Nicolas Aribert
email: nicolas.aribert@ensiacet.fr

Severine Camy
email: severine.camy@ensiacet.fr

Yolande Lucchese
email: yolande.pereslucchese@ensiacet.fr

Jean-Stéphane Condoret
email: jeanstephane.condoret@ensiacet.fr

Patrick Cognet
email: patrick.cognet@ensiacet.fr

FRIEDEL-CRAFTS ACYLATION IN SUPERCRITICAL CO 2

Keywords: Supercritical CO2, Friedel-Crafts acylation, zeolite

Friedel-Crafts acylation is among the most fundamental and useful reactions to yield aromatic ketones but it is one of the less acceptable in terms of unwanted polluting by-products or atom economy because of overconsumption of conventional catalyst used in stoichiometric quantities. This route is nevertheless still widely used in the fine chemicals industry. In recent years, awareness of the impact of industrial activities on the environment has lead chemists to work on new chemical routes, less dangerous and more environmentally friendly. We considered here the acylation of benzofurane derivative by an acid chloride. In this study, one of the first solutions was to replace conventional catalysts by reusable solid catalysts. Indeed, a wide variety of new solid catalysts, more efficient and less polluting, has now emerged (zeolites, ion-exchange resins…). These catalysts were first tested in "conventional" conditions, i.e., using an organic solvent (dichlorobenzene in our case), to determine the best one, in terms of reactivity, lifetime and reusability (regeneration). The zeolite Y was found the most appropriate. However, the use of an organic solvent remaining questionable, we have also considered the use of supercritical carbon dioxide as the solvent. Its inherent properties include non-flammability, mild critical conditions, tuneable solubility near to the critical point and very low environmental impact. The reaction was then operated using a continuous high pressure fixed bed and results concerning yield and selectivity are presented.

INTRODUCTION

The Friedel-Crafts acylation of aromatic compounds is among the most important and useful reactions for the synthesis of aromatic ketones. This acylation is an electrophilic substitution by acyl halides, anhydrides, carboxylic acids or esters with an aromatic substrate and is carried out using overstoichiometric amounts of metal halides as catalysts. Despite the high production of unwanted polluting and corrosive by-products, Friedel-Crafts chemistry is widely used in the manufacture of fine chemicals intermediates (pharmaceuticals, agrochemicals and fragrances) [START_REF] Gore | Friedel-Crafts and Related Reactions[END_REF] .

In recent years, the concept of "green chemistry" has lead chemical industries to develop new products, materials and chemical processes, less dangerous and more environmentally friendly. One of the first solutions is to find alternatives to the catalysts (Lewis acids: AlCl 3 , BF 3 , ZnCl 2 ... or Brønsted acids: H 2 SO 4 , HF ...). It can be done by the use of reusable solid catalysts [START_REF] Corma | Inorganic solid acids and their use in acid-catalyzed hydrocarbon reactions[END_REF][START_REF] Jasra | Solid acid catalyst for acylation of aromatics[END_REF] . Indeed, a wide variety of new solid catalysts, more efficient and less polluting can be proposed: zeolites, ion exchange resins, heteropoly acids... Such solid catalysts can be easily separated from liquid products and they offer in addition the possibility of carrying out Friedel-Crafts reactions using a continuous fixed bed reactor, rather than in batch reactors. For the same production target, continuous flow reactors can be smaller than the batch reactors, thus increasing safety and reducing capital cost.

Although carbon dioxide does not satisfy the social and economic requirements of the world's everincreasing population, its use as a solvent reaction is a good alternative to conventional organics solvents, because, in this case, it has no environmental impact. Up to date its usefulness and particularly in the field of chemical synthesis has been amply demonstrated [START_REF] Beckman | Supercritical and near-critical CO 2 in green chemical synthesis and processing[END_REF][START_REF] Rayner | The Potential of Carbon Dioxide in Synthetic Organic Chemistry[END_REF][START_REF] Ghosh | Polymerisation in Supercritical Carbon Dioxide: A Review[END_REF][START_REF] Patcas | Supercritical carbon dioxide as an alternative reaction medium for hydroformylation with integrated catalyst recycling[END_REF][START_REF] Arunajatesan | Fixed-bed hydrogenation of organic compounds in supercritical carbon dioxide[END_REF][START_REF] Hitzler | Friedel-Crafts alkylation in supercritical fluids: continuous, selective and clean[END_REF] . This article presents briefly the best catalyst found for use in a conventional solvent, and mainly the work made in supercritical conditions with the continuous high pressure fixed bed.

EXPERIMENTAL Materials

Different kinds of catalysts were used to perform the acylation: zeolites, ion exchange resins, metallic oxides, heteropoly acids (HPA) and clay. The choice of these catalysts was made according to their reported efficiency for different aromatic acylations [START_REF] Jasra | Solid acid catalyst for acylation of aromatics[END_REF][START_REF] Arata | Friedel-Crafts acylation of toluene catalyzed by solid superacids[END_REF][START_REF] Choudhary | Highly active Si-MCM-41 Supported Ga 2 O 3 and In 2 O 3 Catalysts for Friedel-Crafts-type benzylation and acylation reactions in Presence or Absence of Moisture[END_REF] . In the context of this study, which is done through a collaborative industrial contract, it is not possible to disclose the exact name of the involved molecules.

Reaction procedure

Conventional solvent: The acylation reaction was carried out in a 250 cm 3 batch reactor equipped with a condenser and magnetic stirrer where the acyl choride (AC), the aromatic substrate (AS) and the solvent (dichlorobenzene, S) were mixed in the reactor. The catalyst (zeolites were used directly after calcination * ) was introduced in the reactor and the medium was heated up to the reaction temperature. The resulting suspension was stirred under reflux for 6h. The initial amounts of each compound are given for each reaction. Samples were taken out periodically during the reaction and assayed by HPLC (Waters 600E apparatus with Symmetry C8 column). * Calcination procedure: the sample of catalyst is heated in a furnace with a heating rate of 4°C min -1 from room temperature to 500°C and then maintained at this temperature during 6 hours Supercritical CO 2 :

The catalytic acylation reactions in supercritical conditions were performed in a continuous fixed-bed reactor, consisting of: (i) a medium supply unit, (ii) a reaction unit; and (iii) an analytical loop (Fig. 1). The medium supply unit comprised two HPLC pump, one for the liquid reactant mixture with a given ratio AS/AC and one for liquid CO 2 . A static mixer (heated around 60°C) before the reaction unit allows a good mixing of reactants and CO 2 . Our system allows in-situ calcination of the catalyst, where nitrogen is used at a flow-rate of 250 cm 3 /min (STP). The calcination was performed at 500°C during 6 h. The reaction unit comprised a tubular reactor with an internal cartridge (14mL, 0.50 m length, 6mm inner diameter) loaded with the solid catalyst, a safety rupture disc, a pressure transducer and a K-type thermocouple located inside the catalytic bed, near the outlet. In addition a PID temperature controller and electrically heated shells allow operating at the desired temperature. Continuous reactors for SCFs have the advantage over batch reactors that they do not require depressurisation to feed in the reactants or to recover the products. Samples can be taken out periodically during the reaction by using a sample loop of known volume. The obtained sample is then decompressed in the HPLC solvent and the loop is rinsed with 10 times the volume of the loop. The HPLC apparatus is the same than for the reaction in conventional solvent.

The procedure for operation of the continuous reaction is as follow:

The cartridge is loaded with 4.3g of catalyst and calcined during the first day. The next day, the CO 2 is pumped and then the reactor is heated up between 150-180°C, according to the experiment. When the temperature at the outlet is stable the mixture of reactants is pumped into the system. Samples are taken each half hour. The flow of CO 2 was fixed to 1g/min or 2g/min, and the flow of reactants to 0.1g or 0.05g depending from experiments. The pressure was fixed at 300 bar, to compensate the loss of dissolving power, induced from the use of high temperature.

RESULTS AND DISCUSSION Conventional solvent

Table 1 shows the results of acylation at 180°C over various solid acid catalysts. Among these different kinds of catalysts, Zeolite Y and Indium oxide seem to be the most appropriate solid catalysts. Indeed, K-10 clay, HPA and H-Mor are completely inactive. Zeolite β, Amberlyst-15, bismuth oxide and iron oxide are barely active. Even if gallium oxide is better, its activity is not sufficient enough.

Indium oxide showed a very good activity, nearly 46% in 3 hours, but reusability tests have shown that it was not reusable. During the reaction, we observed that the oxide was solubilized in the medium, which is redhibitory for our studies. The same reaction with the acid form or the anhydrid form does not give any yield.

Finally, only the zeolite Y (SiO 2 /Al 2 O 3 =15) give interesting yield and behaviour during the reaction. Some reactions have been carried out using zeolite Y with different ratio SiO 2 /Al 2 O 3 (6 and 200), but in the same conditions, the maximal yield is smaller (respectively 18% and 1%).

According to the industrial aspect, the reusability of the catalyst is a decisive parameter. Then other reactions have been carried out to optimise the operating conditions (ratio AS/AC, catalyst loading, reusability…) before using it in the supercritical fixed bed reactor. The figures 2 and 3 summarize the best results in terms of yield and reusability of the zeolite Y(15). The maximal yield reaches 80% with 2g of zeolite in 6 hours and the reactant ratio is the main parameter.

The best one was around 1 mol of substrate for 1.5 mol of acylating agent. The results are equivalent when the ratio AS/AC is increased to 1/4. Although with an excess of AC superior to 1/4 (ratio 1/10 fig 2), the activity of the zeolite seems to be inhibited: active sites may be saturated by the AC. In fact this inhibition already appeared with a ratio of 1/4 at the end of the reaction (around 4h). The kinetic stabilizes unlike the experiment carried out with a 1/1.5 ratio, where the kinetics slowed down but did not stop.

