V Romero-Garcia

Nathalie Lamothe

Georgios Theocharis

O Richoux

Miquel Lluis

Garcia-Raffi

Lluis Miquel

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Stealth acoustic materials

V. Romero-García, 1, * N. Lamothe, 1 G. Theocharis, 1 O. Richoux, 1 and L. M. García-Raffi We report the experimental design of a 1D stealth acoustic material, namely a material that suppresses the acoustic scattering for a given set of incident wave vectors. The material consists of multiple scatterers, rigid diaphragms, located in an air-filled acoustic waveguide. The position of the scatterers has been chosen such that in the Born approximation a suppression of the scattering for a broad range of frequencies is achieved and thus a broadband transparency. Experimental results are found in excellent agreement with the theory despite the presence of losses and the finite size of the material, features that are not captured in the theory. This robustness as well as the generality of the results motivates realistic potential applications for the design of transparent materials in acoustics and other fields of wave physics.

I. INTRODUCTION

Wave scattering represents one of the most studied phenomena in wave physics. When a system with refractive index contrast between the obstacle and the surrounding material is radiated by an incident wave, the incident energy is both radiated and absorbed by the obstacle, forming a scattering pattern which strongly depends on the geometry and size of the obstacle as well as on the frequency-dependent properties of its constituents. The control of the scattering of waves represents a major topic of interest in acoustics 1,2 , photonics [START_REF] Shen | Anomalously weak scattering in metal-semiconductor multilayer hyperbolic metamaterials[END_REF] and electromagnetism [START_REF] Asadchy | Broadband reflectionless metasheets: Frequency-selective transmission and perfect absorption[END_REF] due to the vast possibilities in fundamental physics with potential applications. Complex media [START_REF] Martin | Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstcles[END_REF] and recently metamaterials [START_REF]Metamaterials: Physics and Engineering Explorations[END_REF] , have been shown as potential candidates to control the scattering of different types of waves. In particular it has been shown that metamaterials in both electromagnetism [START_REF] Alù | Achieving transparency with plasmonic and metamaterial coatings[END_REF] and acoustics 8 can be used to induce a dramatic drop in the scattering cross section of cylindrical and spherical objects, making them nearly invisible or transparent to an outside observer.

Ordered media, known as photonic [START_REF] Yablonovitch | Inhibited spontaneous emission in solidstate physics and electronics[END_REF][START_REF] John | Strong localization of photons in certain disordered dielectric superlattices[END_REF][START_REF] Joannopoulos | Photonic Crystals. Molding the Flow of Light[END_REF] or phononic crystals [START_REF] Sigalas | Elastic and acoustic wave band structure[END_REF][START_REF] Martínez-Sala | Sound attenuation by sculpture[END_REF][START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF] in optics and acoustics respectively, have been exploited due to their particular dispersion relation, presenting band gaps for different ranges of frequencies. In periodic structures, order is present on every scale, allowing for multiple Bragg scattering and localized states to form a band gap when the wavelength of the propagating wave is of the order of a multiple of the distance between scatterers. In the Born approximation, periodic structures are transparent at low frequencies [START_REF] Kittel | Introduction to Solid State Physics[END_REF] . Among other potential applications, these systems have motivated tunable frequency filters [START_REF] Sánchez-Pérez | Sound attenuation by a two-dimensional array of rigid cylinders[END_REF][START_REF] Romero-García | Tunable wideband bandstop acoustic filter based on two-dimensional multiphysical phenomena periodic systems[END_REF] , beam forming devices [START_REF] Pérez-Arjona | Theoretical prediction of the nondiffractive propagation of sonic waves through periodic acoustic media[END_REF] , waveguides [START_REF] Khelif | Guided elastic waves along a rod defect of a two-dimensional phononic crystal[END_REF] , wave traps [START_REF] Sigalas | Defect states of acoustic waves in a twodimensional lattice of solid cylinders[END_REF][START_REF] Khelif | Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[END_REF][START_REF] Romero-García | Evidences of evanescent bloch waves in phononic crystals[END_REF] and slow wave systems [START_REF] Baba | Slow light in photonic crystals[END_REF][START_REF] Olgun | Selfcollimated slow sound in sonic crystals[END_REF][START_REF] Theocharis | Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures[END_REF][START_REF] Groby | Use of slow sound to design perfect and broadband passive sound absorbing materials[END_REF] .

On the other hand, disordered media are also good alternatives to control wave scattering in a different manner [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF] . Waves entering in a disordered material are scattered several times before exiting in random direc-tions, producing interferences leading to interesting, and sometimes unexpected, physical phenomena [START_REF] Diederik | Disordered photonics[END_REF] . For example, in the acoustical [START_REF] Hu | Localization of ultrasound in a threedimensional elastic network[END_REF] or optical [START_REF] Sperling | Direct determination of the transition to localization of light in three dimensions[END_REF] analogy of Anderson localization, multiple scattering creates modes with a high level of spatial confinement in which the wave remains in the sample for a long time. Contrary to the periodic case, configurations of multiple scatterers with random and independent coordinates have no order at any scale and as a consequence no characteristic length appears in the system. Therefore no band gap is formed in such kind of structured materials and wave is randomly scattered.

Other approaches based on complex structures with correlated disorder [START_REF] Shen | Anomalously weak scattering in metal-semiconductor multilayer hyperbolic metamaterials[END_REF][START_REF] Fan | Constraints on collective density variables: one dimension[END_REF][START_REF] Kuhl | Experimental observation of the mobility edge in a waveguide with correlated disorder[END_REF][START_REF] Torquato | Random Heterogeneous Materials: Microstructure and macroscopic properties[END_REF][START_REF] Obioma | Constraints on collective density variables: Two dimensions[END_REF][START_REF] Kuhl | Enhancement of localization in one-dimensional random potentials with long-range correlations[END_REF][START_REF] Batten | Classical disordered ground states: Super-ideal gases and stealth and equi-luminous materials[END_REF][START_REF] Florescu | Designer disordered materials with large, complete photonic band gaps[END_REF][START_REF] Dietz | Transmission in waveguides with compositional and structural disorder: experimental effects of disorder crosscorrelations[END_REF][START_REF] Man | Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids[END_REF][START_REF] Man | Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast[END_REF][START_REF] Torquato | Hyperuniformity and its generalizations[END_REF][START_REF] Torquato | Ensemble theory for stealthy hyperuniform disordered ground states[END_REF][START_REF] Leseur | High-density hyperuniform materials can be transparent[END_REF][START_REF] Gkantzounis | Hyperuniform disordered phononic structures[END_REF] , created by introducing local correlations between positions of scatterers in an otherwise random ensemble, have been used to control the overall scattering strength of the material and create a strong wavelength dependence of the transmission. If the structure is properly designed, the system can even present isotropic band gaps [START_REF] Man | Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids[END_REF][START_REF] Man | Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast[END_REF] in the absence of periodicity. Such disordered materials are known as hyperuniform materials [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and macroscopic properties[END_REF][START_REF] Torquato | Ensemble theory for stealthy hyperuniform disordered ground states[END_REF] and nearly suppress all scattering at low frequencies. Hyperuniform materials are a subset of the systems known as stealth materials, referring to configurations of multiple scatterers that completely suppress scattering of incident radiation for a set of wave vectors, and thus, are transparent at these wavelengths. The hyperuniform materials represent the low-k limit of the stealth materials and are, counterintuitively, disordered and highly degenerate.

In this work we construct disordered stealth configurations by engineering the material properties in a controlled manner in such a way that the system prevents scattering only at prescribed wavelengths with no restrictions on any other wavelengths. In particular we focus on the suppression of the Bragg scattering of the corresponding periodic case and make the structured system transparent for a broadband range of frequencies. The inverse problem to generate the configuration of multiple scatterers with a specific scattering properties is non-trivial, as multiple solutions are conceivable. The approach is based on the optimization of the structure factor which, in the Born approximation, can be used as the outcome of a scattering experiment. The configuration of multiple scatterers is used to construct one dimensional (1D) stealth structures made of rigid diaphragms embedded in an air-filled waveguide. The system is analyzed by using the Transfer Matrix Method (TMM). Full wave numerical simulations have been done by using Finite Element Method (FEM). The system is analyzed considering the intrinsic viscothermal losses of the system. Experiments are performed to validate the theoretical results.

-3L/2 -L/2 L/2 3L/2 -3L/2 -L/2 L/2 3L/2 2 /L 0 -2 /L -2 /L 0 2 /L

II. SCATTERING BY MULTIPLE SCATTERERS

A. Structure factor

We are interested in the study of the scattering of waves by disordered materials made of a discrete distribution of point scatterers. In order to theoretically and numerically deal with such distribution of scatterers, the space is periodically divided using a lattice F which unit cell has a volume V and in which N scatterers are distributed at the positions r i (i = 1, . . . , N ) as schematically shown in a two dimensional example in Fig. 1(a). The lattice F has a reciprocal lattice F * in which the lattice sites are specified by the reciprocal lattice vector, G, with the property G R = 2πm for all lattice places R, where m = ±1, ±2, .... The reciprocal fundamental unit cell has a volume V * = (2π) d /V (where d is the dimension of the d-dimensional Euclidean space in which the problem is defined). Although this choice introduces anisotropy, the periodic boundary conditions do not affect the generality of the properties discussed in this work.

Let us consider the scattering of a beam with wavelength λ by the assembly of N scatterers. We assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (Born approximation), and only scattered waves of the first order are considered; absorption, refraction and higher order scattering can be neglected (kinematic diffraction). The direction of any scattered wave is defined by its scattering vector G = k s -k 0 , where k s and k 0 are the scattered and incident beam wavevectors, being θ incidence angle. For elastic scattering,

| k s | = | k 0 | = | k| = 2π/λ and then G = | G| = 4π
λ sin(θ). The amplitude and phase of this scattered wave will be the vector sum of the scattered waves from all the scatterers Ψ s ( q) = N i=1 f i e -ı G ri , with f i the atomic structure factor. The scattered intensity reads as

I( G) = Ψ s ( G).Ψ * s ( G) = N j=1 f j e -i G rj × N k=1 f k e i G r k = N j=1 N k=1 f j f k e -ı G( rj -r k ) . (1)
The structure factor, S( G), is defined as this intensity normalized by 1/

N j=1 f 2 j S( G) = 1 N j=1 f 2 j N j=1 N k=1 f j f k e -ı G( rj -r k ) . (2)
If all the scatterers are identical, then

I( G) = f 2 N j=1 N k=1 e -ı G( rj -r k ) , (3)
so

S( G) = 1 N N j=1 N k=1 e -ı G( rj -r k ) = 1 N N j=1 e ı G rj 2 . (4)
Therefore the structure factor S( G), is proportional to the intensity of scattering of incident radiation from a configuration of N scatterers. It is worth noting here that the structure factor can be also related with the scattering cross section as follows

dσ dΩ = f 2 N j=1 N k=1 e -ı G( rj -r k ) = f 2 N S( G), (5)
where σ is the total cross section and Ω is the solid angle. Notice that the Laue condition [START_REF] Kittel | Introduction to Solid State Physics[END_REF][START_REF] David Mermin | Solid State Physics[END_REF] implies that the constructive interferences will occur provided that the change in wave vector between the incident and reflected waves was a vector of the reciprocal lattice. Therefore the Bragg scattering condition is | k| = | G| 2 sin θ . In the case 1D (θ = π/2), the wavevectors are collinear and then,

| k| = | G|/2.

B. Design of the discrete materials with prescribed scattering properties

The optimization procedure of prescribing the scattering characteristics by the structure factor and constructing configurations of multiple scatterers that give rise to these targeted value. In contrast to the real-space methods, we target information in reciprocal space to construct configurations whose structure factor exactly matches the candidate structure factor for a set of wavelengths.

The objective function is based on the structure factor S( G). For a given limit of wave vectors, fixing the optimization domain Ω in the reciprocal space, the structure factor should have a target value, S op , for all the wave vectors in Ω. Then, the objective functions will be based on the following expression

φ( r 1 , . . . , r N ) = G∈Ω S( G) -S op , (6)
and a standard deviation function

σ = 1 N -1 G∈Ω |S( G) -µ| 2 , (7)
where µ = 1 N G∈Ω S( G) is the mean value in the target range of wave vectors. The optimization algorithm will look for configurations of multiple scatterers r i that simultaneously minimizes the potential and the standard deviation function.

