dj |F (φ j,0

)(l)| 2 ≤ C2 -j , Card(D j ) ≤ C2 j , (A * g ) and the fact that sup l∈Dj |g l | -2 ≤ C sup l∈Dj e 2a|l| b ≤ Ce (8π3 -1 a)2 bj , we have η 2 j = l∈Dj |g l | -2 |F (φ j,0 )(l)| 2 ≤ C2 -j l∈Dj |g l | -2 ≤ C sup l∈Dj |g l | -2 ≤ Ce (8π3 -1 a)2 bj . The decomposition f 2 2 = 2 j * -1 k=0 α 2 j * ,k + ∞ j=j * 2 j -1 k=0 β 2 j,k
and the elementary inequality: (x + y) 2 ≤ 2(x 2 + y 2 ), (x, y) ∈ R 2 , we have

E Q * n -f 2 2 2 ≤ 2(E + F), (16)
where

E = E      2 j * -1 k=0 α 2 j * ,k -n -1 η 2 j * -α 2 j * ,k   2    , F =   ∞ j=j * 2 j -1 k=0 β 2 j,k   2 .
The upper bound for F. Since f ∈ B s 2,∞ (M ), we have

F =   ∞ j=j * 2 j -1 k=0 β 2 j,k   2 ≤ C   ∞ j=j * 2 -2js   2 ≤ C2 -4j * s ≤ C(log n) -4s/b . ( 17
)
The upper bound for E. It follows from the decomposition α j * ,k = α j * ,k + n -1/2 l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l and again: (x+y

) 2 ≤ 2(x 2 +y 2 ), (x, y) ∈ R 2 , that E ≤ 2 (G + H) , where G = n -2 E      2 j * -1 k=0   l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l 2 -η 2 j *     2    and H = 4n -1 E      2 j * -1 k=0 |α j * ,k | l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l   2    .
The upper bound for G. Using the Minkowski inequality, the fact that the random variables (e l ) l∈Z are independent N (0, 1), (A * g ) and Lemma 4, we obtain

G ≤ n -2     2 j * -1 k=0   E      l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l 2 -η 2 j *   2       1/2     2 = n -2    2 j * -1 k=0   V   l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l 2     1/2    2 ≤ n -2    2 j * -1 k=0   E   l∈Z g -1 l e 2iπlk/2 j * F (φ j * ,0 )(l)e l 4     1/2    2 ≤ Cn -2   2 j * -1 k=0 η 2 j *   2 = Cn -2 2 2j * η 4 j * ≤ Cn -2 2 2j * e (16π3 -1 a)2 bj * ≤ Cn -2 (log n) 2/b n = Cn -1 (log n) 2/b ≤ C(log n) -4s/b .
The upper bound for H. Using the Hölder inequality, the inequality

2 j * -1 k=0 α 2 j * ,k ≤ ||f || 2
2 ≤ M * , the fact that the random variables (e l ) l∈Z are independent N (0, 1), (A * g ) and Lemma 4, we obtain

H = 4n -1 E      2 j * -
Putting ( 16), ( 17) and ( 18) together, we obtain

E Q * n -f 2 2 2 ≤ 2(E + F) ≤ C(log n) -4s/b .
This ends the proof of Theorem 3.

Proof of Theorem 4. The proof is similar to the proof of Theorem 2 in the case s ∈ (0, δ + 4 -1 ]. We consider the two functions

f 0 (x) = 0, f 1 (x) = γ j * 2 j * k * =0 w k * ψ j * ,k * (x), x ∈ [0, 1], (19)
where j * is an integer to be chosen below, γ j * is a quantity to be chosen below, and w 0 , . . . , w 2 j * -1 are i. In what follows, we consider the function f 1 described by ( 19) with the integer j * defined by ( 20) and the quantity γ j * = M 1/2 * 2 -j * (s+1/2) . In order to apply Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF], we aim to bound the chi-square divergence χ 2 (P f1 , P f0 ) by a constant. It follows from ( 14 We deduce that χ 2 (P f1 , P f0 ) ≤ Theorem 2.12 (iii) of [START_REF] Tsybakov | Introduction à l'estimation nonparametrique[END_REF] yields

inf e Qn sup f ∈B s 2,∞ (M ) E Q n -f 2 2 2 ≥ cκ 2 n = c(log n) -4s/b .
Theorem 4 is proved.

  Cn -1 2 j * η 2 j * ≤ Cn -1 2 j * e (8π3 -1 a)2 bj * ≤ Cn -1 (log n) 1/b n 1/2 = Cn -1/2 (log n) 1/b ≤ C(log n) -4s/b .It follows from the obtained upper bounds for G and H thatE ≤ 2 (G + H) ≤ C(log n) -4s/b .

  i.d. Rademacher random variables. Clearly, f 0 belongs to B s 2,∞ (M ). If γ 2 j * = M * 2 -j * (2s+1) , then f 1 ∈ B s 2,∞ (M ). Let j * be an integer such that (8π3 -1 a) -1/b (2 log n -((4s + 1)/b) log(log n)) 1/b ≤ 2 j * < 2(8π3 -1 a) -1/b (2 log n -((4s + 1)/b) log(log n)) 1/b .(20)With the previous value of γ j * , the orthonormality of the wavelet basis gives 2 j * 2 j * ≥ C2 -2j * s ≥ C(log n) -2s/b = κ n .

  j * ,k ) (l)| 2 |g l | 2 ≤ Ce -(4π3 -1 a)2 bj * l∈Cj * |F (ψ j * ,k ) (l)| 2 = Ce -(4π3 -1 a)2 bj * 1 0 ψ 2 j * ,k (x)dx = Ce -(4π3 -1 a)2 bj * .

2 j * - 1 k=0exp

 1 Cn 2 γ 4 j * e -(8π3 -1 a)2 bj * = exp Cn 2 γ 4 j * 2 j * e -(8π3 -1 a)2 bj * .Thanks to the definitions of j * and γ j * , we haven 2 γ 4 j * 2 j * e -(8π3 -1 a)2 bj * ≤ Cn 2 2 -j * (4s+1) e -(8π3 -1 a)2 bj * ≤ C (log n) (4s+1)/b (2 log n -((4s + 1)/b) log(log n)) (4s+1)/b ≤ C.It follows the existence of a constant C > 0 such that χ 2 (P f1 , P f0 ) = dP f1 dP f0 2 dP f0 -1 ≤ C < ∞.