In term of reusability, figure 3 shows that the deactivation of the zeolite is relatively fast. After a single use and a simple filtration on a fritted disc the yield drops significantly to 18%. The deactivation of the zeolite has many reasons, notably due to the deposition of heavy products usually termed as "coke". According to Richard et al [START_REF] Richard | Zeolite catalyzed acylation of heterocyclic compounds. Part II. Acylation of benzofuran over Y zeolites[END_REF] this "coke" probably results from the acylation of compounds obtained by condensation of aromatic substrate, thus blocking the actives sites. As expected, the calcination of the used zeolite allows complete recovery of its catalytic activity. Indeed, the need for zeolite regeneration was already observed by Guignard et al and Moreau et al [START_REF] Guignard | Acylation of veratrole by acetic anhydride over Hβ and HY zeolites: Possible role of Di-and Triketone by-products in the deactivation process[END_REF][START_REF] Moreau | Acetylation of dimethoxybenzenes with acetic anhydride in the presence of acidic zeolites[END_REF] . In this kind of experiments the regeneration (here by calcination) of the catalyst becomes a prominent parameter.

Supercritical conditions

These previous experiments in a conventional solvent, the dichlorobenzene, have indicated which solid catalyst has the researched properties: good activity, reusable and which recovers all its activity by calcination.

Thus zeolite Y(15) was tested, first in its powder form. Preliminary calculations, using the Darcy law, have estimated the expected pressure drop (16 bars with a powder around 6-8µm) and this value was found acceptable for an operating pressure above 200 bar. Experiments confirmed the value of the predicted pressure drop. However use of zeolite powder proved to be very problematic, because all reaction runs ultimately lead to clogging of the reactor, very likely because these catalyst particles were indeed too small to be used in a fixed-bed reactor. Although duration of experiments were note sufficient (a few hours) to be really representative, a 10% yield was observed (2g/min of CO 2 , 0.05g/min of reactants with a ratio AS/AC=1/2, T=180°C, P=200 bar).

Fortunately, zeolite H-Y is also available in pelletized form: 1.5 mm in diameter and 2-5mm in length pellets, made with 20% inactive binder clay ("Montmorillonite K-10"). These pellets are more suitable and, indeed, their use did not lead to clogging inside the reactor. The pressure drop of the bed of pellets is expected to be much smaller, the observed pressure drop being mainly due to the fritted disk at the ends of the tube, and did not exceed 5 bar. 4 presents the time course of the product concentration at the output of the fixed bed. The influence of the calcination pre-treatment on the activity of the zeolite, where 15% yield against 5% with the non-calcined zeolite is observed. In a conventional solvent, the same behaviour was observed, i.e., an increase of yield when the zeolite is calcined. But Figure 4 also shows that the activity of the calcined zeolite declines rapidly after 60 minutes (maximal yield reached at this time). The study of the reusability of the zeolite by calcination is now under progress. Different operating conditions are planned to increase the yield and mainly to increase the life time of the zeolite.

CONCLUSION

From experiments in a conventional organic solvent, zeolite Y(15), among all the tested catalysts, was shown to be the only solid catalyst having the researched properties for this reaction (activity of the zeolite) and for its operation in a continuous fixed bed reactor (solid catalyst reusable by calcination). The different experiments carried out with this zeolite have given promising results, where few by-products were generated. The only detected one was the anhydride form of the AC, which could be easily eliminated by simple post reactional hydrolysis of the medium.

The experiments with supercritical CO 2 as the solvent, in a continuous fixed bed, yielded interesting and encouraging results, even if rapid deactivation of the zeolite was observed. Note that a specific formulation of the catalyst (pelletized form) was necessary to operate the fixed bed rector without clogging. This evolution has to be understood and improved very likely by varying temperature or pressure. In the continuous fixed bed, improvement of the yield can be done by increase of the residence time. Batch operation of the reaction in a high pressure agitated vessel is also under progress to better understand the different phenomenon, by direct comparison with results of batch experiments in dichlorobenzene.

Figure 1 .

 1 Figure 1. Scheme of the continuous high pressure fixed bed (length 50 cm, inner diameter 6mm)

Table 1 :

 1 The acylation over different types of solid acid catalysts

	Catalyst (SiO 2 /Al 2 O 3 )	Surface area (m	2 /g) Yield	a (time)
	360 HUA H-Y(15)		600	45% (6h)
	930 NHA H-β (27)		630		4% (6h)
	TZM-1013 H-Mor (12)		475	NR	b (6h)
	Resin	Amberlyst-15		53	6,3% (6h)
	HPA	H 3 [P(W 3 O 10 ) 4 ]		4,6	0,2% (6h)
		Ga 2 O 3	10-50 16,3% (3h)
	Oxide	In 2 O 3 Fe 2 O 3	25-50 45,5% (3h) 30-50 6,5% (3h)
		Bi 2 O 3	10-50	1,6% (3h)
	Clay	Mont. K-10		220	NR	b (6h)
	a Reaction conditions: 10mmol of AS; 10 mmol of AC; 50mL of S; catalyst
	loading 1,0g; temperature 180°C

b NR: No Reaction

• calcined zeolite.


Laurène Despènes

Sébastien Elgue

Christophe Gourdon

Michel Cabassud

Jean Marie Autret

Transposition to continuous and process intensification

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID : 2980

To link to this article : URL : http://www.gpe-epic2009.org/ To cite this version : Autret, Jean Marie and Cabassud, Michel and Despenes, Laurent and Elgue, Sébastien and Gourdon, Christophe Transposition to continuous and process intensification of a pharmaceutical production. In: 2nd International Congress on Green Process Engineering,2nd European Process Intensification Conference, 14-17 June 2009, Venise,Italie.

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TRANSPOSITION TO CONTINUOUS AND PROCESS INTENSIFICATION OF A PHARMACEUTICAL PRODUCTION

Abstract

In the field of Process Intensification, manufacturers offer many technologies of heat exchanger reactors in terms of design, of material and of operating conditions range which make the choice of the optimal solution difficult to be performed. Moreover, such apparatuses combine a continuous operating with strongly coupled features of heat transfer, hydrodynamics, mixing, mass transfer and reaction. In this way and to assess the feasibility and potentialities of applications carried out in this kind of apparatus, a specific methodology has been developed by the Laboratoire de Génie Chimique. This methodology aims to define the suitable process and the associated operating conditions which comply for a given application with Process Intensification criteria. To achieve this goal, the methodology, based at the same time on experimental and theoretical methods, defines and limits the experiments to be carried out to obtain the optimal process. Such a feature is of particular interest in application cases which involve productivity, operating and safety issues. The methodology could be divided in three different parts. The first one deals with the equipment characterisation in terms of hydrodynamics, of mixing and of thermal exchanges and results in an accurate modelling of the considered reactors. The second part gathers all the aspects related to the considered application: physical properties of components, reaction kinetics, heat generated, etc. The last step of the methodology combines the results of the twofirst parts in a simulation/optimisation tool that provides relevant information in order to define the suitable intensified process (optimal design) and the associated operating conditions (optimal control). From a production viewpoint, this tool also offers information on dynamics that allows the safety, the start-up and shut-down operations and the regulation system to be studied. Related to this methodology, the present study aims to transpose to continuous and to intensify a Pierre Fabre's pharmaceutical application. In fact, this application currently carried out in batch offers productivity limitations that could be get round using a continuous intensified reactor. In this way, a complete reaction characterisation based on calorimetric experiments has been performed and provided to the optimisation tool. The results highlight unconventional conditions compared to the batch operating and the need to control the pH level all along the reactor by side-injections. The application exothermicity also leads to the use of a very efficient heat exchanger reactor made of silicon carbide, to maintain an isothermal temperature profile.


R Ionescu
email: raluca.ionescu@ensiacet.fr

Y Lucchese

S Camy

D Riboul
email: david.riboul@ensiacet.fr

J-S Condoret ♦ Bp

HETEROGENEOUS CATALYSIS IN SUPERCRITICAL CO 2 CONDITIONS APLICATION IN DIMETHYL CARBONATE SYNTHESIS

Keywords: dimethyl carbonate, cyclic carbonate, supercritical carbon dioxide

In the present work, the development of a new catalytic system based on the synthesis of dimethyl carbonate (DMC) starting from methanol and carbon dioxide in supercritical conditions was investigated. Various metallophthalocyanines (MPc: ZnPc, CoPc, AlClPc, CuPc(SO3Na)4, CoPc(SO3H)2-3, ZnPc(SO3H)2-3) were used as catalysts. In a first step reactions were carried out in homogeneous phase. No reaction was observed with non substituted phthalocyanines. In the case of Co and Zn substituted by SO3H moieties a DMC formation was observed. Tests are carried out in a 95 mL autoclave; the organics were separated from the catalyst via a vacuum distillation with a liquid nitrogen trap. The reaction products are analyzed by a GC-FID and GC-MS. Key-words.

INTRODUCTION

From the appearance of the concept of "green chemistry" in 1990 and the emergence of twelve recommendations that now govern the research in the field of chemical industry, the work on the synthesis of green molecules have turned to new high performance techniques that avoid the use of hazardous products. In the specific case of dimethyl carbonate, the global production is about 100 Kt/year [1] but could easily reach 1Mt/year, if its full potential was exploited. Indeed, currently, the DMC, a non-toxic, non-carcinogenic, nonmutagenic reagent, has a great environmental interest. For example it is used as source of polycarbonates [2,3]. In the EniChem process for the synthesis of aromatic polycarbonates, the use of the DMC and diphenylcarbonate (DPC) as intermediates avoids, not only the use of phosgene, but also the use of methylene chloride as a solvent, and prevents the formation of NaCl as a co-product. DMC is also employed in alkylation and methoxycarbonilation reactions in the pharmaceutical industry [4]. For example, the mono-Cmethylation of aryl derivatives, conducted in DMC, gives higher conversion rate and selectivity, respectively 90% and 99%, to produce anti-inflammatory precursors as naproxen, ketoprofen and ibuprofen [5,6]. It replaces in this case, very toxic compounds such as phosgene, dimethyl sulphate and alkyl halides. As oxidizer, it reacts with cyclic aliphatic ketones to form unsaturated diesters, intermediates in the production of polyesters and polyamides [7]. It has also solvent utilizations. Dimethyl carbonate is an alternative to the use of solvents such as acetone and other carbonyl compounds.