C. Different kind of materials based on S( G).

Examples in 1D

The interest of this work is focused on stealth materials, which are built from a configuration of multiple scatterers whose structure factor is exactly zero for some set of wavelengths. In the low-k limit, several disordered systems, known as hyperuniform [START_REF] Torquato | Random Heterogeneous Materials: Microstructure and macroscopic properties[END_REF][START_REF] Florescu | Designer disordered materials with large, complete photonic band gaps[END_REF][START_REF] Man | Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids[END_REF][START_REF] Man | Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast[END_REF][START_REF] Torquato | Hyperuniformity and its generalizations[END_REF][START_REF] Torquato | Ensemble theory for stealthy hyperuniform disordered ground states[END_REF][START_REF] Gkantzounis | Hyperuniform disordered phononic structures[END_REF][START_REF] Torquato | Local density fluctuations hyperuniform systems and order metrics[END_REF] materials, nearly suppress all scattering. These systems have the property that lim k→0 S( G) = 0, i.e., infinitewavelength density fluctuations vanish. Following this definition, we can see that periodic , i.e. crystalline configurations, are by definition, in the Born approximation, hyperuniform since they suppress scattering for all wavelengths except those associated with Bragg scattering. It is worth noting here that other kind of materials can be defined by using the structure factor. For example, equiluminous materials scatter waves with the same intensity for a set of wave vectors. The structure factor for this class of materials is simply a constant for a set of wave vectors. Subsets of equiluminous materials include super-ideal gases S( G) = 1 as well as the stealth materials S( G) = 0. Super-ideal gases are single configurations of multiple scatterers whose scattering exactly matches that of an ensemble of ideal gas configurations, or Poisson point distributions, for a set of wave vectors, being S( G) = cte.

Figure 2 shows the 1D structure factor for several configurations made of N = 30 point scatterers in a unit cell of length L. The first analysis is done for the periodic case in which the system has a periodicity of L/N (see Fig. 2(a)). In this case, the structure factor can be obtained analytically. If the Bragg condition is fulfilled the structure factor takes the value N for k B = N π/L. For wavelengths out of the Bragg condition, the structure factor is zero. This means that a periodic distribution of scatterers suppress scattering for all wavelengths except those associated with Bragg scattering, corresponding to a structure factor with a form of Dirac comb as shown in Fig. 2(a). As soon as the structure becomes fully random, the periodic behavior of the structure factor is broken and random scattering is produced (see Fig. 2(b)). The structure factor for a hyperuniform distribution of scatterers is shown in Fig. 2(c). The range of frequencies in which scattering is suppressed translates with the cancellation of the structure factor. As it can be seen, the suppression of the scattering is produced around k = 0. Moreover, the structure factor peaks at wave vectors corresponding to the characteristics length scales of the periodic distribution, among them the first Bragg peak meaning that the system presents some hints of periodicity. This result is in agreement with those obtained in recent works showing that hyperuniform materials present isotropic band gaps. [START_REF] Florescu | Designer disordered materials with large, complete photonic band gaps[END_REF][START_REF] Man | Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids[END_REF][START_REF] Man | Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast[END_REF] . Figure 2(d) shows the results for a stealth material optimized in the range k B /5 ≤ |k| ≤ k B /1.5. Effectively, the structure factor for such configuration of multiple scatterers is exactly zero for the prescribed set of wavelengths. This means that this structure completely suppress scattering of incident radiation for this set of wave vectors, and thus, is transparent at these wavelengths.

III. STEALTH MATERIALS FOR ACOUSTIC TRANSPARENCY A. Stealth distribution of point scatterers

In this Section, we design a 1D stealth configuration made of N = 25 scatterers embedded in a fluid medium of length L = 1 m for making the system transparent in a broad range of frequencies around the corresponding Bragg frequency of the equivalent periodic distribution of scatterers with periodicity L/N . The range of wave numbers to be optimized is 0.92k B < k < 1.06k B . In particular, if we consider air as the fluid medium of the waveguide (with c = 340 m/s the speed of sound), the range of frequencies to be optimized corresponds to the domain f = [3910, 4504] Hz being f B = cN/L = 4250 Hz the Bragg frequency.

The resulting configuration of multiple scatterers obtained from our optimization procedure is shown in the upper panel of Fig. 3 with blue dots. The structure factor of the periodic case presents a peak at the Bragg frequency of the system, as shown in Fig. 3. The optimized stealth structure has a minimum of the structure factor in the optimization range of frequencies (yellow region).

B. Physical system: 1D waveguide with diaphragms

To implement the stealth material we consider the propagation of sound waves within an air-filled cylindrical tube (with density and sound velocity of air defined as ρ and c respectively) of section S = πR 2 with diaphragms of length d n and section s n = πr 2 n placed along the waveguide at positions x n , as shown in Fig. 4. The tube and the diaphragms are made of aluminium that can be considered acoustically rigid for the purpose of this work. The target frequency range is well below the first cutoff frequency of higher propagation modes in the waveguide, therefore the problem can be considered as one-dimensional. Moreover the size of the diaphragms can be considered small enough to be in the Born approximation.

Theoretical model: Transfer Matrix Method

The sound waves that propagate in the proposed acoustic structure are subject to the viscothermal effects on the walls which are taken into account by considering a complex expression for both the impedance and the wave number. In our case, we used the model of losses from Ref. [25], namely we replace the wave number and the characteristic impedance by the following expressions

k q = ω c 1 + (1 + ı) √ 2R q /δ (1 + (γ -1)) √ P r , (8)
Z q = ρc S q 1 + (1 + ı) √ 2R q /δ (1 -(γ -1)) √ P r , (9)
where δ = 2µ/ρω is the viscous boundary layer thickness, µ = 1.839 10 -5 kg m -1 s 1 being the viscosity of air, P r = 0.71 the Prandtl number and γ = 1.4 the heat capacity ratio of air. Notice that these expressions correspond for the wavevector and impedance of a cylindrical tube of radius R q and section S q . The transfer matrix between the faces of the system, T, extending from x = 0 to thickness x = L, presenting N diaphragms placed at position x n , relates the sound pressure, p, and normal acoustic flux, v, between its two faces by using

p v x=0 = T p v x=L , (10)
where

T = T 11 T 12 T 21 T 22 = N i=1 T w i T d i , (11)
with

T w i = cos(k w w i ) -ıZ w sin(k w w i ) -ı Zw sin(k w w i ) cos(k w w i ) (12)
and

T d i = cos(k d d i ) -ıZ d sin(k d d i ) -ı Z d sin(k d d i ) cos(k d d i ) (13)
describing the propagation through a waveguide of length w i and a diaphragm of thickness d i .

The reflection and transmission coefficients can be directly calculated from the elements of the matrix given in Eq. ( 11) as 47

T = 2e ıkL T 11 + T 12 /Z w + Z w T 21 + T 22 , (14)
R -= T 11 + T 12 /Z w -Z w T 21 -T 22 T 11 + T 12 /Z w + Z w T 21 + T 22 , (15)
R + = -T 11 + T 12 /Z w -Z w T 21 + T 22 T 11 + T 12 /Z w + Z w T 21 + T 22 , (16)
where the superscripts (+, -) denote the incidence direction, i.e., the positive and negative x-axis incidence respectively. We notice that, if the system was symmetric, then, T 11 = T 22 , and as a consequence, R + = R -. The reciprocal behaviour of the system can be seen from the fact that its determinant is one (T 11 T 22 -T 12 T 21 = 1).

Experimental set-up

The experimental set-up was made of a cylindrical waveguide with radius R = 1.25 cm, which cut-off frequency is around 8000 Hz well above the analyzed range of frequencies. Two types of pieces were built: the connections [Fig. 4(c)] and the separators [Fig. 4(d)]. Two types of connections were built, the ones with no diaphragm, in order to measure the properties of an empty waveguide, and the ones with a diaphragm, in order to obtain the scattering properties of the stealth material. The dimensions of the diaphragms were d n = d = 2 mm and r n = 1.04 cm, dimensions for which the Born approximation is fulfilled. The separators had different lengths in order to fit all the positions x n in the stealth material. A picture of the stealth material once the different pieces are properly assembled is shown in Fig. 4

(e).

A loudspeaker is used to generate a plane wave field, and a single microphone was used to measure the transfer functions between the signal provided to the loudspeaker and the sound pressure at four locations in order to experimentally obtain the reflection and transmission coefficients of the stealth material (the 1 microphone technique was used, see Ref. [47]). On the other side of the tube, an anechoic termination with less than 5% of reflection in the analyzed frequency range is used.

The stealth material is experimentally analyzed in the forward and in the backward incidence directions. To do that, we have to reverse the source and the anechoic termination in order to experimentally obtain all the scattering parameters, i.e., R + , R -and T .

IV. RESULTS

In this Section, we analyze and present the theoretical results obtained from the TMM and the experimental results for the reflection, transmission and absorption coefficients of the designed stealth material. In addition to these results, we have used a numerical approach based on the Finite Element Method (FEM) to perform full wave numerical simulations in order to check that the possible evanescent coupling between scatterers is weak and can be neglected. In the numerical model the viscothermal losses were accounted for using the effective parameters defined above for each domain. An incident plane wave is used as excitation of the system and perfectly matched layers were placed at the boundaries of the numerical domain to avoid unexpected reflections. The mesh is designed such as to ensure a maximum element size of λ B /20 (being λ B = k -1 B = L/N/π). We start by the analysis of the lossless case. In Fig. 5(a) we represent the scattering coefficients of the periodic distribution of scatterers. The Bragg scattering is shown in the target range of frequencies with an strong reduction of the transmission and an enhancement of the reflection, representing the band gap of the structure. The fact that we are dealing with finite structures explains the lobes of the reflection and transmission coefficients out of the band gap that corresponds to Fabry-Perot resonances.

Figure 5(b) shows the scattering coefficients of the designed stealth material in the lossless case. The reflection coefficient is dramatically reduced and the transmission is almost one for the target range of frequencies. Therefore, the stealth material can be considered as transparent, even if N scatterers are placed. For the frequencies out of the optimization range the system is not transparent. We notice that in this lossless case, R + =R -. The differences between the numerical and the analytical results out of the target frequencies shows that the evanescent coupling between the scatterers are not negligible. However in the optimization range, where the scattering is suppressed, the coupling is weak and the transparency of the system is shown by the numerics and analytics. Therefore, the optimization procedure is robust with respect to the evanescent coupling.

We analyze now the lossy case in order to show their effects on the scattering properties of the stealth material. Numerical, theoretical and experimental results are compared in Figs. 5(c) and 5(d). In this case, as the stealth material is not symmetric, R + = R -, but being reciprocal T + = T -= T . We start by analyzing the results for the system excited from the right side (forward direction, Fig. 5(c)). In the optimized frequency band, the values of the reflection coefficient are nearly zero while the transmission coefficient has been reduced with respect to the lossless case. However the coefficients keep the flat behavior also shown in the lossless case. In order to check the transparency of the stealth material, we compare its transmission coefficients with the corresponding empty waveguide. In the optimized range of frequencies the transmission of the stealth material matches the transmission of the empty tube in very good agreement between numerics, analytics and experiments. As in the lossless case, out of the optimization range, the system is not transparent.

One of the properties of the structure factor is that S(-G) = S( G), i.e. the scattered intensity should be the same in opposite directions even if the configuration of multiple scatterers conforming the stealth material is not symmetric, i.e., the reflection coefficient from both sides of the structure should be equal. Figure 5(d) analyzes the same coefficients as Fig. 5(c) but evaluated for the backward incidence direction, having a similar behavior as the opposite direction. Figure 5(e) shows that the difference between the amplitudes of the reflection coefficients evaluated from each side of the stealth material is less than 3% in the whole frequency range. The slightly differences are due to the fact that the analyzed system is finite and the optimization procedure is based on an infinite one. Out of the target frequency range, the differences are bigger because the scattering is stronger than in the optimization range due to finite size effects of the system. However, the differences between the two sizes remain small than 3%.

Finally, we want to point out that the reflection coefficient of all the analyzed systems in Figs. 5(b-d) follows the same trends as the theoretical value of the structure factor, see Fig. 2. This proves that the size of the diaphragms allows the system to work in the Born approximation.

V. CONCLUSIONS

By capturing a configuration of multiple scatterers in the direct space and Fourier transforming it to the reciprocal space, the structure factor has been used to design configurations of multiple scatterers with prescribed scattering characteristics in the Born approximation. The methodology proposed in this work allows us to engineer different kinds of materials: hyperuniform, steath, equiluminous and super-ideal materials. We have paid attention to the case of stealth materials, made from a configuration of multiple scatterers which structure factor is exactly zero for a target set of wavelengths. There is a main property of such materials: the broadband transparency in the target range of frequencies. The physical system used to check the theoretical predictions consists of an air-filled 1D waveguide with diaphragms. The positions of the centers of the diaphragms, which are small enough to fulfill the Born approximation, correspond to a stealth point distribution to avoid Bragg scattering and to achieve broadband transparency around this frequency. The system is analyzed analytically, numerically and experimentally along the two incidence directions showing very good agreement and reproducing the prescribed scattering properties of the system. Moreover and interestingly enough, we have proven that the optimization procedure is robust enough to be independent of the presence of losses. The optimization is also very robust to the finite size effects, showing that the property S( G) = S(-G) is fulfilled with differences less than 3% between the reflection coefficients evaluated from each side of the structure in the whole range of frequencies. These preliminary results could be the basis for future applications in acoustics and more generally in wave physics. It is worth noting here that the methodology is general and can be applied for any type of wave. Extension to higher dimensions could provide very promising applications for the broadband control of waves.

FIG. 1 .

 1 FIG. 1. (Color online) Schematic example for the case of square periodicity in 2D with a lattice period L. (a) Direct space. (b) Reciprocal space.

5 FIG. 2 .

 52 FIG. 2. (Color online) Representation of the structure factor for (a) a periodic, (b) a random, (c) a hyperuniform and (d) a stealth distributions of N = 30 scatterers. The hyperuniform configuration is obtained to reduce scattering in the range |k| ≤ |kB/2.5| (where kB = πN/L is the Bragg condition of the corresponding periodic configuration). The stealth configuration is optimized for kB/5 ≤ |k| ≤ kB/1.5. The target range of frequencies of the two last cases are shown by the in (c) and (d).

FIG. 4 .

 4 FIG. 4. (Color online) Schematics of both a three dimensional view (a) and the cross section (b) of the cylindrical waveguide along the n-th diaphragm. (c-e) Experimental set-up. (c) Connections: left diaphragme and flat connection for empty tube. (d) Separators. (e) Picture of the stealth material.

2 FIG. 5 .

 25 FIG. 5. (Color online) Analysis of the scattering produced by the stealth material. In the Figure the transmission, T 2 , [reflection, R 2 ,] analytically and numerically calculated are represented by the shaded pink ( ) and red dashed (--) lines respectively [shaded blue ( ) and blue continuous (-) lines respectively] and the experimental values are represented by red squares ( ) [blue triangles ( )]. (a) Scattering coefficients of (a) the periodic distribution in the lossless case, (b) the stealth material in the lossless case, (c) and (d) the stealth material in the lossy case excited from the forward and backward incidence directions respectively. Black (-) continuous line and black circles ( ) represent the transmission coefficient of the empty waveguide, theoretical and experimental values respectively. (e) |R + | -|R -| from the TMM ( ) and experimental results ( ).

2 1

 2 Laboratoire d'Acoustique de l'Université du Mans, LAUM -UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France 2 Universitat Politècnica de València, Camíŋ de vera s/n, 46022, València, Spain (Dated: November 15, 2019)

C. Shen, J. Xu, N. X. Fang, and Y. Jing, "Anisotropic complementary acoustic metamaterial for canceling out aberrating layers," Phys. Rev. X 4, 041033 (2014).

ACKNOWLEDGMENTS

This work has been funded by RFI Le Mans Acoustique (Région Pays de la Loire) in the framework of the APAMAS project, by the project HYPERMETA funded under the program Étoiles Montantes of the Région Pays de la Loire as well as by the Ministerio de Economía y Competitividad (Spain) and European Union FEDER through project FIS2015-65998-C2-2-P. V. Romero-García and L. M. García-Raffi acknowledge the short term scientific mission (STSM) funded by the COST (European Cooperation in Science and Technology) Action DENORMS -CA15125.


Ouerdia Arkoun
email: ouerdia.arkoun@gmail.com

Jean-Yves Brua
email: jean-yves.brua@univ-rouen.fr

Serguei Pergamenshchikov
email: serge.pergamenshchikov@univ-rouen.fr

Adaptive efficient robust estimation for nonparametric autoregressive models *

Keywords: primary 62G08, secondary 62G05 Non-parametric estimation, Non parametric autoregresion, Nonasymptotic estimation, Robust risk, Model selection, Sharp oracle inequalities

 for non parametric autoregressions an adaptive efficient model selection procedure is proposed, i.e. such for which the upper bound of its robust quadratic risk coincides with the obtained Pinsker constant.

1 Introduction

Model

In this paper we consider the nonparametric autoregression model defined as

y k = S(x k )y k-1 + ξ k and x k = a + k(b -a) n , (1.1)
where S(•) ∈ L 2 [a, b] is unknown function, a < b are fixed known constants, 1 ≤ k ≤ n, the initial value y 0 is a constant and the noise (ξ k ) k≥1 is i.i.d. sequence of unobservable random variables with Eξ 1 = 0 and Eξ 2 1 = 1. In the sequel we denote by p the distribution density of the random variable ξ 1 .

The problem is to estimate the function S(•) on the basis of the observations (y k ) 1≤k≤n under the condition that the noise distribution p is unknown and belongs to some noise distributions class P. There is a number of papers which consider these models such as [START_REF] Dahlhaus | Maximum Likelihood Estimation and Model Selection for Locally Stationary Processes[END_REF], [START_REF] Dahlhaus | On the Kullback-Leibler Information Divergence of Locally Stationary Processes[END_REF] and [START_REF] Belitser | Recursive Estimation of a Drifted Autoregressive Parameter[END_REF]. In all these papers, the authors propose some asymptotic (as n → ∞) methods for different identification studies without considering optimal estimation issues. Firstly, minimax estimation problems for the model (1.1) has been treated in [START_REF] Arkoun | Nonparametric Estimation for an Autoregressive Model[END_REF] and [START_REF] Moulines | On Recursive Estimation for Time Varying Autoregressive Processes[END_REF] in the nonadaptive case, i.e. for the known regularity of the function S. Then, in [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF] and [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF] it is proposed to use the sequential analysis method for the adaptive pointwise estimation problem in the case when the Hölder regularity is unknown.

Main contributions

In this paper we consider the adaptive estimation problem for the quadratic risk defined as

R p ( S n , S) = E p,S S n -S 2 , S 2 = b a S 2 (x)dx , (1.2)
where S n is an estimator of S based on observations (y k ) 1≤k≤n and E p,S is the expectation with respect to the distribution law P p,S of the process (y k ) 1≤k≤n given the distribution density p and the coefficient S. Moreover, taking into account that the distribution p is unknown, we use the robust nonparametric estimation approach proposed in [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF]. To this end we set the robust risk as

R * ( S n , S) = sup p∈P R p ( S n , S) , (1.3)
where P is a family of the distributions defined in Section 2.

To estimate the function S in model (1.1) we make use of the model selection procedures proposed in [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF] based on the family of the optimal poinwise truncated sequential estimators from [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF] for which using the model selection method developed in [START_REF] Galtchouk | Sharp nonasymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF] a sharp oracle inequality is shown. In this paper, using this inequality we show that the model selection procedure is efficient in adaptive setting for the robust quadratic risks (1.3). To this end, first of all we have to study the sharp lower bound for the these risks, i.e. we have to study the best potential accuracy estimation for the model (1.1) which is called the Pinsker constant. For this we use the approach proposed in [START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF] - [START_REF] Galtchouk | Adaptive Sequential Estimation for Ergodic Diffusion Processes in Quadratic Metric[END_REF] which is based the Van-Trees inequality. It turns out that for the model (1.1) the Pinsker constant has the same form as for the filtration signal problem in the "signal -white noise" model studied in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian noise[END_REF] but with new coefficient which equals to the optimal variance given by the Hajek -Le Cam inequality for the parametric model (1.1). This is the new result in the efficient non parametric estimation theory for the statistical models with dependent observations. Then, using the oracle inequality from [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF] and the weight least square estimation method we show that for the model selection procedure with the Pinsker weight coefficients the upper bound asymptotically coincides with the obtained Pinsker constant without using the regularity properties of the unknown functions, i.e. it is efficient in adaptive setting with respect to the robust risks (1.3).

Plan of the paper

The paper is organized as follows. In Section 2 we give all conditions and construct the sequential point-wise estimation procedures. to pass from the auto-regression model to the corresponding regression model. In Section 3 we construct the model selection procedure based on the sequential estimators from Section 2. In Section 4 we announce the main results. In Section 5 we show the Van -Trees inequality for the model (1.1). In section 6 we obtain the lower bound for the robust risks. In Section 7 we obtain the upper bounds for the robust risks. In Appendix A we give the all auxiliary and technic tools.

Sequential procedures.

As in [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF] we assume that in the model (1.1) the i.i.d. random variables (ξ k ) k≥1 have a density p (with respect to the Lebesgue measure) from the functional class P defined as

P := p ≥ 0 : +∞ -∞ p(x) dx = 1 , +∞ -∞ x p(x) dx = 0 , +∞ -∞ x 2 p(x) dx = 1 and sup k≥1 +∞ -∞ |x| 2k p(x) dx ς k (2k -1)!! ≤ 1 , (2.1)
where ς ≥ 1 is some fixed parameter, which may be a function of the number observation n, i.e. ς = ς(n), such that for any b > 0

lim n→∞ ς(n) n b = 0 . (2.2)
Note that the (0, 1)-Gaussian density belongs to P. In the sequel we denote this density by p 0 . It is clear that for any q > 0

m * q = sup p∈P E p |ξ 1 | q < ∞ , (2.3)
where E p is the expectation with respect to the density p from P. To obtain the stable (uniformly with respect to the function S ) model (1.1), we assume that for some fixed 0 < < 1 and L > 0 the unknown function S belongs to the ε -stability set introduced in [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF] as

Θ ,L = S ∈ C 1 ([a, b], R) : |S| * ≤ 1 - and | Ṡ| * ≤ L , (2.4)
where

C 1 [a, b] is the Banach space of continuously differentiable [a, b] → R functions and |S| * = sup a≤x≤b |S(x)|.
We will use as a basic procedures the point wise procedure from [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF] at the points (z l ) 1≤l≤d defined as

z l = a + l d (b -a) , (2.5)
where d is an integer value function of n, i.e.

d = d n , such that lim n→∞ d n √ n = 1 . (2.6)
So we propose to use the first ι l observations for the auxiliary estimation of S(z l ). We set

S l = 1 A ι l ι l j=1 Q l,j y j-1 y j , A ι l = ι l j=1 Q l,j y 2 j-1 , (2.7)
where Q l,j = Q(u l,j ) and the kernel

Q(•) is the indicator function of the interval [-1; 1], i.e. Q(u) = 1 [-1,1] (u).
The points (u l,j ) are defined as

u l,j = x j -z l h . (2.8)
Note that to estimate S(z l ) on the basis of the kernel estimate with the kernel Q we use only the observations (y j ) k 1,l ≤j≤k 2,l from the h -neighbor of the point z l , i.e.

k 1,l = [n z l -n h] + 1 and k 2,l = [n z l + n h] ∧ n , (2.9)
where z l = (z l -a)/(b -a) and h = h/(b -a). Note that, only for the last point z d = b the k 2,d = n. We chose ι l in (2.7) as

ι l = k 1,l + q and q = q n = [(n h) µ 0 ] (2.10)
for some 0 < µ 0 < 1. In the sequel for any 0 ≤ k < m ≤ n we set

A k,m = m j=k+1 Q l,j y 2 j-1 and A m = A 0,m . (2.11)
Next, similarly to [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF], we use a some kernel sequential procedure based on the observations (y j ) ι l ≤j≤n . To transform the kernel estimator in the linear function of observations and we replace the number of observations n by the following stopping time

τ l = inf{ι l + 1 ≤ k ≤ k 2,l : A ι l ,k ≥ H l } , (2.12)
where inf{∅} = k 2,l and the positive threshold H l will be chosen as a positive random variable measurable with respect to the σ -field {y 1 , . . . , y ι l }. Now we define the sequential estimator as

S * l = 1 H l   τ l -1 j=ι l +1 Q l,j y j-1 y j + κ l Q(u l,τ l ) y τ l -1 y τ l   1 Γ l , (2.13)
where Γ l = {A ι l ,k 2,l -1 ≥ H l } and the correcting coefficient 0 < κ l ≤ 1 on this set is defined as

A ι l ,τ l -1 + κ 2 l Q(u l,τ l )y 2 τ l -1 = H l . (2.14)
Note that, to obtain the efficient kernel estimate of S(z l ) we need to use the all k 2,l -ι l -1 observations. Similarly to [START_REF] Konev | Estimate of the Number of Observations in Sequential Identification of the Parameters of Dynamical Systems[END_REF], one can show that τ l ≈ γ l H l as H → ∞, where γ l = 1 -S 2 (z l ) .