The limitation of the production of DMC is due to the existing synthesis technologies, which are extremely dangerous: the use of phosgene, as for the SNPE process, the mixture of carbon monoxide and oxygen, as for the EniChem process [8], nitric oxide and carbon monoxide, as for the UBE process [9].

To bring it into line with the demands of the green chemistry, research has been carried out to find a cleaner way of synthesis that could be also applied on an industrial scale. Other important environmental problem is the increasing quantity of carbon dioxide, one of the Green House gases, in the atmosphere. Various studies have shown that an utilization of carbon dioxide in the chemical industry together with its storage will help in decreasing the danger concerning the global heating. The reaction between methanol and the CO 2 molecule in the DMC synthesis showed a great interest in chemistry since the only by-product is water (equation 1). One of the most studied series of catalysts is the tin complexes [10,11,12,13] but the main problem is the water contamination of the catalyst witch annihilates its activity. Although the acid-base properties of ZrO 2 catalysts [14] are of big interest in the reaction mechanism and no poisoning effect by water, there is still the low yield inconvenient. Another example is the nickel acetates/ methyl iodide system. Although a base co-catalyst is used the yields are low and the reaction generates HI.

2 CH 3 OH + CO 2 catalyst C O O O CH 3 C H 3 + H 2 O
Research has been developed in order to determine the methoxycarboxylation mechanism and to eliminate the thermodynamic constraint and so increasing the yield in DMC. Since carbon dioxide is hardly reactive an efficient catalytic system must be found to activate the CO 2 molecule and make it react with methyl and methoxy moieties. Our work is based on the research of a new heterogeneous catalytic system for the methoxycarboxylation process. The proposed catalyst is a transition metal phthalocyanine. These compounds are considered as non-dangerous [15] and present a high stability towards acids, bases and temperature. The Metal-Nitrogen bond is strong [16] preventing the loss of metal ions in solution, and hence the risk of pollution by heavy metals.

MPcs find interesting properties as the tunable solubility by changing the R moiety with groups like: H, SO2Cl, X, OH, OCH3, CN, NO2, SO3Na…Along with the solubility there is also a tunable reactivity depending on the metal center: Al, Fe, Mn, Co, Ni, Cu, Zn. Figure 1 shows the configuration of the metallophtahlocyanine molecule.

MPcs have found many applications, such as dyes [17] in paints, inks, textiles, photosensitizers in cancer treatment [18], bleaching agents for detergents [19,20], desulphurization catalysts for petroleum products and oxidation catalysts for organic pollutants.

As part of the synthesis of organic carbonates, there are several examples of catalytic systems based on metallophthalocyanines. Thus, copper [21] and aluminum [22] phthalocyanines catalyze the formation of cyclic carbonates from propylene oxide and carbon dioxide. Reaction rates are high but with certain metal centers the reactivity increases if a co-catalyst (ionic liquid, tertiary amine) is added. In addition, there is evidence that metallophthalocyanines and similar complexes activate carbon dioxide and make it react with different substrates [23,24,25].

The aim of our study is to perform the applicability of this catalyst in the reaction between methanol and carbon dioxide under supercritical conditions. Supercritical fluids exhibit a tunable density by controlling its temperature and pressure. In our case the utilization of supercritical conditions is motivated by the low solubility of metallophthalocyanines in methanol. It is thought that by searching appropriate conditions (temperature and pressure) to get the reaction system in a monophasic state the catalyst solubility will increase significantly.

EXPERIMENTAL Materials

All chemicals used were of reagent grade. AlClPc, ZnPc, CuPc(SO 3 Na) 4 were delivered by Acros Organics and CoPc by… CoPc(SO 3 H) 2-3 , ZnPc(SO 3 H) 2-3 were kindly supplied by Europhthal. Methanol and carbon dioxide were delivered from Panreac and Linde respectively.

Tests were carried out in a TopIndustry high pressure autoclave with an inner volume of 95 mL and a magnetically stirrer, 1600 rpm. The maximum temperature and pressure were of 250°C and 340 bar respectively. The reactor was electrically heated and equipped with a Temperature Security High to control the reaction temperature. An ISCO pump was used to control the carbon dioxide volume that was injected in the reactor. The high pressure system layout and photography are shown in figure 2 and figure 3 respectively.

The separation of the catalyst from the organic products was realized with a vacuum distillation vessel using a liquid nitrogen double trap. Analyses were realized by GC using a capillary column Restek 502.2 (30 m, Ø 0.25 mm, 1.4 µ film) on a Trace GC Ultra gas chromatography supplied by Thermo Scientific. Indeed, a flame ionization detector and a TriPlus autosampler were used.

Preliminary high pressure tests

First tests were carried out in homogeneous. Calculations have been made in order to choose reagents amounts and to establish reaction conditions (temperature and pressure). This preliminary procedure is of great importance because realizing a monophasic reaction system increases the contact between the reagents and the possibility of getting great conversion. Typical procedure: 20 mL (500 mmol) of methanol and 8.8 mg (0.01 mmol) of catalyst were charged into the autoclave. Then the reactor was heated up to 140°C. The ISCO pump was charged in liquid CO 2 followed by a compression to the desired pressure (220 bar). Whereafter, carbon dioxide was injected into the autoclave. After 7 hours the system was cooled down and depressurized with the sampler extractor. Then the reactor was opened to recuperate the reaction products. The organics were separated from the catalyst by vacuum distillation and then submitted to a GC analysis.

RESULTS AND DISSCUTION

A first observation is that the solubility of all metallophtahlocyanines employed in preliminary tests varies as follows: ZnPc, CoPc, AlClPc, CuPc(SO 3 Na) 4 << CoPc(SO 3 H) 2-3 , ZnPc(SO 3 H) 2-3 . The H and SO 3 Na substituted phthalocyanines remain in suspension in the presence of methanol while the SO 3 H moiety gets the Co and Zn amount totally soluble in CH 3 OH.This demonstrates that the aromatic ring substitute plays an important role in phthalocyanine solubility. The enhancement of solubility means a better contact between the reagents and this is also a reason for using supercritical fluid properties.

In previous papers metallophthalocyanines showed good activity in the cyclic carbonates synthesis. An enhancement of the activity is brought by the addition of a base co-catalyst like a tertiary amine or an ionic liquid. In this work it is thought that the reaction mechanism passes through the same steps: the coordination of OCH 3 moiety on the metallic center of the catalyst, then the activation of CO 2 and its insertion in the metal-OCH 3 bond. The last step of the mechanism will be the breaking of CH 3 -O bond and the formation of DMC molecule. In addition, breaking of methyl-oxygen bond is more easily realized in the presence of a base cocatalyst. In this study is analyzed the activity of proposed catalysts without co-catalyst.

After separation of organic species a GC-FID followed. Results and some chromatograms are From the first results we can easily observe that the methalophthalocyanines non-substituted or having SO 3 Na moieties doesn't catalyze the reaction; the chromatography results show no reaction product than methanol. By the way the two Co and Zn methallophthalocyanines substituted by SO 3 H moieties show a certain activity, so that we can see that the turnover frequency number is of 160 and 96 h -1 respectively. We may say that we can get an activation of the reaction only with soluble catalysts. The yields are not very high but the reaction seems to be catalytic since the molar ratio of DMC versus the catalyst is 50/1 in the case of CoPc(SO 3 H) 2-3 .

The chromatogram of organic products issued from the test where CoPc(SO 3 H) 2-3 was used is displayed in the figure 4. We can see that there are many signals in this chromatogram but only two of them are reaction products. A comparison of DMC, methanol and test chromatogram shows that the signal at 2.98 min represents a rezidu from dimethyl carbonate. The signal at 3.23 min is an acetonitrile impurity coming from the methanol as well as the 5.35 min signal. The signal that appears at 6.27 min represents the dimethyl carbonate molecule. Another important signal is at 3.92 min and it is surely a byproduct of the reaction. The determination of this secondary product is o great importance in increasind the dimethyl carbonate yield. Furthermore, with this information reaction conditions and amounts of reagents can be changed to get reaction selective. For bringing information about this second organic product a GC-MS analysis is in now in progress.

Once the second product is determined, a mechanistic study will help for the understanding the reaction. We don't know yet if either the presence of the dimeric form of the phthalocyanines decreases the catalyst activity or the difficulty of reaching the last step of the reaction (breaking the O-CH 3 bond in methanol to generate a CH 3 moiety).

CONCLUSIONS AND PERSPECTIVES

In this work, catalytic activity of ZnPc(SO 3 H) 2-3 and CoPc(SO 3 H) 2-3 towards the dimethyl carbonate formation from methanol and supercritical carbon dioxide has been evidenced. Other metallophthalocyanines, less soluble in methanol like ZnPc, CoPc, AlClPc, CuPc(SO 3 Na) 4 show no activity in the proposed reaction conditions. Along with the dimethyl carbonate, another secondary product has been found but its identification by GC-MS analysis is in progress. Even the yield is low the reaction is catalytic and a mechanistic study must be undertaken. Furthermore, the modulation of the reaction conditions along with the addition of a base co-catalyst can achieve a greater conversion of methanol.

Catalyst

  Figure 1. Metallophthalocyanine

Figure 3 .

 3 Figure 3. Photography of magnetic stirrer (up) and high pressure system for DMC synthesis (bottom)

Figure 4 .

 4 Figure 4. Chromatogram of reaction products issued from the test where CoPc(SO 3 H) 2-3 was used

Table 1 .

 1 TOF (h -1 ) The formation rates of dimethyl carbonate from CH 3 OH and CO 2 with various metallophthalocyanine complexes (reaction conditions: 220 bar, 140°C, 7h, 500mmol CH 3 OH, 0.01mmol catalyst and 300mmol CO 2 )

	Cu (SO 3 Na) 4	No reaction
	AlClPc	No reaction
	ZnPc	No reaction
	CoPc	No reaction
	ZnPc(SO 3 H) 2-3	96
	CoPc(SO 3 H) 2-3	160


Fabienne Comte
email: fabienne.comte@parisdescartes.fr

Jan Johannes
email: johannes@statlab.uni-heidelberg.de

Adaptive estimation in circular functional linear models

Keywords: Orthogonal series estimation, model selection, derivatives estimation, mean squared error of prediction, minimax theory AMS 2000 subject classifications: Primary 62G05, secondary 62J05, 62G08

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y 1 , . . . , Y n are modeled in dependence of 1periodic, second order stationary random functions X 1 , . . . , X n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L 2 -risk. This means that we provide adaptive estimators of both β and its derivatives.