(2.15)

Therefore, one needs to chose H as (k 2,l -ι l -1)/γ l . Taking into account that the coefficients γ l are unknown we define the threshold H l as

H l = 1 - γ l (k 2,l -ι l -1) and = 1 2 + ln n , (2.16)
where

γ l = 1 -S 2 ι l
and S ι l is the projection of the estimator S ι l in the interval ] -1 + , 1 -[, i.e.

S ι l = min(max( S ι l , -1 + ), 1 -) .

(2.17)

To obtain the uncorrelated stochastic terms in the kernel estimators for S(z l ) we chose the bandwidth h as

h = b -a 2d . (2.18)
As to the estimator S ι l , we can show the following property.

where 0 = 0 (n) → 0 as n → ∞ such that lim n→∞ n δ 0 = ∞ for any δ > 0. Now we set Y l = S * H,h (z l )1 Γ and Γ = ∩ d l=1 Γ l . ( 2
Y l = S(z l ) + ζ l and ζ l = ξ * l + l , (2.22)
in which

ξ * l = τ l -1 j=ι l +1 Q l,j y j-1 ξ j + κ l Q(u l,τ l ) y τ l -1 ξ τ l H l (2.23)
and l = 1,l + 2,l , where

1,l = τ l -1 j=ι l +1 Q l,j y 2 j-1 ∆ l,j + κ 2 l Q(u l,τ l ) y 2 τ l -1 ∆ l,τ l H l , ∆ l,j = S(x j ) -S(z l )
and

2,l = (κ l -κ 2 l ) Q(u l,τ l ) y 2 τ l -1 S(x τ l ) H l .
Note that in the model sec:In.1-11-1R the random variables (ξ * j ) 1≤j≤d are defined only on the set Γ. By the technical reasons we need the definitions for these variables on the set Γ c was well. To this end for any j ≥ 1 we set Ql,j = Q l,j y j-1 1 {j<k 2,l } + H l Q l,j 1 {j=k 2,l } (2.24)

and Ǎι l ,m = m j=ι l +1 Q2 l,j . Note, that for any j ≥ 1 and l = m Ql,j Qm,j = 0 .

(2.25)

and Ǎι l ,k 2,l ≥ H l . So we can modify now stopping time (2.12) as τl = inf{k ≥ ι l + 1 : Ǎι l ,k ≥ H l } .

(2.26)

Obviously, τl ≤ k 2,l and τl = τ l on the set Γ for any 1 ≤ l ≤ d. Now similarly to (2.14) we define the correction coefficient as

Ǎι l ,τ l -1 + κ2 l Q2 l,τ l = H l . (2.27)
It is clear that 0 < κl ≤ 1 and κl = κ l on the set Γ for 1 ≤ l ≤ d. Using this coefficient we set

η l = τl -1 j=ι l +1 Ql,j ξ j + κl Ql,τ l ξ τl H l .
(2.28) Note that on the set Γ for any 1 ≤ l ≤ d the random variables η l = ξ * l . Moreover (see Lemma A.2 in [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF]), for any 1 ≤ l ≤ d and p ∈ P

E p,S (η l |G l ) = 0 , E p,S η 2 l |G l = σ 2 l and E p,S η 4 l |G l ≤ mσ 4 l , (2.29)
where

σ l = H -1/2 l , G l = σ{η 1 , . . . , η l-1 , σ l } and m = 4(144/ √ 3) 4 m * 4 . It is clear that σ 0, * ≤ min 1≤l≤d σ 2 l ≤ max 1≤l≤d σ 2 l ≤ σ 1, * , (2.30)
where

σ 0, * = 1 -2 2(1 -)nh and σ 1, * = 1 (1 -)(2nh -q -3) . Now, taking into account that | 1,l | ≤ Lh, for any S ∈ Θ ,L we obtain that sup S∈Θ ,L E p,S 1 Γ 2 l ≤ L 2 h 2 + υn (nh) 2 , (2.31)
where υn = sup p∈P sup S∈Θ ,L E p,S max 1≤j≤n y 4 j . The behavior of this coefficient is studied in the following Proposition. when n → ∞. We will use this in the oracle inequalities below.

Remark 2.2. Note, that to estimate the function S in (1.1) we use the approach developed in [START_REF] Galtchouk | Adaptive Sequential Estimation for Ergodic Diffusion Processes in Quadratic Metric[END_REF] for the diffusion processes. To this end we use the efficient sequential kernel procedures developed in [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF][START_REF] Arkoun | Nonparametric Estimation for an Autoregressive Model[END_REF][START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF]. It should be emphasized that to obtain an efficient estimator, i.e. an estimator with the minimal asymptotic risk, one needs to take only indicator kernel as in (2.13).

Remark 2.3. Ii should be noted also that the sequential estimator (2.13) has the same form as in [START_REF] Arkoun | Sequential Robust Estimation for Nonparametric Autoregressive Models[END_REF], but except the last term, in which the correction coefficient is replaced by the square root of the coefficient used in [START_REF] Konev | On One Property of Martingales with Conditionally Gaussian Increments and Its Application in the Theory of Nonasymptotic Inference[END_REF]. We modify this procedure to calculate the variance of the stochastic term (2.23).

Model selection

In this section we consider the nonparametric estimation problem in the non asymptotic setting for the regression model sec:In.1-11-1R for some set Γ ⊆ Ω. The design points (z l ) 1≤l≤d are defined in (2.5). The function S(•) is unknown and has to be estimated from observations Y 1 , . . . , Y d . Moreover, we assume that the unobserved random variables (η l ) 1≤l≤d satisfy the properties (2.29) with some nonrandom constant m > 1 and the known random positive coefficients (σ l ) 1≤l≤d satisfy the inequlity (2.30) for some nonrandom positive constants σ 0, * and σ 1, * Concerning the random sequence = ( l ) 1≤l≤d we suppose that

E p,S 1 Γ 2 d < ∞ . (3.1)
The performance of any estimator S will be measured by the empirical squared error

S -S 2 d = ( S -S, S -S) d = b -a d d l=1 ( S(z l ) -S(z l )) 2 . (3.2)
Now we fix a basis (φ j ) 1≤j≤d which is orthonormal for the empirical inner product:

(φ i , φ j ) d = b -a d d l=1 φ i (z l )φ j (z l ) = 1 {i=j} . (3.3)
For example, we can take the trigonometric basis (φ j ) j≥ 1 in L 2 [a, b], i.e.

φ 1 = 1 √ b -a , φ j (x) = 2 b -a Tr j (2π[j/2]l 0 (x)) , j ≥ 2 , (3.4)
where the function Tr j (x) = cos(x) for even j and Tr j (x) = sin(x) for odd j, [x] denotes integer part of x. and l 0 (x) = (x -a)/(b -a). Note that, in this case to obtain the property (3.3) the numbers of points d must be odd.

To obtain the property (2.6) we can choose, for example,

d = 2[ √ n/2] + 1, where [a] is the integer part of a ∈ R.
Note that, using the orthonormality property (3.3) we can represent for any 1 ≤ l ≤ d the function S as

S(z l ) = d j=1 θ j,d φ j (z l ) and θ j,d = S, φ j d . (3.5)
So, to estimate the function S we have to estimate the Fourrier coefficients (θ j,d ) 1≤j≤d . To this end we reply the the function S by the observations, i.e.

θ j,d = b -a d d l=1 Y l φ j (z l ) . (3.6)
From sec:In.1-11-1R we obtain immediately the following regression shceme

θ j,d = θ j,d + ζ j,d with ζ j,d = b -a d η j,d + j,d , (3.7)
where

η j,d = b -a d d l=1 η l φ j (z l ) and j,d = b -a d d l=1 l φ j (z l ) .
Note that the upper bound (2.30) and the Bounyakovskii-Cauchy-Schwarz inequality imply that

| j,d | ≤ d φ j d = d .

Now we set

B n = n 2 d . (3.8)
Note here that Proposition 3.3 from [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF] implies directly that for any b > 0

lim n→∞ 1 n b sup p∈P sup S∈Θ ε,L E p,S B n 1 Γ = 0 . (3.9)
We estimate the function S on the sieve (2.5) by the weighted least squares estimator

S λ (z l ) = d j=1 λ(j) θ j,d φ j (z l ) 1 Γ , 1 ≤ l ≤ d , (3.10)
where the weight vector λ = (λ(1), . . . , λ(d)) belongs to some finite set Λ ⊂ [0, 1] d , the prime denotes the transposition. We set for any A 1 ) For any δ > 0

a ≤ t ≤ b S λ (t) = d l=1 S λ (z l )1 {z l-1 <t≤z l } . ( 3
lim n→∞ φ * n + ν n n δ = 0 and lim n→∞ Λ * (n) n 1/6+ δ = 0 , (3.12)
where φ * n = max 1≤j≤n max x 0 ≤x≤x 1 |φ j (x)|. In order to obtain a good estimator, we have to write a rule to choose a weight vector λ ∈ Λ in (3.10). We define the empirical squared risk as

Err d (λ) = S λ -S 2 d .
Using (3.5) and (3.10) we can rewire this risk as

Err d (λ) = d j=1 λ 2 (j) θ 2 j,d -2 d j=1 λ(j) θ j,d θ j,d + d j=1 θ 2 j,d . (3.13)
Since the coefficient θ j,d is unknown, we need to replace the term θ j,d θ j,d by some its estimator which we choose as

θ j,d = θ 2 j,d - b -a d s j,d with s j,d = b -a d d l=1 σ 2 l φ 2 j (z l ) . (3.14)
Note that from (2.30) -(3.3) it follows that

s j,d ≤ σ 1, * . (3.15)
Finally, we define the cost function of the form

J d (λ) = d j=1 λ 2 (j) θ 2 j,d -2 d j=1 λ(j) θ j,d + δP d (λ) , (3.16)
where the penalty term is defined as

P d (λ) = b -a d d j=1 λ 2 (j)s j,d (3.17)
and 0 < δ < 1 is some positive constant which will be chosen later. We set

λ = argmin λ∈Λ J d (λ) and S * = S λ . (3.18)
To study the efficiency property we specify the weight coefficients (λ(j)) 1≤j≤n as it is proposed, for example, in [START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF]. First, for some 0 < ε < 1 introduce the two dimensional grid to adapt to the unknown parameters (regularity and size) of the Sobolev bull, i.e. we set

A = {1, . . . , k * } × {ε, . . . , mε} , (3.19)
where m = [1/ε 2 ]. We assume that both parameters k * ≥ 1 and ε are functions of n, i.e.

k * = k * (n) and ε = ε(n), such that      lim n→∞ k * (n) = +∞ , lim n→∞ k * (n) ln n = 0 , lim n→∞ ε(n) = 0 and lim n→∞ n δ ε(n) = +∞ (3.20)
for any δ > 0. One can take, for example, for n ≥ 2

ε(n) = 1 ln n and k * (n) = k * 0 + √ ln n , (3.21)
where k * 0 ≥ 0 is some fixed constant. For each α = (β, l) ∈ A, we introduce the weight sequence

λ α = (λ α (j)) 1≤j≤p
with the elements

λ α (j) = 1 {1≤j<j * } + 1 -(j/ω α ) β 1 {j * ≤j≤ω α } , (3.22)
where

j * = 1 + [ln n], ω α = ( β l n) 1/(2β+1) , β = (β + 1)(2β + 1) π 2β β = 2β π 2β ι β and ι β = 2β 2 (β + 1)(2β + 1)
.

Now we define the set Λ as

Λ = {λ α , α ∈ A} . (3.23)
Note, that these weight coefficients are used in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] for continuous time regression models to show the asymptotic efficiency. It will be noted that in this case the cardinal of the set Λ is ν = k * m. It is clear that the properties (3.20) imply the condition (3.12).

In [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF] we shown the following result.

Main results

For any fixed r > 0 and k ≥ 1 we define the Sobolev ellipse as

W k,r = {f ∈ Θ ,L : k j=0 a j θ 2 j ≤ r} , (4.1)
where a j = k l=0 (2π[j/2]) 2l , (θ j ) j≥1 are the trigonometric Fourier coefficients, i.e.

θ j = b a f (x)φ j (x)dx
and (φ j ) j≥1 is the trigonometric basis defined in (3.4). It is clear we can represent this functional class as

W k,r = {f ∈ Θ ,L : k j=0 f (j) 2 ≤ r} , (4.2)
In order to formulate the asymptotic results we define the following normalizing coefficients. First, for any r > 0 we set

l * (r) = ((1 + 2k)r) 1/(2k+1) k π(k + 1) 2k/(2k+1) (4.3) and ς(S) = b a (1 -S 2 (u))du . (4.4)
It is well known that for any S ∈ W k,r the optimal rate of convergence is n -2k/(2k+1) . First we study the lower bound for the asymptotic risks in the class of all estimators E n , i.e. any measurable function with respect to the observations σ{y 1 , . . . , y n }.

lim inf n→∞ inf S n ∈E n n 2k/(2k+1) sup S∈W k,r υ(S)R * ( S n , S) ≥ l * (r) , (4.5)
where υ(S) = (ς(S)) -2k/(2k+1) .

Now we stady the asymptotic upper bound for the quadratic risk of the estimator S * . To this end we assume the following condition for the penalty coefficient δ in the objective function (3.16).

A 2 ) Assume that the parameter δ is a function of n, i.e. δ = δ n such that for any b > 0 Remark 4.1. Note that the limit equalties (4.8) and (4.9) imply that the function l * (r)/υ(S) is the minimal value of the normalised asymptotic quadratic robust risk, i.e. Pinsker constant in this case. We remind that the coefficient l * (r) is the well known Pinsker constant for the "signal+standard white noise" model obtained in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian noise[END_REF]. Therefore, the Pinsker constant for the model (1.1) is represented by the Pinsker constant for the "signal+white noise" model in which the noise intensity is given by the function (4.4).

lim n→∞ δ n n b = 0 . ( 4

The van Trees inequality

In this section we consider the following continuous time parametric model (1.1) with the (0, 1) gaussian i.i.d. random variable (ξ j ) 1≤j≤n and the parametric linear function S, i.e.

S θ (x) = d l=1 θ l ψ l (x) , θ = (θ 1 , . . . , Ξ d ) ∈ R n . (5.1)
The functions (ψ i ) 1≤i≤d are orthogonal with respect to the scalar product (3.3). Let now P n θ be the distribution in R n of the observations y = (y 1 , . . . , y n ) in the model (1.1) with the function (5.1) and ν n ξ be the distribution in R n of the gaussian vector (ξ 1 , . . . , ξ n ). In this case the Radon -Nykodim density is given as

f n (y, θ) = dP (n) θ dν n ξ = exp    n j=1 S θ (x j )y j-1 y j - 1 2 n j=1 S 2 θ (x j )y 2 j-1    . (5.2)
Let u be a prior distribution density on R d for the parameter θ of the following form:

u(θ) = d j=1 u j (θ j ) ,
where u j is some continuously differentiable probability density in R with the support ] -L j , L j [, i.e. u j (z) > 0 for any -L j < z < L j and u j (z) = 0 for all |z| ≥ L j , such that the Fisher information is finite, i.e.

I j = L j -L j u2 l (z) u j (z) dz < ∞ . (5.3) Now, we set Ξ =] -L 1 , L 1 [ × . . . × ] -L d , L d [ ⊆ R d . (5.4)
Let g(θ) be a continuously differentiable Ξ → R function such that, for each 1 ≤ j ≤ d,

lim |θ j |→L j g(θ) u j (θ j ) = 0 and R d |g j (θ)| u(θ) dθ < ∞ , (5.5)
where g j (θ) = ∂g(θ)/∂θ j .

For any B(X )×B(R d )-measurable integrable function H = H(y, θ) we denote

E H = Ξ R n H(y, θ) dP n θ u(θ)dθ = Ξ R n H(y, θ) f n (y, θ) u(θ)dν (n) ξ dθ .
Now we obtain an lower bound for the corresponding bayesian risks in the case when the model (1.1) is gaussian with the function (5.1).

Lemma 5.1. For any F y n -measurable square integrable function g n and for any 1 ≤ j ≤ d, the following inequality holds

E( g n -g(θ)) 2 ≥ g 2 j E Ψ n,j + I j , (5.6)
where Ψ n,j = n j=1 ψ 2 j (x l ) y 2 l-1 and g j = Ξ g j (θ) u(θ) dθ.

Proof. First, for any θ ∈ Ξ we set

U j = U j (y, θ) = 1 f (y, θ)u(θ) ∂ (f (y, θ)u(θ)) ∂θ j .
Taking into account the condition (5.5) and integrating by parts we get

E ( g n -g(θ)) U j = R n ×Ξ ( g n (y) -g(θ)) ∂ ∂θ j (f (y, θ)u(θ)) dθ dν (n) ξ = R n × Ξj L j -L j g j (θ) f (y, θ)u(θ)dθ j   i =j dθ i   dν (n) ξ = g j , where Ξj = i =j ] -L i , L i [ .
Now by the Bouniakovskii-Cauchy-Schwarz inequality we obtain the following lower bound for the quadratic risk

E( g n -g(θ)) 2 ≥ g 2 j E U 2 j .
To study the denominator in the left hand of this inequality note that in view of the representation (5.2)

1 f n (y, θ) ∂ f n (y, θ) ∂θ j = n l=1 ψ j (x l ) y l-1 (y l -S θ (x l )y l-1 ) .
Therefore, for each θ ∈ Ξ,

E (n) θ 1 f n (y, θ) ∂ f n (y, θ) ∂θ j = 0 E (n) θ 1 f n (y, θ) ∂ f n (y, θ) ∂θ j 2 = E (n) θ n l=1 ψ 2 j (x l ) y 2 l-1 = E (n) θ Ψ n,l .
Using the equality

U j = 1 f n (y, θ) ∂ f n (y, θ) ∂θ j + 1 u(θ) ∂ u(θ) ∂θ j ,
we get E U 2 j = E Ψ n,j + I j , where the Fisher information I j is defined in (5.3). Hence Lemma 5.1.

Remark 5.1. It should be noted that in the definition of the prior distribution the bound L j may be equal to infinity either for some 1 ≤ j ≤ d or for all 1 ≤ j ≤ d.

Low bound

First, note that R * ( S n , S) ≥ R p 0 ( S n , S) , (6.1) where p 0 is the (0, 1) gaussian density. Now for any fixed 0 < ε < 1 we set

d = d n = k + 1 k (ςn) 1/(2k+1) l * (r ε ) (6.2)
where ς = 1/(b -a), r ε = (1 -ε)r and

l * (r ε ) = ((1 + 2k)r ε ) 1/(2k+1) k π(k + 1) 2k/(2k+1) = (1 -ε) 1/(2k+1) l * (r) .
For any vector θ = (θ j ) 1≤j≤d ∈ R d , we set

S θ (x) = d n j=1 θ j φ j (x) , (6.3)
where (φ j ) 1≤j≤d n is the trigonometric basis defined in (3.4). As ir is shown in [START_REF] Golubev | On the second order minimax estimation of distribution functions[END_REF] there exist continuously differentiable density pL with the support on

[-L, L]] such that L -L xp L (x)dx = 0, L -L x 2 pL (x)dx = 1 and ǏL = L -L (p (x)) 2 p(x) p(x)dx = 1 + ˇ L ,
where ˇ L → 0 as L → ∞. To define the bayesian risk we choose a prior distribution on R d as θ = (θ j ) 1≤j≤d n and κ j = s j ηj , (6.4)

where ηj are i.i.d. random variables with the density pL ,

s j = s * j ςn and s * j = d n j k -1 .
Furthermore, for any function f , we denote by h(f ) its projection in

L 2 [0, 1] onto W k,r , i.e. h(f ) = Pr W k,r (f ) .
Since W k,r is a convex set, we obtain, that for any function

S ∈ W k,r S -S 2 ≥ h -S 2 with h = h( S) .
From the definition of the prio distribution (6.4) we obtain that a.s.

max a≤x≤b |S θ (x)| + | Ṡθ (x)| ≤ 2 b -a b -a + 1 b -a n := * n , (6.5)
where

n = L √ n d n j=1 d n j k/2 j → 0 as n → ∞
for any k ≥ 2. Therefore, for sufficiently large n the function (6.3) belongs to the class (2.4) and the last property yileds sup

S∈W k,r υ(S) R p 0 ( S, S) ≥ {z∈R d : S z ∈W k,r } υ(S z )E p 0 ,S z h -S z 2 µ κ (dz) ≥ υ * {z∈R d : S z ∈W k,r } E p 0 ,S z h -S z 2 µ κ (dz)
,

where υ * = inf |S| * ≤ * n υ(S) → ς 2k/(2k+1) as n → ∞ .
Using the distribution µ κ we introduce the following Bayes risk

R 0 ( S) = R d R p 0 ( S, S z ) µ κ (dz) .
Taking into account now that h 2 ≤ r we obtain sup

S∈W k,r υ(S) R p 0 ( S, S) ≥ υ * R 0 ( h) -2 υ * R 0,n (6.6)
with R 0,n =

{z∈R d : S z / ∈W k,r } (r + S z 2 ) µ κ (dz) .
In Lemma A.2 we studied the last term in this inequality. Now it is easy to see that

h -S z 2 ≥ d n j=1 ( z j -z j ) 2 ,
where

z j = 1 0 h(t) φ j (t)dt. So, in view of Lemma 5.1, we obtain R 0 ( h) ≥ 1 ςn d n j=1 1 1 + ǏL (s * j ) -1 ≥ 1 ςn max(1, ǏL ) d n j=1 1 - j k d k n .
Therefore, using now the definition (6.2), Lemma A.2 and the inequality (6.1) we obtain that lim inf

n→∞ inf S∈Π n n 2k 2k+1 sup S∈W k,r υ(S) R * ( S n , S) ≥ (1 -ε) 1 2k+1 1 max(1, ǏL ) l * .
Taking here limit as ε → 0 and L → ∞ we come to the Theorem 4.1.

7 Upper bound

Known regularity

We start with the estimation problem for the functions S from W k,r with known parameters k, r and ς(S) defined in (4.4). In this case we use the estimator from family (3.23)

S = S α , (7.1)
where α = (k, t n ), l n = [r(S)/ε] ε and r(S) = r/ς(S). We remind, that ε = 1/ ln n. Note that for sufficiently large n, the parameter α belongs to the set (3.19). In this section we obtain the upper bound for the empiric risk (3.2).