Introduction

Functional linear models have become very important in a diverse range of disciplines, including medicine, linguistics, chemometrics as well as econometrics (see for instance [START_REF] Ramsay | Functional Data Analysis[END_REF] and [START_REF] Ferraty | Nonparametric Functional Data Analysis: Methods, Theory, Applications and Implementations[END_REF], for several case studies, or more specific, [START_REF] Forni | Let's get real: A factor analytical approach to disaggregated business cycle dynamics[END_REF] and [START_REF] Preda | Pls regression on a stochastic process[END_REF] for applications in economics). Roughly speaking, in all these applications the dependence of a response variable Y on the variation of an explanatory random function X is modeled by

Y = 1 0 β(t)X(t)dt + σε, σ > 0, (1.1)
for some error term ε. One objective is then to estimate nonparametrically the slope function β based on an independent and identically distributed (i.i.d.) sample of (Y, X).

1

In this paper we suppose that the random function X is taking its values in L 2 [0, 1], which is endowed with the usual inner product •, • and induced norm • , and that X has a finite second moment, i.e., E X 2 < ∞. In order to simplify notations we assume that the mean function of X is zero. Moreover, the random function X and the error term ε are uncorrelated, where ε is assumed to have mean zero and variance one. This situation has been considered, for example, in [START_REF] Cardot | Spline estimators for the functional linear model[END_REF], Müller and Stadtmüller [2005] or most recently [START_REF] James | Functional linear regression that's interpretable[END_REF]. Then multiplying both sides in (1.1) by X(s) and taking the expectation leads to g(s) := E[Y X(s)] = 1 0 β(t) cov(X(t), X(s))dt =: [Γβ](s), s ∈ [0, 1],

(1.2)

where g belongs to L 2 [0, 1] and Γ denotes the covariance operator associated to the random function X. We shall assume that there exists a unique solution β ∈ L 2 [0, 1] of equation (1.2). Estimation of β is thus linked with the inversion of the covariance operator Γ and, known to be an ill-posed inverse problem (for a detailed discussion in the context of inverse problems see chapter 2.1 in [START_REF] Engl | Regularization of inverse problems[END_REF], while in the special case of a functional linear model we refer to [START_REF] Cardot | Spline estimators for the functional linear model[END_REF]).

In this paper we consider a circular functional linear model (defined below), where the associated covariance operator Γ admits a spectral decomposition {λ j , ϕ j , j 1} given by the trigonometric basis {ϕ j } as eigenfunctions and a strictly positive, possibly not ordered, zero-sequence λ := (λ j ) j 1 of corresponding eigenvalues. Then the normal equation can be rewritten as follows

β = ∞ j=1 [g] j λ j
• ϕ j with [g] j := g, ϕ j , j 1.

(1.3)

For estimation purpose, we replace the unknown quantities g j and λ j in equation (1.3) by their empirical counterparts. That is, if (Y 1 , X 1 ), . . . , (Y n , X n ) denotes an i.i.d. sample of (Y, X), then for each j 1, we consider the unbiased estimator

[ g] j := 1 n n i=1 Y i [X i ] j , and
λ j := 1 n n i=1 [X i ] 2 j
with [X i ] j := X i , ϕ j for [g] j and λ j respectively. The orthogonal series estimator β m of β is then defined by

β m := m j=1 g j λ j
• ½{ λ j 1/n} • ϕ j .

(1.4)

Note that we introduce an additional threshold 1/n on each estimated eigenvalue λ j , since it could be arbitrarily close to zero even in case that the true eigenvalue λ j is sufficiently far away from zero. Moreover, the orthogonal series estimator keeps only m coefficients; this is an alternative to the popular Tikhonov regularization (c.f. [START_REF] Hall | Methodology and convergence rates for functional linear regression[END_REF]), where in (1.3) the factor 1/λ j is replaced by λ j /(α + λ 2 j ). Thresholding in the Fourier domain has been used, for example, in a deconvolution problem in [START_REF] Mair | Statistical inverse estimation in Hilbert scales[END_REF] or [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] and coincides with an approach called spectral cut-off in the numerical analysis literature (c.f. [START_REF] Tautenhahn | Error estimates for regularization methods in Hilbert scales[END_REF]).

In this paper we shall measure the performance of an estimator β of β by the F ω -risk, that is E ββ 2 ω , where for some strictly positive sequence of weights ω := (ω j ) j 1

f 2 ω := ∞ j=1 ω j | f, ϕ j | 2 for all f ∈ L 2 [0, 1].
This general framework allows us with appropriate choices of the weight sequence ω to cover the estimation not only of the slope parameter itself (c.f. [START_REF] Hall | Methodology and convergence rates for functional linear regression[END_REF]) but also of its derivatives as well as the optimal estimation with respect to the mean squared prediction error (c.f. [START_REF] Cardot | Spline estimators for the functional linear model[END_REF] or [START_REF] Crambes | Smoothing splines estimators for functional linear regression[END_REF]). For a more detailed discussion, we refer to [START_REF] Cardot | Thresholding projection estimators in functional linear models[END_REF]. It is well-known that the obtainable accuracy of any estimator in terms of the F ω -risk is essentially determined by the regularity conditions imposed on both the slope parameter β and the eigenvalues λ. In the literature the a-priori information on the slope parameter β such as smoothness is often characterized by considering ellipsoids (see definition below) in L 2 [0, 1] with respect to a weighted norm • γ for a pre-specified weight sequence γ. Moreover, it is usually assumed that the sequence λ of eigenvalues of Γ has a polynomial decay (c.f. [START_REF] Hall | Methodology and convergence rates for functional linear regression[END_REF] or [START_REF] Crambes | Smoothing splines estimators for functional linear regression[END_REF]). However, it is well-known that this restriction may exclude several interesting cases, such as an exponential decay. Therefore, we do not impose a specific form of a decay. It is shown in [START_REF] Johannes | Nonparametric estimation in circular functional linear model[END_REF] that the estimator β m given in (1.4) is optimal in a minimax sense if the parameter m = m(n) is appropriately chosen. Roughly speaking, the introduction of a dimension reduction implies a bias in addition to the classical variance term which leads the statistician to perform a compromise. The optimal choice of the dimension parameter m requires an a-priori knowledge about the sequences γ and λ, which is unknown in practice. However, useful elements of this previous work are recalled in Section 2.

Our aim in this paper, is to provide a data driven method to select the dimension parameter m, in such a way that the bias and variance compromise is automatically reached by the resulting estimator. The methodology is inspired by the works of [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], now extensively described in [START_REF] Massart | Concentration inequalities and model selection[END_REF] whose results, like ours, are in a non asymptotic setting. By re-writing the estimator β m as a minimum contrast estimator over the function space S mcalled modellinearly spanned by ϕ 1 , . . . , ϕ m , we can propose a model selection device by defining a penalty function. We obtain a selected m in an admissible set of values of m. We first define and study in Section 3, the resulting estimator β m with deterministic penalty and deterministic set of admissible m's: this requires to assume that the degree of ill-posedness of the problem is known. In other words, information are first supposed to be available about the order of the decay of the eigenvalues λ j . This study gives the tools to the next and final step: we define in Section 4 a completely data driven estimator, built by using a random penalty function and a random set of admissible dimensions m. We can provide a general risk bound for this estimator and show that it can automatically reach the optimal rate of convergence, without requiring any a-priori knowledge. All proofs are gathered in the Appendix section.

2 Background to the methodology.

Notations and basic assumptions

Circular functional linear model. In this paper we suppose that the regressor X is 1-periodic, that is X(0) = X(1), and second order stationary, i.e., there exists a positive definite covariance function c :

[-1, 1] → R such that cov(X(t), X(s)) = c(t -s), s, t ∈ [0, 1].
Then it is straightforward to see that the covariance function c(•) is 1-periodic too. In this situation applying the covariance operator Γ equals a convolution with the covariance function. Since c(•) is 1-periodic it is easily seen that due to the classical convolution theorem, the eigenfunctions of the covariance operator Γ are given by the trigonometric basis

ϕ 1 (s) :≡ 1, ϕ 2k (s) := √ 2 cos(2πks), ϕ 2k+1 (s) := √ 2 sin(2πks), s ∈ [0, 1], k 1
and the corresponding eigenvalues satisfy

λ 1 = 1 0 c(s)ds, λ 2k = λ 2k+1 = 1 0 cos(2πks)c(s)ds, k 1.
Notice that the eigenfunctions are known to the statistician and only the eigenvalues depend on the unknown covariance function c(•), i.e., have to be estimated.

Moment assumptions. The results derived below involve additional conditions on the moments of the random function X and the error term ε, which we formalize now. Let X be the set of all centered 1-periodic and second order stationary random functions X ∈ L 2 [0, 1] with finite second moment, i.e., E X 2 < ∞, and strictly positive covariance operator Γ.

If λ := (λ j ) j 1 denotes the sequence of eigenvalues associated to Γ, then given X ∈ X the random variables {[X] j / λ j , j ∈ N} are centered with variance one. Here and subsequently, we denote by X k η , k ∈ N, η 1, the subset of X containing only random functions X such that the k-th moment of the corresponding random variables [X] j / λ j , j ∈ N are uniformly bounded, that is

X k η := X ∈ X such that sup j∈N E [X] j / λ j k η .
It is worth noting that in case X ∈ X is a Gaussian random function the corresponding random variables [X] j / λ j , j ∈ N, are Gaussian with mean zero and variance one. Hence, if η 3 then any Gaussian random function X ∈ X belongs also to X k η for each k ∈ N.

Minimal regularity conditions. Given a strictly positive sequence of weights w := (w j ) j 1 , denote by F c w the ellipsoid with radius c > 0, that is,

F c w := f ∈ L 2 [0, 1] : ∞ j=1 w j | f, ϕ j | 2 =: f 2 w c .
Furthermore, let F w := {f ∈ L 2 [0, 1] : f 2 w < ∞} and f, g w := ∞ j=1 w j f, ϕ j ϕ j , g . Note that this weighted inner product induces the weighted norm • w .

Here and subsequently, given strictly positive sequences of weights γ := (γ j ) j 1 and ω := (ω j ) j 1 we shall measure the performance of any estimator β by its maximal F ω -risk over the ellipsoid F ρ γ with radius ρ > 0, that is sup β∈F ρ γ E ββ 2 ω . We do not specify the sequences of weights γ and ω, but impose from now on the following minimal regularity conditions.

Assumption 2.1. Let ω := (ω j ) j 1 and γ := (γ j ) j 1 be positive sequences of weights with ω 1 = 1 and γ 1 = 1 such that (1/γ j ) j 1 and (ω j /γ j ) j 1 are non increasing zero-sequences.

Note that under Assumption 2.1 the ellipsoid F ρ γ is a subset of F ρ ω , and hence the F ω -risk a well-defined risk for β. Roughly speaking, if F ρ γ describes p-times differentiable functions, then the Assumption 2.1 ensures that the F ω -risk involves maximal s < p derivatives.

Minimax optimal estimation.

The objective of the paper is to construct an estimator which attains the minimal rate of convergence of the maximal F ω -risk over the ellipsoid F ρ γ for wide range of sequences γ and ω satisfying Assumption 2.1, without using an a-priori knowledge of neither γ nor ρ. Therefore, let us first recall a lower bound which can be found in [START_REF] Johannes | Nonparametric estimation in circular functional linear model[END_REF]. Let m * := (m * n ) ∈ N for some △ 1 be chosen such that

1/△ γ m * n n ω m * n m * n j=1 ω j λ j △, i.e. (1/n) m * n
j=1 ω j /λ j and ω m * n /γ m * n have the same orders. Given an i.i.d. n-sample of (Y, X) obeying (1.1) with σ > 0 and X ∈ X with associated sequence of eigenvalues λ, we have then for any estimator β that sup

β∈F ρ γ E β -β 2 ω 1 4 △ min σ 2 2 , ρ △ max(ω m * n /γ m * n , 1/n) for all n 1. (2.1)
On the other hand consider the estimator β m defined in (1.4) with dimension parameter m = m * n . If in addition X ∈ X 16 ξ , then it is shown in [START_REF] Johannes | Nonparametric estimation in circular functional linear model[END_REF] that there exists a numerical constant C > 0 such that sup

β∈F ρ γ E β m * n -β 2 ω C △ 3 ξ [ρ E X 2 + σ 2 ] max(ω m * n /γ m * n , 1/n).
Therefore, the minimax-optimal rate of convergence is of order O(max(ω m * n /γ m * n , 1/n)). As a consequence, the orthogonal series estimator β m * n attains this optimal rate and hence is minimax-optimal. However, the definition of the dimension parameter m * n used to construct the estimator involves an a-priori knowledge of the sequences γ, ω and λ. Throughout the paper our aim is to construct a data-driven choice of the dimension parameter not requiring this a-priori knowledge and automatically attaining the optimal rate of convergence.

Example of rates

We compute in this section the rates that we can obtain in three configurations for the sequences γ, ω and λ. These cases will be referred to in the following. In all three cases, we take the sequence ω with ω j = j 2s , j 1, for s ∈ R.

Case [P-P] Polynomial-Polynomial. Consider sequences γ and λ with γ j = j 2p , j 1, for p > max(0, s), and λ j ≍ j -2a , j 1, for a > 1/2 respectively, where the notation u j ≍ v j , j 1, means that there exists a constant d > 0 such that u j /d v j du j for all j 1. Then it is easily seen that (m

* n ) 2(s-p) = ω m * n γ m * n ≍ m * n j=1 ω j nυ j ≍ n -1 m * n j=1 j 2s+2a and hence m * n ≍ n 1/(2p+2a+1) if 2s + 2a + 1 > 0, m * n ≍ n 1/[2(p-s)] if 2s + 2a + 1 < 0 and m * n ≍ (n/ log(n)) 1/[2(p-s)] if 2a + 2s + 1 = 0.
Finally, the optimal rate attained by the estimator is max(n -(2p-2s)/(2a+2p+1) , n -1 ), if 2s + 2a + 1 = 0 (and log(n)/n if 2s + 2a + 1 = 0). Observe that an increasing value of a leads to a slower optimal rate of convergence. Therefore, the parameter a is called degree of ill-posedness (c.f. [START_REF] Natterer | Error bounds for Tikhonov regularization in Hilbert scales[END_REF]).

Remark 2.1. Obviously the rate is parametric if 2a + 2s + 1 < 0. The case 0 s < p can be interpreted as the L 2 -risk of an estimator of the s-th derivative of the slope parameter β.

On the other hand the case, s = -a, corresponds to the mean-prediction error (c.f. [START_REF] Cardot | Thresholding projection estimators in functional linear models[END_REF]).

Case [E-P] Exponential-Polynomial. Consider sequences γ and λ with γ j = exp(j 2p ), j 1, for p > 0, and (as previously) 2p) is the optimal rate attained by the estimator. Furthermore, if 2a+2s+1 < 0, then m * n ≍ (log(n)+(s/p) log(log(n))) 1/(2p) and the rate is parametric, while if 2a + 2s + 1 = 0, the rate is of order log(log(n))/n.

λ j ≍ j -2a , j 1, for a > 1/2 respectively. Then m * n is such that exp(-(m * n ) 2p )(m * n ) 2s = ω m * n γ m * n ≍ m * n j=1 ω j nυ j ≍ n -1 m * n j=1 j 2s+2a . In case 2a + 2s + 1 > 0 this is equivalent to exp(-(m * n ) 2p ) ≍ (m * n ) 2a+1 n -1 and hence m * n ≍ (log n -2a+1 2p log(log n)) 1/(2p) . Thereby, n -1 (log n) (2a+1+2s)/(
Case [P-E] Polynomial-Exponential. Consider sequences γ and λ with γ j = j 2p , j 1, for p > max(0, s), and

λ j ≍ exp(-j 2a ), j 1, for a > 0 respectively. Then (m * n ) 2(s-p) = ω m * n γ m * n ≍ m * n j=1 ω j nυ j ≍ n -1 m j=1 j 2s exp(j 2a ) and hence m * n ≍ (log n-2p+(2a-1) ∨0 2a log(log n)) 1/(2a)
with (q) ∨0 := max(q, 0). Thereby, (log n) -(p-s)/a is the optimal rate attained by the estimator. The parameter a reflects again the degree of ill-posedness since an increasing value of a leads also here to a slower optimal rate of convergence.

A model selection approach: known degree of ill-posedness

In the previous section, we have recalled an estimation procedure that attains the optimal rate of convergence in case the slope parameter belongs to some ellipsoid F ρ γ and its accuracy is measured by a F ω -risk. In this section, we suppose that there exists an a-priori knowledge concerning the degree of ill-posedness, that is the asymptotic behavior of the sequence of eigenvalues λ is known. The objective is the construction of an adaptive estimator which depends neither on the sequence of weights γ nor on the radius ρ but still attains the optimal rate over the ellipsoid F ρ γ . In this section, we use the following assumption. Assumption 3.1. Let λ := (λ j ) j 1 denote the sequence of eigenvalues associated to the regressor X and let ω := (ω j ) j 1 be a sequence satisfying Assumption 2.1 such that (i) there exist non decreasing sequences δ :

= δ(λ, ω) := (δ m (λ, ω)) m 1 and ∆ := ∆(λ, ω) := (∆ m (λ, ω)) m 1 with δ m m j=1 ω j /λ j and ∆ m max 1 j m ω j /λ j for all m 1 such that for some Σ > 0, m 1 ∆ m exp(- δ m 6∆ m ) Σ. (3.1) (ii) the sequence M := (M n ) n 1 given by M n := arg max 1 M n {δ M δ 1 n(ω M ) ∧1 }, n 1, with (q) ∧1 := min(q, 1), satisfies min 1 j Mn λ j 2/n for all n 1. (3.2)
It is worth to note that both sequences δ and M depend on the eigenvalues λ.

3.1 Definition of the estimator.

Consider the orthogonal series estimator β m defined in (1.4). In what follows we construct an adaptive procedure to choose the dimension parameter m based on a model selection approach. Therefore, let Φ u = j 1 λ -1 (3.4)

j ½{ λ j 1/n}[u] j ϕ j for u ∈ L 2 [0, 1] with Fourier coefficients [u] j := u, ϕ j . Then we consider the contrast Υ(t) := t 2 ω -2 t, Φ b g ω . ( 3
Note that we can compute

Υ( β m ) = - m j=1 ω j [ g] 2 j λ 2 j ½{ λ j 1/n}.
Remark 3.1. Throughout the paper we ignore that also the value σ 2 Y and η are unknown in practice. Obviously σ 2 Y can be estimated straightforwardly by its empirical counterpart. An estimator of the value η is not a trivial task. However, if in addition the regressor X and the error term ε are Gaussian, then Y ∼ N (0, σ 2 Y ) and hence η = 3 is a-priori known. We may take an other point of view if we chose a-priori a sufficiently large η 3 (the Gaussian case is included) then the following assertions apply as long as the unknown data generating process satisfies the conditions X ∈ X 4 η and E|Y /σ Y | 4 η.

An upper bound.

We derive first an upper bound of the adaptive estimator β b m by assuming an a-priori knowledge of appropriate sequences δ and M which are used in the construction of the penalty and the admissible set of values of m.