Theorem 7.1. The estimator S constructed on the trigonometric basis satisfies the following asymptotic upper bound

lim sup n→∞ n 2k/(2k+1) sup S∈W k,r υ(S) E p,S S -S 2 d 1 Γ ≤ l * (r) . (7.2)
Proof. We denote λ = λ α and ω = ω α . Now we recall that the Fourier coefficients on the set Γ

θ j,d = θ j,d + ζ j,d with ζ j,d = b -a d η j,d + j,d ,
Therefore, on the set Γ we can represent the empiric squared error as

S -S 2 d = d j=1 (1 -λ(j)) 2 θ 2 j,d -2M n -2 d j=1 (1 -λ(j)) λ(j)θ j,d j,d + d j=1 λ 2 (j) ζ 2 j,d ,
where

M n = b -a d d j=1 (1 -λ(j)) λ(j)θ j,d η j,d . Now for any 0 < ε < 1 2 d j=1 (1 -λ(j)) λ(j)θ j,d j,d ≤ ε d j=1 (1 -λ(j)) 2 θ 2 j,d + ε-1 d j=1 2 j,d .
Taking into account here the definition (3.8), we can rewrite this inequality as

2 d j=1 (1 -λ(j)) λ(j)θ j,d j,d ≤ ε d j=1 (1 -λ(j)) 2 θ 2 j,d + B n εn .
Therefore,

S -S 2 d ≤ (1 + ε) d j=1 (1 -λ(j)) 2 θ 2 j,d -2M n + B n εn + d j=1 λ 2 (j)ζ 2 j,d .
By the same way we estimate the last term on the right-hand side of this inequality as

d j=1 λ 2 (j) ζ 2 j,d ≤ (1 + ε)(b -a) d d j=1 λ 2 (j) η 2 j,d + (1 + ε-1 ) B n n .
Thus, on the set Γ we find that for any 0 < ε < 1

S n -S 2 d ≤ (1 + ε)Υ n (S) -2M n + (1 + ε)U n + 3B n εn , (7.3)
where

Υ n (S) = d j=1 (1 -λ(j)) 2 θ 2 j,d + ς(S) d 2 d j=1 λ 2 (j) (7.4)
and

U n = 1 d 2 d j=1 λ 2 (j) d(b -a)η 2 j,d -ς(S) .
First, note that in view of Lemma A.7

E p,S M 2 n ≤ σ 1, * (b -a) d d j=1 θ 2 j,d = σ 1, * (b -a) d S 2 d ≤ σ 1, * (b -a) 2 d ,
where the constant σ 1, * is defined in (2.30). Moreover, taking into account here that E p,S M n = 0, we get

|E p,S M n 1 Γ | = |E p,S M n 1 Γ c | ≤ (b -a) σ 1, * P p,S (Γ c ) d .
Therefore, Proposition 2.2 yields

lim n→∞ n 2k/(2k+1) sup S∈Θ ,L |E p,S M n 1 Γ | = 0 . (7.5)
Now, the property (3.9) Lemma A.6 and Lemma A.8 imply the inequality (7.2). Hence Theorem 7.1.

Unknown smoothness

A Appendix

A.1 Properties of the prior distribution (6.4)

Lemma A.1. For any 1 ≤ j ≤ d lim n→∞ (b -a) n E Ψ n,j = 1 . (A.1)
Proof. First, note that for any k ≥ 1

y k = y 0 k j=1 S θ (x j ) + k l=1 k j=l+1 S θ (x j )ξ l .
Using the distribution (6.4), we obtain that

E y 2 m = y 2 0 E m j=1 S 2 θ (x j ) + m l=1 E m j=l+1 S 2 θ (x j ) .
Therefore, due to the property (6.5) we obtain that for any m ≥ 1 and for any n ≥ 1 for which * n < 1 we get, that

| E y 2 m -1| ≤ y 2 0 ( * n ) m + m-1 l=1 ( * n ) m-l ≤ * n y 2 0 + 1 1 - * n .
Taking into account that for any n ≥ 1 Proof. First note, that the bound (6.5) implies that for sufficiently large n the function (6.3) with the random coefficients defined through the prior distribution (6.4) almost sure belongs to the class Θ ε,L . Therefore, the definition (4.1) we obtain that

S θ / ∈ W k,r = {ζ n > r} ,
where ζ n = d n j=1 κ 2 j a j . So, it suffices to show that for any b > 0 lim

n→∞ n b P(ζ n > r) = 0 . (A.3) Note now, that lim n→∞ E ζ n = lim n→∞ 1 (b -a)n d n j=1 a j s * j = r ε = (1 -ε)r .
So, for sufficiently large n we obtain that

{ζ n > r} ⊂ ζ n > r 1 ,
where r 1 = rε/2 and

ζ n = ζ n -E ζ n = 1 n d n j=1
s * j a j η j .

Using again the correlation inequality from [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observe at discrete times[END_REF] we get that for any p ≥ 2 there exists some constant C p > 0 for which

E ζ p n ≤ C p 1 v p n   d j=1 (s * j ) 2 a 2 j   p/2 ≤ C p n -p 4k+2 ,
i.e. the expectation E ζ p n → 0 as n → ∞. Therefore, using the Chebychev inequality we obtain that for any b > 0

n b P( ζ n > r 1 ) → 0 as n → ∞ .
This implies (A.3), and therefore, Lemma A.2.

A.2 Relations between the norms • and • d .

Lemma A.3. Let f be an absolutely continuous [0, 1] → R function with ḟ < ∞ and g be a simple [0, 1] → R function of the form

g(t) = d j=1 c j χ (t j-1 ,t j ] (t),
where c j are some constants and t j = j/d. Then for any ε > 0

f -g 2 ≤ (1 + ε) f -g 2 d + (1 + ε -1 ) ḟ 2 d 2 and f -g 2 d ≤ (1 + ε) f -g 2 + (1 + ε -1 ) ḟ 2 d 2 . Proof. Setting ∆(t) = f (t) -g(t)
, we obtain, that for any ε > 0

∆ 2 = ∆ 2 d + d l=1 t l t l-1 2∆(t l ) (∆(t) -∆(t l )) + (∆(t) -∆(t l )) 2 dt ≤ (1 + ε) ∆ 2 d + (1 + ε -1 ) d l=1 t l t l-1 [∆(t l ) -(∆(t))] 2 dt = (1 + ε) ∆ 2 d + (1 + ε -1 ) d l=1 t l t l-1 |f (t l ) -f (t)| 2 dt .
Noting that, for t l-1 < t ≤ t l , one has the estimate

|f (t l ) -f (t)| 2 ≤ t l t l-1 | ḟ (u)|du 2 ≤ 1 p t l t l-1 | ḟ (u)| 2 du ,
one comes to the first inequality. Similarly, one can verify the second inequality. Hence Lemma A.3.

A.3 Properties of the trigonometric basis.

Lemma A.4. For any 1 ≤ j ≤ d the trigonometric Fourier coefficients (θ j,d ) 1≤j≤p for the functions S from the class W k,r satisfy, for any ε > 0, the following inequality

θ 2 j,d ≤ (1 + ε) θ 2 j + (1 + ε -1 ) 2r d 2k . (A.4)
Proof. First we represent the function S as

S(x) = d l=1 θ l φ l (x) + ∆ d (x) , where ∆ d (x) = l>d θ l φ l (x) .
Therefore,

θ j,d = (S, φ j ) d = θ j + (∆ d , φ j ) d
and for any 0 < ε < 1

θ 2 j,d ≤ (1 + ε)θ 2 j + (1 + ε -1 ) ∆ d 2 d
. By applying Lemma A.3 with g = 0, we obtain that

∆ d 2 d ≤ 2 ∆ d 2 + 2 ∆d 2 d 2 . Note here, that for any N ≥ 1 ∆N 2 = (2π) 2 l>N θ 2 l [l/2] 2 .
Taking into account here that 2π[l/2] ≥ l for l ≥ 2 ,

we obtain that ∆N 2 ≤ l>N a l l 2(k-1) θ 2 l ≤ r N 2(k-1) . (A.5)
Hence Lemma A.4

Lemma A.5. For any d ≥ 2, 1 ≤ N ≤ d and r > 0, the coefficients (θ j,d ) 1≤j≤d of functions S from the class W 1 r satisfy, for any ε > 0, the following inequality

d j=N θ 2 j,d ≤ (1 + ε) j≥N θ 2 j + (1 + ε -1 ) r d 2 N 2(k-1) . (A.6)
Proof. First we note that

d j=N θ 2 j,d = min x 1 ,...,x N -1 S - N -1 j=1 x j φ j 2 d ≤ ∆ N 2 d ,
where ∆ N (t) = j≥N θ j φ j (t). By applying Lemma A.3 and taking into account the inequality (A.5), we obtain the bound (A.6). Hence Lemma A.5 (A.7)

A.4 Technical lemmas

Proof. First of all, note that

0 < ε 2 (b -a) ≤ inf S∈Θ ε,L ς(S) ≤ sup S∈Θ ε,L ς(S) ≤ b -a . (A.8)
This implies directly that

lim n→∞ sup S∈Θ ε,L l n r(S) -1 = 0 , (A.9)
where r(S) = r/ς(S). Moreover, note that

n 2k/(2k+1) υ(S) Υ n (S) ≤ n 2k/(2k+1) υ(S) Ξ d + (ς(S)) 1/(2k+1) n 1/(2k+1) d j=1 λ 2 (j)
and

Ξ d = d j=1 (1 -λ(j)) 2 θ 2 j,d = Ξ 1,d + Ξ 2,d ,
where

Ξ 1,d = [ ω] j=j * (1 -λ(j)) 2 θ 2 j,d and Ξ 2,d = d j=[ ω]+1 θ 2 j,d .
We recall that ω = ω α = n l n k

1/(2k+1)

.

Lemma A.4 and Lemma A.5 yield

Ξ 1,d ≤ (1 + ε) [ ω] j=j * (1 -λ(j)) 2 θ 2 j + 2r(1 + ε -1 ) ω d 2k . and Ξ 2,d ≤ (1 + ε) j≥[ ω]+1 θ 2 j + (1 + ε -1 ) r d 2 ω 2(k-1) , i.e. Ξ d ≤ (1 + ε) j≥1 (1 -λ(j)) 2 θ 2 j + 2r(1 + ε -1 ) Ξ n , (A.10)
where

Ξ * n = j≥1 (1 -λ(j)) 2 θ 2 j = j≤ ω (1 -λ(j)) 2 θ 2 j + j> ω θ 2 j := Ξ * 1,n + Ξ * 2,n and Ξ n = ω d 2k + 1 d 2 ω 2(k-1) . Note, that n 2k/(2k+1) υ(S)Ξ * 1,n = [ ω] j=j * (1 -λ(j)) 2 θ 2 j = υ(S) ( k l n ) 2k/(2k+1) [ ω] j=j * j 2 θ 2 j ≤ υ(S) ( k l n ) 2k/(2k+1) a * j * [ ω] j=j * a j θ 2 j ,
where a * n = sup j≥n j 2k /a j . It is clear that

lim n→∞ a * n = 1 π 2k .
Therefore, from (A.9) we obtain that lim sup

n→∞ sup S∈Θ ε,L n 2k/(2k+1) υ(S) Ξ * 1,n [ ω] j=j * a j θ 2 j ≤ 1 π 2k ( k r) 2k/(2k+1) = ι 2k/(2k+1) k (2kπ r) 2k/(2k+1) .
Next note, that for any 0 < ε < 1 and for sufficiently large n

Ξ * 2,n ≤ 1 π 2k ω 2k j≥[ ω]+1 a j θ 2 j ≤ (1 + ε) (ς(S)ι k ) 2k/(2k+1) π 2k/(2k+1) (2krn) 2k/(2k+1) j≥[ ω]+1 a j θ 2 j ,
i.e. Moreover, according to the inequality (A.4) from [START_REF] Arkoun | Sequential Model Selection Method for Nonparametric Autoregression[END_REF] we have that E p,S η 4 j,q ≤ 64 mσ This implies the quality (A.12), and we obtain Lemma A.8.

Proposition 2 . 1 .