Theorem 3.1. Assume an n-sample of (Y, X) satisfying (1.1). Let E|Y /σ Y | 4 η and X ∈ X 4
η be 1-periodic and second order stationary with associated eigenvalues λ. Suppose that the sequences γ and ω satisfy Assumption 2.1. Let δ, △ and M be sequences satisfying Assumption 3.1 for some constant Σ. Consider the estimator β b m defined in (1.4) with m given by (3.4). If in addition X ∈ X 24 ξ and E|Y /σ Y | 24 ξ, then there exists a numerical constant C such that for all n 1 and 1 m M n , we have

sup β∈F ρ γ E β b m -β 2 ω C ω m γ m ρ + δ m n (ρE X 2 + σ 2 )η + K n (ρE X 2 + σ 2 ) [δ 1 + ρ][1 + (E X 2 ) 2 ],
where K = K(Σ, η, ξ, δ 1 ) is a constant depending on Σ, η, ξ and δ 1 only.

It is worth noting, that in the last assertion we do not impose a complete knowledge of the sequence of eigenvalues λ associated to the regressor X. In the next Corollary we state the upper bound when balancing the terms depending on m, which is obviously a trivial consequence of Theorem 3.1.

Corollary 3.2. Let the assumptions of Theorem 3.1 be satisfied. If in addition the sequence

m ⋄ := (m ⋄ n ) n 1 is chosen such that γ m ⋄ n δ m ⋄ n /(n ω m ⋄ n ) ≍ 1, n 1, then we have sup β∈F ρ γ E β b m -β 2 ω = O max(ω m ⋄ n /γ m ⋄ n , 1/n) as n → ∞.
Remark 3.2. Comparing the last assertion with the lower bound given in (2.1), we see that the adaptive estimator attains the optimal rate of convergence, as long as

sup n 1 ω m ⋄ n γ m * n /(γ m ⋄ n ω m * n ) < ∞.
Obviously a sufficient condition is given if the sequence δ satisfies in addition sup m 1 δ m /( m j=1 ω j /λ j ) < ∞. The polynomial case below provides an example. However, this condition is not necessary as can be seen in the exponential case.

Convergence rate of the theoretical adaptive estimator.

We described in Section 2.3 three different cases where we could choose the model m such that the resulting estimator reaches the optimal minimax rate. The following result shows that, in case of known degree of ill-posedness, we can propose choices of sequences δ, ∆ and M such that the penalized estimator automatically attains the optimal rate. Proposition 3.3. In cases [P-P] and

[E-P] with 2a+2s+1 > 0, let δ m ≍ m 2a+2s+1 , ∆ m ≍ m (2a+2s) ∨0 and M n ≍ n 1/(2a+1+(2s) ∨0 ) with (q) ∨0 := max(q, 0). While in case [P-E], choose δ m ≍ m 2a+1+(2s) ∨0 exp(m 2a ), ∆ m ≍ m (2s) ∨0 exp(m 2a ) and M n ≍ (log n/(log n) (2a+1+(2s) ∨0 )/(2a) ) 1/(2a) .
Then Assumption 3.1 is fulfilled and, under the additional assumptions of Theorem 3.1, the adaptive estimator β b m reaches the optimal rate. In cases [P-P] and [E-P], if 2a + 2s + 1 < 0, then the sequence δ can be taken of order 1. The collection of models must be reduced to {[ √ n], . . . , n} since M n can be taken equal to n. It appears then that the rate is parametric in this case. In fact, no model selection is necessary in this case, a large m (m = n for instance) can be chosen. Now, we have in mind to prepare the case where the degree of ill-posedness of the λ j 's, and more precisely δ m and M n , are unknown. We propose hereafter a more intrinsic choice of δ m , which does not require anything but the λ j 's (which can be estimated). In this spirit, we can prove the following assertion.

Proposition 3.4. In cases [P-P] and [E-P] with a + s 0 or in case [P-E], choose ∆ m := max 1 j m ω j /λ j , κ m := max 1 j m (ω j ) ∨1 /λ j with (q) ∨1 := max(q, 1) and 4 A model selection approach: unknown degree of ill-posedness

δ m := m∆ m log(κ m ∨ (m + 2)) log(m + 2) . ( 3
In this section, the objective is the construction of a fully adaptive estimator which does not depend on the sequence γ and λ. Nevertheless the resulting estimator still attains the optimal rate in case the slope parameter belongs to some ellipsoid F ρ γ and the sequence of eigenvalues λ associated to the covariance operator of X has a given (unknown) rate of decrease.

The configuration given in Proposition 3.4 is now the right reference and the choice that the estimator is going to mimic. In particular, it is easily seen that there exists always a constant Σ > 0 such that the sequences δ and △ given in Proposition 3.4 satisfy Assumption 3.1 (i). Observe that in this situation we have

∆ m exp(- δ m 6∆ m ) = ∆ m exp(- m 6 log(κ m ∨ (m + 2)) log(m + 2) ) (κ m ∨ (m + 2)) exp(- m 6 log(κ m ∨ (m + 2)) log(m + 2) ) exp -m 1 6 - log(m + 2) m log(κ m ∨ (m + 2)) log(m + 2)
where the last term is obviously summable.

Assumption 4.1. Let λ denote the sequence of eigenvalues associated to the regressor X, let δ and △ be the sequences defined in Proposition 3.4 and let γ and ω be sequences satisfying Assumption 2.1 such that (i) the sequence M := (M n ) n 1 given in Assumption 3.1 satisfies in addition to (3.2) also

log n 2n max m>Mn λ m m(ω m ) ∨1 for all n 1; (ii) the sequence m ⋄ := (m ⋄ n ) n 1 given by 1/c γ m ⋄ n δ m ⋄ n /(n ω m ⋄ n ) c for all n 1 and some c 1 satisfies min 1 m m ⋄ n λ m m(ω m ) ∨1
2(log n)/n for all n 1;

(iii) the sequence N := (N n ) n 1 given by N n := arg max

1 N n { max 1 j N ω j /n 1}, n 1, satisfies M n N n n for all n 1.
Remark 4.1. The last assumption is technical but satisfied in the interesting case. Note that (i) and (ii) together imply m ⋄ n M n for all n 1. The condition (iii) is rather weak, observe that the sequence ω is a-priori known and thus also the sequence of upper bounds N . In particular, recall that in case ω ≡ 1 the F ω -risk corresponds to the L 2 -risk. If ω m 1 for all m 1, then F ω -risk is weaker than the L 2 -risk and N n = n. Only if the F ω -risk is stronger than the L 2 -risk, that is ω is monotonically increasing, we choose N n such that ω Nn ≍ n. Then it is not hard to see that in these situations (iii) is satisfied at least for sufficiently large n.

Definition of the estimator

We follow the model selection approach presented in the last section. Define

∆ m := max 1 j m ω j λ j ½ { b λ j 1/n}
and κ m := max

1 j m (ω j ) ∨1 λ j ½ { b λ j 1/n} .
We shall refer to δ m as defined in (3.5) and consider its estimator given by

δ m := m ∆ m log( κ m ∨ (m + 2)) log(m + 2) .
If X ∈ X 4 η and E|Y /σ Y | 4 η, then we define a random penalty function

pen(m) = 1920σ 2 Y η δ m n .
Moreover, we consider a random upper bound for the collection of models given by

M n := arg max 1 M Nn λ M M (ω M ) ∨1 (log n)/n . (4.1)
The adaptive estimator β b m is obtained from (1.4) by choosing the dimension parameter m := arg min

1 m c Mn Υ( β m ) + pen(m) (4.2)
We shall emphasize that the proposed estimator does not depend on an a-priori knowledge of neither the sequence γ nor the sequence λ.

An upper bound.

In the next assertion we provide an upper bound of the fully adaptive estimator β b m by assuming that the sequences λ, ω and γ satisfy Assumption 4.1.

Theorem 4.1. Assume an n-sample of (Y, X) satisfying (1.1). Suppose that E|Y /σ Y | 4 η and that X ∈ X 4 η is 1-periodic and second order stationary. Let Assumption 4.1 be satisfied. Consider the estimator β b m defined in (1.4) with m given by (4.2). If in addition X ∈ X 28 ξ and E|Y /σ Y | 28 ξ, then there exists a numerical constant C > 0 such that for all n 1 sup

β∈F ρ γ E β b m -β 2 ω C ω m ⋄ n γ m ⋄ n (ρ + cη[ρE X 2 + σ 2 ]) + K n [ρE X 2 + σ 2 ] [1 + δ 1 + ρ][1 + (E X 2 ) 2 ],
where m ⋄ n and c are defined in Assumption 4.1, K = K(Σ, η, ξ, δ 1 ) is a constant only depending on η, ξ, δ 1 and Σ such that the sequences δ and △ given in Proposition 3.4 satisfy Assumption 3.1.

Remark 4.2. Comparing the last assertion with Theorem 3.1, we see that under Assumption 4.1 the proposed adaptive estimator obtains the same rate as in case of known degree of ill-posedness. We only have to impose in addition slightly stronger moment conditions.

It is easily verified that in all the examples discussed above the fully adaptive estimator attains the optimal rate, which is summarized in the next assertion. Conclusion. Assuming a circular functional linear model we derive in this paper a fully adaptive estimator of the slope function β or its derivatives, which attains the minimax optimal rate of convergence. It is worth to note, that in this paper not only the penalty is chosen randomly but also the collection of models. In this way the proposed estimator is adaptive also with respect to the degree of ill-posedness of the underlying inverse problem. We can thereby face both, the mildly and the severely ill-posed case.

It is not clear that the ideas in this paper can be straightforwardly adapted to treat the case of noncircular functional models. We are currently exploring this issue.

A Appendix

A.1 Proof of Theorem 3.1

We begin by defining and recalling notations to be used in the proof. Given u ∈ L 2 [0, 1] we denote by [u] the infinite vector of Fourier coefficients [u] j := u, ϕ j . In particular we use the notations

[X i ] j = X i , ϕ j , [β] j = β, ϕ j , σ 2 Y = Var(Y ), β m = m j=1 λ -1 j ½{ λ j 1/n}[ g] j ϕ j , β m := m j=1 λ -1 j [ g] j ϕ j , β m := m j=1 [β] j ϕ j , Φ u = j 1 λ -1 j ½{ λ j 1/n}[u] j ϕ j , Φ u := j 1 λ -1 j [u] j ϕ j .
Given m 1 we have then for all t ∈ S m = span{ϕ 1 , . . . , ϕ m }

t, β ω = m j=1 ω j [t] j [β] j = m j=1 ω j [t] j [g] j λ j = t, Φ g ω , t, β m ω = t, Φ b g ω = 1 n n i=1 Y i t, Φ X i ω = 1 n n i=1 Y i m j=1 ω j λ j [X i ] j [t] j , t, β m ω = t, Φ b g ω = 1 n n i=1 Y i t, Φ X i ω = 1 n n i=1 Y i m j=1 ω j λ j ½{ λ j 1/n}[X i ] j [t] j . (A.1) Furthermore, define the event Ω Y,X := {|Y /σ Y | n 1/6 , |[X] j / λ j | n 1/6 , 1 j M n }
and denote its complement by Ω c Y,X . Then consider the functions h and f with Fourier coefficients given by

[ h] j := 1 n n i=1 {Y i [X i ] j ½Ω Y i ,X i -E(Y i [X i ] j ½Ω Y i ,X i )}, [ f ] j := 1 n n i=1 {Y i [X i ] j ½Ω c Y i ,X i -E(Y i [X i ] j ½Ω c Y i ,X i )}. Obviously we have [ g] j -[g] j = [ h] j + [ f ] j and hence for all t ∈ S m t, Φ b g -β ω = t, Φ b g -Φ g ω = t, Φ b g -Φ g ω + t, Φ b g -Φ b g ω = t, Φ b h ω + t, Φ b f ω + t, Φ b g -Φ b g ω . (A.2)
We shall prove in the end of this section three technical Lemmas (A.2 -A.4) which are used in the following steps of the proof.

Consider now the contrast Υ then by using (3.3) and (3.4) it follows that

Υ( β b m ) + pen( m) Υ( β m ) + pen(m) Υ(β m ) + pen(m), ∀1 m M n ,
which in particular implies by using the notations given in (A.1) that

β b m 2 ω -β m 2 ω 2{ β b m , Φ b g ω -β m , Φ b g ω } + pen(m) -pen( m) = 2 β b m -β m , Φ b g ω + pen(m) -pen( m).
Rewriting the last estimate by using (A.2) we conclude that

β b m -β 2 ω = β -β m 2 ω + β b m 2 ω -β m 2 ω -2 β b m -β m , β ω β -β m 2 ω + pen(m) -pen( m) + 2 β b m -β m , Φ b g -β ω β -β m 2 ω + pen(m) -pen( m) + 2 β b m -β m , Φ b h ω + 2 β b m -β m , Φ b f ω + 2 β b m -β m , Φ b g -Φ b g ω . (A.3)
Consider the unit ball B m := {f ∈ S m : f ω 1} and let m∨m := max( m, m). Combining for τ > 0 and f ∈ S m the elementary inequality

2| f, g ω | 2 f ω sup t∈Bm | t, g ω | τ f 2 ω + 1 τ sup t∈Bm | t, g ω | 2 with (A.3) and β b m -β m ∈ S b m∨m ⊂ S Mn we obtain β b m -β 2 ω β -β m 2 ω + 6τ β b m -β m 2 ω + pen(m) -pen( m) + 2 τ sup t∈B b m∨m | t, Φ b h ω | 2 + 2 τ sup t∈B Mn | t, Φ b f ω | 2 + 2 τ sup t∈B Mn | t, Φ b g -Φ b g ω | 2 .
Then, noting that pen(m ∨ m ′ ) pen(m) + pen(m ′ ) and

β b m -β m 2 ω 2 β b m -β 2 ω + 2 β m -β 2
ω , we get, together for τ = 1/16 and pen(m

) = 192σ 2 Y ηδ m /n that (1/4) β b m -β 2 (7/4) β -β m 2 + 32 sup t∈B b m∨m | t, Φ b h ω | 2 -(1/32) pen( m ∨ m) + + 32 sup t∈B Mn | t, Φ b f ω | 2 + 32 sup t∈B Mn | t, Φ b g -Φ b g ω | 2 + pen( m ∨ m) + pen(m) -pen( m) (7/4) β -β m 2 + 32 Mn m ′ =1 sup t∈B m ′ | t, Φ b h ω | 2 -6σ 2 Y ηδ m ′ /n + + 32 sup t∈B Mn | t, Φ b f ω | 2 + 32 sup t∈B Mn | t, Φ b g -Φ b g ω | 2 + 2 pen(m). (A.4)
Combining the last bound with (A.5) in Lemma A.2, (A.9) and (A.10) in Lemma A.3 we conclude that there exist a numerical constant C and a constant K(Σ, η) depending on Σ and η only, such that for all n 1 and for all 1 m M n we have

E β b m -β 2 ω 7 β -β m 2 ω +8 pen(m)+ 1 n [Cξ(σ 2 Y δ 1 + β 2 ω }{1+(E X 2 ) 2 }+σ 2 Y K(Σ(6), η)].
Since (ω/γ) is monotonically non increasing we obtain in case

β ∈ F ρ γ that β 2 ω ρ and β -β m 2 ω
(ω m /γ m )ρ. Moreover, by using that X and ε are uncorrelated it follows

σ 2 Y = Var( X, β ) + σ 2 Var(ε) E X, β 2 + σ 2 β 2 E X 2 + σ 2 . Hence, σ 2 Y ρE X 2 + σ 2 because
γ is monotonically non decreasing. The result follows now by combining the last estimates with the definition of the penalty, that is, pen(m) = 192σ 2 Y ηδ m /n, which completes the proof of Theorem 3.1.

Technical assertions.

The following lemmas gather technical results used in the proof of Theorem 3.1. We begin by recalling an inequality due to [START_REF] Talagrand | New concentration inequalities in product spaces[END_REF], which can be found e.g. in [START_REF] Comte | Penalized contrast estimator for density deconvolution[END_REF].

Lemma A.1 (Talagrand's Inequality). Let T 1 , . . . , T n be independent T -valued random variables and ν * n (r) = (1/n) n i=1 r(T i ) -E[r(T i )] , for r belonging to a countable class R of measurable functions. Then, for ε > 0,

E[sup r∈R |ν * n (r)| 2 -2(1 + 2ε)H 2 ] + C v n exp(-K 1 ε nH 2 v )) + h 2 n 2 C 2 (ε) exp(-K 2 C(ε) √ ε nH h ) with K 1 = 1/6, K 2 = 1/(21 √ 2), C(ε) = √ 1 + ε -1
and C a universal constant and where

sup r∈R sup t∈T |r(t)| h, E sup r∈R |ν * n (r)| H, sup r∈R 1 n n i=1 Var(r(T i )) v.
Lemma A.2. Let λ be the eigenvalues associated to X ∈ X 4 η and E|Y /σ Y | 4 η. Suppose sequences δ, △ and M satisfying Assumption 3.1. Then there exists a constant K(Σ, η, δ 1 ) only depending on Σ, η and δ 1 such that

Mn m=1 E sup t∈Bm | t, Φ b h ω | 2 -6σ 2 Y η δ m n + K(Σ, η, δ 1 ) σ 2 Y n for all n 1. (A.5) Proof. Given m ∈ N and t ∈ B m := {f ∈ S m : f ω 1} denote v t (Y, X) := Y ½ Ω Y,X t, Φ X ω = m j=1 ω j [t] j λ j Y ½ Ω Y,X [X] j ,
then it is easily seen that t,

Φ b h ω = (1/n) n i=1 {v t (Y i , X i ) -Ev t (Y i , X i )}.
Below we show the following three bounds

sup t∈Bm sup y∈R,x∈L 2 [0,1] |v t (y, x)| σ Y n 1/3 δ 1/2 m =: h, (A.6) E sup t∈Bm | t, Φ b h ω | 2 σ 2 Y η δ m n =: H 2 , (A.7) sup t∈Bm 1 n n i=1 Var(v t (Y i , X i )) σ 2 Y η △ m =: v. (A.8)
From Talagrand's inequality (Lemma A.1) with ε = 1 we obtain by combining (A.6)-(A.8)

E sup t∈Bm | t, Φ b h ω | 2 -6H 2 C v n exp - nH 2 6v + h 2 n 2 exp - c n H h = C σ 2 Y η △ m n exp - δ m 6△ m + σ 2 Y n 2/3 δ m n 2 exp -c η n 1/6 with c = (1 -1/ √
2)/21 and some numerical constant C > 0. By using Assumption 3.1, that is δ m /n δ Mn /n δ 1 and M n /n 1, together with

H 2 = σ 2 Y ηδ m /n it follows that Mn m=1 E sup t∈Bm | t, Φ b h ω | 2 -6σ 2 Y ηδ m /n C σ 2 Y η n Mn m=1 △ m exp - δ m 6△ m + σ 2 Y δ 1 n 2/3 exp -c η n 1/6 C σ 2 Y n η Σ + δ 1 exp -c η n 1/6 + (5/3) log n ,
where condition (3.1) in Assumption 3.1 implies the last inequality. It follows that there exists a constant K(Σ, η, δ 1 ) only depending on Σ, η and δ 1 such that

Mn m=1 E sup t∈Bm | t, Φ b h ω | 2 -6σ 2 Y ηδ m /n σ 2 Y n K(Σ, η, δ 1 ), for all n 1, which proves the result. Proof of (A.6). From sup t∈Bm | t, g ω | 2 = m j=1 ω j [g] 2 j and the definition of Ω Y,X follows sup y∈R,x∈L 2 [0,1],t∈Bm |v t (y, x)| 2 = sup y∈R,x∈L 2 [0,1] m j=1 ω j σ 2 Y λ j ½ Ωy,x y 2 σ 2 Y [x] 2 j λ j σ 2 Y n 2/3 m j=1 ω j λ j
and, hence the definition of δ m implies (A.6).