 21 The convergence rate in probability of the estimator (2.17) is more rapid than any power function, i.e. for any b > 0 lim n→∞ n b max 1≤l≤d sup S∈Θ ,L sup p∈P P p,S | S ι l -S(z l )| > 0 = 0 , (2.19)

Proposition 2 . 3 .Remark 2 . 1 .

 2321 For any b > 0 the sequence (υ n ) n≥1 satisfies the following limiting equality lim n→∞ n -b υn = 0 . (2.32) It should be noted that the property (2.32) means that the asymptotic behavior of the upper bound (2.31) approximately almost as h -2

. 11 )

 11 Denote by ν be the cardinal number of the set Λ and Λ * = max λ∈Λ d j=1 λ(j) .

Theorem 3 . 1 .Remark 3 . 1 .

 3131 Assume that the conditions (2.2) and (3.12) hold. Then for any n ≥ 3, any S ∈ Θ ,L and any 0 < δ ≤ 1/12, the procedure (3.18) with the coefficients (3.23) satisfies the following oracle inequalityR * ( S * , S) ≤ (1 + 4δ)(1 + δ) 2 1 -6δ min λ∈Λ R * ( S λ , S) + D * n δn ,(3.24)where the term D * n is such that for any b > It sjhould be noted that the weight least square estimators (3.10) with the weight coefficients (3.23) is efficient for the Sobolev ball (see, for example, [? 19]). So, Theorem 3.1 means that this model selection procedure is best among all the effective procedures in the sharp oracle inequality sense (3.24) and below we will use this property to show the efficiency property in adaptive setting, i.e. in the case, when the regularity property of the function S (1.1) is unknown.

Theorem 4 . 1 .

 41 For the model (1.1) with the noise distribution from the class P defined in(2.1)

2 .

 2 For any b > 0 the term R 0,n introduced in (6.6) satisfies the following property lim n→∞ n b R 0,n = 0 . (A.2)

Lemma A. 6 .

 6 The sequence Υ n (S) satisfies the following upper bound lim sup n→∞ sup S∈W k,r n 2k/(2k+1) υ(S) Υ n (S) ≤ l * (r) .

λ 2 2 .Lemma A. 8 . 2 l 1 -( 1 - 2 λ 2

 22821122 (j) n 1/(2k+1) (r(S)) 1/(2k+1) g k -1 = 0 , (A.11)whereg k = 2k 2 ( k ) 2k/(2k+1) (2k + 1)(k + 1).So, taking into account that in (A.10)lim n→∞ sup S∈W k,r n 2k/(2k+1) Ξ n = 0 ,we obtain the limit equality (A.7) and, hence Lemma A.6.Lemma A.7. For any non random coefficients (u j ) 1≤j≤d Using the definition of η j,d in (3.7), we obtain that Now, the orthonormality property of the basis functions (φ j (•)) 1≤j≤d implies this lemma. Now we show thatlim n→∞ n 2k/(2k+1) sup S∈W k,r |E p,S U n 1 Γ | = 0 . (A.12)Proof. First of all, note that, using the definition of s j,d in (3.14), we obtainE p,S η 2 j,d = E p,S s j,d φ j (x l ) and φ j (z) = (b -a)φ 2 j (z) -1 .Therefore, we can represent the expectation of U n asE p,S U n = λ 2 d 2 E p,S U 1,n + b -a d 2 E p,S U 2,n ,whereλ 2 = d j=1 λ 2 (j), U 1,n = b -a d d l=1 d H l -ς(S) and U 2,n = d j=1 λ 2 (j)s j,d .Note now, that using Proposition 2.19 and the dominated convergence theorem in the definition (2.16) we obtain that lim S 2 (z l )) = 0 .Taking into account that for the functions from the class (2.4) their derivatives are bounded in modulus by the fixed constant L > 0, we can easy deduce S 2 (z l )) -ς(S) = 0 ,i.e. lim n→∞ sup S∈Θ ,L sup p∈P |E p,S U 1,n | = 0 .Therefore, taking into account that lim supn→∞ n 2k/(2k+1) sup S∈Θ ,L λ 2 d 2 < ∞ , we obtain that lim n→∞ n 2k/(2k+1) sup S∈Θ ,L sup p∈P |E p,S U n | ≤ (b -a)lim n→∞ U * sup p∈P |E p,S U 2,n | .29Now, using Lemma A.2 from[START_REF] Galtchouk | Sharp nonasymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF] we obtain thatE p,S U 2,n = (j) φ j (z l ) ≤ d σ 1, * (2 2k+1 + 2 k+2 + 1) ≤ 5d σ 1, * 2 2k .Note that from the definition of σ 1, * in (2.30) it follows that lim sup n→∞ d σ 1, * < ∞ , i.e. lim sup n→∞ sup S∈Θ ,L sup p∈P E p,S U 2,n < ∞ . Therefore, the using this bound in (A.13) implies lim n→∞ n 2k/(2k+1) sup S∈Θ ,L sup p∈P |E p,S U n | = 0 .

  where the coefficient m is given in(2.29). From this we obtain, thatE p,S |U n |1 Γ c ≤ (b -a) d d j=1 E p,S η 2 j,d 1 Γ c + ς(S)P p,S (Γ c ) ≤ 8σ 1, * (b -a) √ m d P p,S (Γ c ) + ς(S)P p,S (Γ c ) . So, Proposition 2.2 implies lim n→∞ n 2k/(2k+1) sup S∈W k,r E p,S |U n |1 Γ c = 0 .

  Theorem 3.1 and Theorem 7.1 imply Theorem 4.2. Acknowledgements. The last author was partially supported by by the Russian Federal Professor program (project no. 1.472.2016/1.4, the Ministry of Education and Science of the Russian Federation), the research project no. 2.3208.2017/4.6 (the Ministry of Education and Science of the Russian Federation), by RFBR Grant 16-01-00121 A and by "The Tomsk State University competitiveness improvement program" grant 8.1.18.2018.

* This work was supported by the RSF grant 17-11-01049 (National Research Tomsk State University).


Antonio A Fernández-Marín

Noé Jiménez

Jean-Philippe Groby

José Sánchez-Dehesa

Vicente Romero-García

Aerogel-based metasurfaces for perfect acoustic energy absorption

The unusual viscoelastic properties of silica aerogel plates are efficiently used to design subwavelength perfect sound absorbers. We theoretically, numerically and experimentally report a perfect absorbing metamaterial panel made of periodically arranged resonant building blocks consisting of a slit loaded by a clamped aerogel plate backed by a closed cavity. The impedance matching condition is analyzed using the Argand diagram of the reflection coefficient, i.e., the trajectory of the reflection coefficient as a function of frequency in the complex plane. The lack or excess of losses in the system can be identified via this Argand diagram in order to achieve the impedance matching condition. The universality of this tool can be further exploited to design more complex metasurfaces for perfect sound absorption, thus allowing the rapid design of efficient absorbing metamaterials.

Silica aerogels are extremely-lightweight nanoporous materials 1 . The frame of these materials consists of an assembly of connected small cross-sections beam-like elements resulting from fused nanoparticles. This particular assembly provides silica aerogel a very low elastic stiffness when compared to rigid silica structure of identical porosity. Aerogels possess a wide variety of exceptional properties such as low thermal conductivity, low dielectric constant, low index of refraction or a very large porosity (80-99.8%) thus providing these materials an extremely low density 2 . Because of this large porosity and therefore of the very large available contact area, they have been used as filters, absorbent media or waste containment (see Refs. [2-4] and references therein). They have also been applied as catalysts or even to capture cosmic dust 5,6 . Similarly, their low thermal conductivity, which so far seems to be their most interesting property compared to any other elastic or poroelastic material, has been exploited in various commercial applications including thermal insulation 7 , heat and cold storage devices 5,8 .

In acoustics, aerogels are used as impedance matching materials to develop efficient ultrasonic devices 9,10 or sound absorbing materials for anechoic chambers 5,[START_REF] Gibiat | the Fourth International Symposium on AEROGELS[END_REF] . Beyond these properties, silica aerogel plates are excellent candidates to design new types of membrane metamaterials, since they exhibit subwavelength resonances and present absorption efficiency [START_REF] Guild | [END_REF]13 . Effectively, the use of membrane metamaterials to control acoustic waves has shown an increasing interest in recent years 14 . Mema) Wave Phenomena Group, Departamento de Ingeniería Electrónica, Universitat Politècnica de València, Camino de Vera s/n, 46022 València, Spain b) Laboratoire d'Acoustique de l'Université du Mans, LAUM -UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France c) Corresponding author: vicente.romero@univ-lemans.fr brane and plate metamaterials have been employed in the past to design efficient absorbers [15][16][17] , e.g., using a single membrane backed by a cavity 16,18 , which can present deeper subwavelength resonances as compared with absorbing metamaterials based in air cavities [19][20][21][22] . Moreover, double negative acoustic metamaterials 23 can be achieved by combining a lattice of membranes, which provides a negative effective mass density 24 , with subwavelength side-branch resonators, which provides a negative effective bulk modulus 25 . In addition, periodic arrangements of clamped plates have been efficiently used to control harmonic generation 26 or solitary waves in the nonlinear acoustic regime 27 .

In this work, we make use of the efficient and unusual attenuating properties of silica aerogel plates to design subwavelength perfect sound absorbers. The analyzed system is depicted in Fig. 1(a) and consists of a periodic repetition of the resonant building units illustrated in Fig. 1(b). These units are made of a slit loaded by a clamped aerogel plate backed by a closed cavity. The perfect absorption of this system is comprehensively analyzed both theoretically and experimentally. In a first stage, we model the system using the Transfer Matrix Method (TMM) accounting for the contribution of the losses to the problem, i.e., the viscothermal losses from the slit and cavity and the viscoelastic losses from the aerogel plate. In a second stage, we analyze the impedance matching condition, also known as critical coupling condition, which is obtained once the inherent losses exactly compensates the leakage of the system 28 . The Argand diagram of the reflection coefficient 29 is further employed to evaluate either the lack or excess of inherent losses in the system, thus providing important information on the impedance matching condition. The Argand diagram is revealed as a universal and powerful tool to design perfect absorbers.

We consider a slotted panel of thickness L, whose slits, For wavelengths λ large enough compared to the thickness of the aerogel plate h m and neglecting the effects of rotary inertia and additional deflections caused by shear forces, the transverse plate displacement w m satisfies the Kirchhoff-Love wave equation [START_REF] Graff | Wave motion in elastic solids[END_REF] . The plate can be described by ρ, the density and D = Eh 3 m /12(1 -ν 2 ), the bending stiffness. E is the Young modulus and ν is the Poisson's ratio of the plate. Assuming an implicit time dependence e iωt , with ω the angular frequency, the viscoelastic behavior of silica aerogel can be modeled via a complex Young modulus, E = E 0 (1+iηω), where E 0 and η are the unrelaxed Young modulus and the loss factor respectively. In the subwavelength regime, the silica aerogel disk can be considered as a punctual resonant element located at (x, y) = (L/2, a/2) (note that h m λ 0 , where λ 0 is the wavelength in air). The acoustic impedance of the clamped circular cross-sectional plate thus takes the form [START_REF] Skvor | Vibrating Systems and Their Equivalent Circuits[END_REF]32 ,

Z p = -iωρh m πr 2 m I 1 (kr m )J 0 (kr m ) + J 1 (kr m )I 0 (kr m ) I 1 (kr m )J 2 (kr m ) -J 1 (kr m )I 2 (kr m ) , (1)
where J n and I n are the of n-th order regular and modified Bessel's functions of the first kind respectively and the wavenumber in the plate satisfies k 2 = ω ρh m /D.

Viscothermal losses also occurs in the narrow slits 33 and in the cavity, also offering a useful degree of freedom to tune the losses of the system. Assuming only plane wave propagates in these channels, the viscothermal losses are modeled by effective parameters: complex and frequency-dependent wavenumbers k s = ω ρ s /κ s and k c = ω ρ c /κ c , and impedances Z s = √ κ s ρ s /h s a and Z c = √ κ c ρ c /πr 2 m in the slit and in the cavity respectively. Note that we make use of the effective density ρ s and bulk modulus κ s of a slit 34 for the slotted channel, while we make use of the effective density ρ c and bulk modulus κ c of a cylindrical duct 34 for the cavity.

The scattering properties of the system are studied through the reflection coefficient R obtained by TMM. We relate the sound pressure and normal particle velocity at the surface of the system, [P 0 , V 0 ] = [P (x), V x (x)] x=0 to the ones at the end of the system, [P L , V L ] = [P (x), V x (x)] x=L , by a transfer matrix as

P 0 V 0 = T P L V L , where T = M ∆l M S M R M S . (2)
In Eq. ( 2), transfer matrix over half the slit length M S reads as

M S = cos (k s L/2) iZ s sin (k s L/2) i sin (k s L/2) /Z s cos (k s L/2) , (3)
and M R accounts for the local effect of the aerogel plate together with the back cavity as

M R = 1 0 1/Z R 1 , (4)
where Z R = Z p -iZ c cot(k c l c ). The matrix M ∆l provides the radiation correction of the slit to the free space as

M ∆l = 1 Z ∆l 0 1 , (5)
where Z ∆l = -iω∆lρ 0 /φ s a 2 , with φ s = h s /a the surface porosity of the metasurface, ρ 0 the air density and ∆l the end correction length that can be approximated as

35 ∆l = h s φ s ∞ n=1 sin 2 (nπφ s ) (nπφ s ) 3 . (6)
The surface impedance at x = 0 can thus be directly obtained using Eq. ( 2) and considering the rigid backing condition (V L = 0) as

Z T = P 0 V 0 = Z s (Z ∆l + Z R ) + i tan (k s L/2) iZ s Z R tan (k s L/2) + 2Z ∆l Z R + Z 2 s Z s + 2iZ R tan (k s L/2) . (7)
Finally, we calculate the reflection and absorption co-efficient using Eq. ( 7) as where Z 0 is the impedance of the surrounding medium, i.e., the air [START_REF]× 10 -5 Pa•s, a sound speed c 0 = γP 0 /ρ 0 m/s, and an acoustic[END_REF] . For the aerogel plate, we used an unrelaxed Young modulus E 0 = 197.92 kPa and a loss factor η = 4.47 × 10 -6 Pa•s, a density ρ = 80 kg/m 3 and a Poison's ratio ν = 0.12, as characterized in Ref. [13].

R = Z T -Z 0 Z T + Z 0 , and α = 1 -|R| 2 , (8)
Aerogel plates of r m = 19.5 mm and h m = 10.5 mm were selected and a = 42 mm was fixed by the width of the square cross-sectional impedance tube that was used for the experimental validation. The procedure begins with looking for the geometric parameters giving the most efficient absorption at the lowest resonance frequency. The geometric parameters of the system are thus optimized numerically using a sequential quadratic programming (SQP) method [START_REF] Powell | Numerical analysis[END_REF] . The following parameters were obtained: L = 44 mm, h s = 1.285 mm, and l c = 29.8 mm. The highly efficient absorption peak also appears at 591.2 Hz and is associated with a reflection coefficient amplitude of 10 log 10 |R| = -62 dB. The corresponding perfectly absorbed wavelength is λ/L = 13.1 times larger than he depth of the structure. This subwavelength feature is due to the slow sound properties induced by the presence of the slit loading resonators 20,[START_REF] Groby | [END_REF]39 . Effectively, the resonance frequency of the slit in the absence of these loading resonators is around 1900 Hz. [START_REF]Note that the slow sound properties can be improved by using thinner aerogel plates, thereby[END_REF] Figures 2(a) and 2(b) show a parametric study of the system reflection coefficient around the optimal configuration. The reflection coefficient is significantly reduced when the parameters correspond to the optimal parameters (marked by white crosses). However, the balance between the inherent losses and the leakage of the system is difficult to identify by using this parametric analysis.

A first approach to ensure that these parameters led to perfect absorption of the sound energy consists in representing the reflection coefficient in the complex frequency plane, as shown in Fig. 2(c). Using this representation, the locations of the zero/pole pairs of the reflection coefficient can be studied. In the lossless case, the zeros are complex conjugates of their corresponding poles, both appearing in the opposite half spaces of complex frequency plane (zeros in the lower half space and poles in the upper one with our time Fourier convention). However, the zeros follows a given trajectory towards the pole half space when losses are introduced in the system. Note that the losses are modified when modifying the system geometry. In this way, the trajectory of the lowest frequency zero is depicted Fig. 2(c), when h s , l c and L are modified. For a given set of geometric parameters, the trajectory of the zero crosses the real frequency axis, ensuring the balance of the leakage by the inherent losses, therefore providing the perfect absorption 28 .

The complex frequency plane also gives useful insights to design and tune open lossy resonant systems 18,20,[START_REF] Jiménez | [END_REF] . Such system is characterized by its leakage and the inherent losses; the impedance matching condition corresponds to the critical coupling of the system, i.e., the per- fect balance of the leakage by the inherent losses. On the one hand, the intrinsic losses of the system are too large (too small) compared to the leakage of the system when the zero has (has not) already crossed the real axis, meaning that the absorption is not optimal as the impedance condition is not satisfied. These situations are illustrated Fig. 2(d), where the absorption coefficient is depicted for different values of h s . The red curve corresponds to a narrow slit (height h s = h s /2) providing an excess of losses, while the blue curve corresponds to a wide slit (height h s = 2h s ) providing a lack of losses. The location of the corresponding zero in the complex frequency plane is marked with red and blue crosses in Fig. 2(c). The perfect balance between the leakage by the losses is the situation depicted on the color map Fig. 2(c); the zero of the reflection coefficient is exactly located on the real frequency axis. Therefore, we can conclude that the complex frequency plane is very useful to immediately identify if one particular configuration has a lack or excess of losses. This approach has been recently used to design absorbing materials ranging from porous media 42,43 to different kind of metamaterials 18,39,[START_REF] Jiménez | [END_REF]44,45 . However, the acoustic behavior of all systems cannot necessarily be assessed in the complex frequency plane. Effectively, numerical methods do not usually allow to calculate solutions for complex frequencies, and more importantly, experimental results are usually only provided for real frequencies.

A useful approach to overcome this problem consists in analyzing the reflection coefficient R = |R|e iϕ , with ϕ = arctan[Im(R)/Re(R)], in the complex plane. Figure 2(e) and (f) depict respectively the phase of the reflection coefficient and the corresponding Argand diagram from 400 to 800 Hz. The reflection coefficient is necessarily inscribed within the unitary circle, i.e., |R| ≤ 1. In the lossless case, the reflection coefficient follows the unitary circle counter-clockwise with increasing frequency starting from ϕ = 0 at 0 Hz, as R = e iϕ . When losses are accounted for, the trajectory of R is modified and follows an elliptical trajectory around the resonance, contained inside of the unitary circle and displaced along the real axis in the diagram. On the one hand, the reflection coefficient describes a loop that does not encompass the origin if the losses exceed the optimal ones, e.g. the red ellipse Figure 2(f) calculated for h s = h s /2. On the other hand, the ellipse encompasses the origin if the losses lacks, e.g. the blue ellipse Figure 2(f) calculated for h s = 2h s . Finally, the ellipse must pass through the origin, i.e., R = 0, when perfect absorption is reached, e.g. the black ellipse Figure 2(f) calculated for h s = h s . In this situation, the impedance matching condition is satisfied.

The designed optimal structure was validated experimentally in a square cross-sectional impedance tube. The circular aerogel plate was cut by laser cutting and then inserted in a 3D-printed support manufactured by stereolithography (Form 2, Formlabs, UK). In addition, full wave numerical simulations by finite element method (FEM) were performed. For the FEM simulations, the plate was modeled as an elastic bulk plate of thickness h m considering a Kelvin-Voigt viscoelastic model and viscothermal losses were accounted for in the ducts using effective parameters as previously introduced for TMM calculations. Figure 3(a) shows the 3D printed system together with the aerogel plate before assembling. The measured absorption is shown in Fig. 3(b). A good agreement is observed between the measurements, FEM simulations and TMM predictions. The ripples observed in the experimental data are probably due to nonsymmetrical errors during the manufacturing of the circular plate, as well as to the fact that the plate is not perfectly clamped. Finally, Fig. 3(c) presents the Argand diagram of the reflection coefficient measured and calculated with the TMM. Both curves passe through the origin at a specific frequency that corresponds to the one at which the system is impedance matched.

In summary, we have designed and manufactured a resonant building block made of cavity backed aerogel clamped plates that is suitable and efficient for perfect sound absorbing panel. The experimental data agree with those predicted by both the one-dimensional TMM model and the FEM simulations. We have presented a universal methodology based on the complex representation of the reflection coefficient, i.e., its Argand diagram, to identify the lack or the excess of losses in the system. This tool can be used further to design complex metasurfaces for perfect sound absorption when the system cannot be evaluated at complex frequencies, thus help-ing in the rapid design of novel and efficient absorbing metamaterials.

SUPPLEMENTARY MATERIAL

See supplementary materials for animations and details of the dependence of the acoustic absorption on the different geometrical parameters.

FIG. 1 .

 1 FIG. 1. (color online) (a) Scheme of the panel under consideration constructed by a slit, the aerogel plate, and the cavity. (b) Schematic description of the unit cell. of height h s , are loaded by a circular clamped aerogel plate of radius r m and thickness h m backed by a cylindrical air cavity of the same radius and depth l c . The clamped aerogel plate plays the central role, as it represents the main source of intrinsic losses and mainly governs the resonance of the system. Although the theoretical calculations are only carried out for the building block of size L × a × a shown in Fig. 1(b), the generalization to the case of N resonator unit cell is straightforward.For wavelengths λ large enough compared to the thickness of the aerogel plate h m and neglecting the effects of rotary inertia and additional deflections caused by shear forces, the transverse plate displacement w m satisfies the Kirchhoff-Love wave equation[START_REF] Graff | Wave motion in elastic solids[END_REF] . The plate can be described by ρ, the density and D = Eh 3 m /12(1 -ν 2 ), the bending stiffness. E is the Young modulus and ν is the Poisson's ratio of the plate. Assuming an implicit time dependence e iωt , with ω the angular frequency, the viscoelastic behavior of silica aerogel can be modeled via a complex Young modulus, E = E 0 (1+iηω), where E 0 and η are the unrelaxed Young modulus and the loss factor respectively. In the subwavelength regime, the silica aerogel disk can be considered as a punctual resonant element located at (x, y) = (L/2, a/2) (note that h m λ 0 , where λ 0 is the wavelength in air). The acoustic impedance of the clamped circular cross-sectional plate thus takes the form31,32 ,

FIG. 2 .

 2 FIG. 2. (color online)Reflection coefficient, in logarithmic scale, as a function of the frequency and (a) slit length, L, and (b) cavity depth, lc. The rest of parameters are fixed to the optimal geometry (see main text). (c) Representation of the reflection coefficient in the complex frequency plane for the sample with optimized parameters. The lines show the trajectory of the zero when the corresponding geometrical parameter is modified. (d) and (e) show the absorption coefficient and the phase of the reflection coefficient respectively for three values of the slit thickness, h s , hs (black line) being the optimum value. (f) Argand diagram of the complex reflection coefficient from 0 to 1200 Hz, for the lossless structure (dashed circle), the case of perfect absorption (PA) geometry (black circle), the case h s = hs/2, (red circle) and the case h s = 2hs (blue circle). The small arrows indicate the trajectory from low to high frequencies.

FIG. 3 .

 3 FIG. 3. (color online) (a) Photograph of the experimental configuration. (b) Absorption as a function of the frequency. (c) Complex plane representation of the reflection coefficient. The small arrows indicate the trajectory from low to high frequencies.

ACKNOWLEDGMENTS

This work has been funded by the RFI Le Mans Acoustique, Région Pays de la Loire. N.J. acknowledges financial support from Generalitat Valenciana through grant APOSTD/2017/042. J.-P.G and V.R.G. gratefully acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project and the HYPERMETA project funded under the program Étoiles Montantes of the Région Pays de la Loire. J.S-D. acknowledges the support of the Ministerio de Economía y Competitividad of the Spanish government, and the European Union FEDER through project TEC2014-53088-C3-1-R.