Proof of (A.7). Since (Y i , X i ), i = 1, . . . , n, form an n-sample of (Y, X) we have

E sup t∈Bm | t, Φ b h ω | 2 = m j=1 ω j λ 2 j Var 1 n n i=1 Y i ½ Ω Y i ,X i [X i ] j 1 n m j=1 ω j λ 2 j E Y ½ Ω Y,X [X] j 2 and hence from E|Y /σ Y | 4 η and X ∈ X 4 η it follows that E sup t∈B N | t, Φ b h ω | 2 σ 2 Y n m j=1 ω j λ j E|Y /σ Y | 4 E|[X] j / √ λ j | 4 1/2 σ 2 Y n η m j=1 ω j λ j .
Thereby, the definition of δ m implies also (A.7). Proof of (A.8). Consider z := (z j ) with

z j := (ω j [t] j / λ j )/( m j=1 (ω 2 j [t] 2 j /λ j )) 1/2 and, hence z ∈ S m = {z ∈ R m , m j=1 z 2 j = 1}. Since (Y i , X i ), i = 1, . . . , n, form an n-sample of (Y, X) it follows that sup t∈Bm 1 n n i=1 Var(v t (Y i , X i )) sup t∈Bm E Y ½ Ω Y,X m j=1 ω j [t] j λ j [X] j 2 . Thereby, from E|Y /σ Y | 4 η and X ∈ X 4 η we conclude that sup t∈Bm 1 n n i=1 Var(v t (Y i , X i )) sup t∈Bm σ 2 Y (E|Y /σ Y | 4 ) 1/2 E m j=1 ω j [t] j λ j [X] j √ λ j 4 1/2 σ 2 Y η 1/2 sup t∈Bm m j=1 (ω 2 j [t] 2 j /λ j ) sup z∈S N E m j=1 z j [X] j / √ λ j 4 1/2 σ 2 Y η sup t∈Bm m j=1 (ω 2 j [t] 2 j /λ j ) σ 2 Y η max 1 j m ω j /λ j .
Thus the definition of △ m implies now (A.8), which completes the proof of Lemma A.2.

Lemma A.3. Let λ be the eigenvalues associated to X ∈ X 24 ξ and let E|Y /σ Y | 24 ξ. Suppose sequences δ, △ and M satisfying Assumption 3.1. Then there exists a numerical constant C such that

E sup t∈B Mn | t, Φ b f ω | 2 √ 2 ξ σ 2 Y δ 1 /n and (A.9) E sup t∈B Mn | t, Φ b g -Φ b g ω | 2 C ξ n {σ 2 Y δ 1 + β 2 ω }{1 + (E X 2 ) 2 } for all n 1. (A.10) Proof. Since (Y i , X i ), i = 1, . . . , n, form an n-sample of (Y, X) it follows that E sup t∈B Mn | t, Φ b f ω | 2 = Mn j=1 ω j λ 2 j Var 1 n n i=1 Y ½ Ω c Y,X [X] j Mn j=1 ω j λ 2 j n E Y ½ Ω c Y,X [X] j 2 .
Thereby, from E|Y /σ Y | 24 ξ and X ∈ X 24 ξ we conclude that

E sup t∈B Mn | t, Φ b f ω | 2 σ 2 Y n Mn j=1 ω j λ j E|Y /σ Y | 8 E|[X] j / √ λ j | 8 1/4 P (Ω c Y,X ) 1/2 σ 2 Y ξ 1/2 n Mn j=1 ω j λ j P (Ω c Y,X ) 1/2 σ 2 Y ξ 1/2 δ Mn n P (Ω c Y,X ) 1/2
where the last inequality follows from the property δ m m j=1 ω j λ j for all m 1. Hence by using Assumption 3.1, that is δ Mn /n δ 1 , we obtain

E sup t∈B Mn | t, Φ b f ω | 2 σ 2 Y δ 1 ξ 1/2 P (Ω c Y,X ) 1/2 .
The estimate (A.9) follows now from P (Ω c Y,X ) 2ξ/n 2 , which can be realized as follows.

Since Ω c Y,X = {|Y /σ Y | > n 1/6 } ∪ Mn j=1 {|[X] j / √ λ j | > n 1/6
} it follows by using Markov's inequality together with E|Y /σ Y | 24 ξ and X ∈ X 24 ξ that

P (Ω c Y,X ) P (|Y /σ Y | > n 1/6 ) + Mn j=1 P (|[X] j / √ λ j | > n 1/6 ) E|Y /σ Y | 18 n 3 + Mn j=1 E|[X] j / √ λ j | 18 n 3 ξ n 3 (1 + M n ) Thus, under Assumption 3.1, that is, M n /n
1, we obtain P (Ω c Y,X ) 2ξ/n 2 , which completes the proof of (A.9).

Proof of (A.10). Consider the decomposition sup .11) where we bound each summand separately. First, from (A.16) and (A.19) in Lemma A.4 together with X ∈ X 24 ξ and E|Y /σ Y | 24 ξ it follows that there exists a numeric constant C > 0 such that

t∈B Mn | t, Φ b g -Φ b g ω | 2 = Mn j=1 ω j λ j λ j λ j ½{ λ j 1/n} -1 2 1 n n i=1 Y i [X i ] j √ λ j 2 2 Mn j=1 ω j λ j λ j λ j -1 2 ½{ λ j 1/n} 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2 + 2 Mn j=1 ω j [β] 2 j λ j λ j -1 2 ½{ λ j 1/n} + 2 Mn j=1 ω j λ j 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2 ½{ λ j < 1/n} + 2 Mn j=1 ω j [β] 2 j ½{ λ j < 1/n} (A
E Mn j=1 ω j λ j λ j λ j -1 2 ½{ λ j 1/n} 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2 Mn j=1 ω j λ j E|λ j / λ j -1| 4 ½{ λ j 1/n} 1/2 E 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 4 1/2 C σ 2 Y ξ n Mn j=1 ω j nλ j {λ 2 j + 1}; (A.12) E Mn j=1 ω j [β] 2 j λ j λ j -1 2 ½{ λ j 1/n} C ξ n Mn j=1 ω j [β] 2 j {λ 2 j + 1}. (A.13)
Furthermore, Assumption 3.1 (ii), i.e., 2/n min{λ j : 1 j M n }, implies P ( λ j < 1/n) P ( λ j /λ j < 1/2). Thereby, from (A.16) and (A.18) in Lemma A.4 together with X ∈ X 24 ξ and E|Y /σ Y | 24 ξ it follows that there exists a numeric constant C > 0 such that

E Mn j=1 ω j λ j 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2 ½{ λ j < 1/n} Mn j=1 ω j λ j E 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 4 1/2 P ( λ j /λ j < 1/2) 1/2 C σ 2 Y ξ n Mn j=1 ω j nλ j ; (A.14) E Mn j=1 ω j [β] 2 j ½{ λ j < 1/n} Mn j=1 ω j [β] 2 j P ( λ j /λ j < 1/2) C ξ n Mn j=1 ω j [β] 2 j . (A.15)
Combining the decomposition (A.11) and the bounds (A.12) -(A.15) we obtain

E sup t∈B Mn | t, Φ b g -Φ b g ω | 2 C ξ n Mn j=1 ω j nλ j σ 2 Y {λ 2 j + 2} + Mn j=1 ω j [β] 2 j {λ 2 j + 2} .
Therefore the properties E X 2 max j 1 λ j and δ m m j=1 ω j λ j for all m 1 imply

E sup t∈B Mn | t, Φ b g -Φ b g ω | 2 C ξ n {σ 2 Y δ Mn /n + β 2 ω }{(E X 2 ) 2 + 2}.
Thus (A.10) follows now from δ Mn /n δ 1 (Assumption 3.1), which completes the proof.

Lemma A.4. Suppose X ∈ X 4k η 4k and E|Y /σ Y | 4k η 4k , k 1. Then for some numeric constant C k > 0 only depending on k we have

E 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2k C k σ 2k Y η 4k n -k , (A.16) E| λ j /λ j -1| 2k C k η 4k n -k . (A.17)
If in addition w 1 2 and w 2 1/2, then we obtain

sup j∈N P ( λ j /λ j w 1 ) C k η 4k n -k and sup j∈N P ( λ j /λ j < w 2 ) C k η 4k n -k . (A.18)
Moreover, if X ∈ X 12k η 12k , k 1, then for some numeric constant C k > 0 only depending on k we have

E|λ j / λ j -1| 2k ½{ λ j 1/n} C k η 12k {λ 2k j + 1}n -k . (A.19)
Proof. Since EY [X] j = λ j [β] j the independence within the sample of (Y, X) implies by using Theorem 2.10 in [START_REF] Petrov | Limit theorems of probability theory. Sequences of independent random variables[END_REF] for some generic constant

C k that E 1 n n i=1 Y i [X i ] j √ λ j -λ j [β] j 2k C 2k σ 2k Y n -k E|Y /σ| 2k |[X] j / √ λ j | 2k C k σ 2k Y n -k E|Y /σ| 4k E|[X] j / √ λ j | 4k 1/2 .
Then the last estimate together with X ∈ X 4k η 4k and E|Y /σ Y | 4k η 4k implies (A.16). Furthermore, since {(|[X i ] j | 2 /λ j -1) i } are independent and identically distributed with mean zero, it follows by applying again Theorem 2.10 in [START_REF] Petrov | Limit theorems of probability theory. Sequences of independent random variables[END_REF]

that E| λ j /λ j -1| 2k C k n -k E||[X] j / √ λ j | 2 -1| 2k
. Thus, the condition X ∈ X 4k η 4k implies (A.17). Proof of (A.18). If w 2 then P ( λ j /λ j w) P (| λ j /λ j -1| 1). Thus applying Markov's inequality together with (A.17) implies the first bound in (A.18), while the second follows in analogy.

Proof of (A.19). By using twice the elementary inequality | λ j /λ j -1| 2k + | λ j /λ j | 2k 1/2 2k-1 we conclude that

E|λ j / λ j -1| 2k ½{ λ j 1/n} 2 2k-1 {E| λ j /λ j -1| 4k λ 2k j λ 2k j ½{ λ j 1/n} + E| λ j /λ j -1| 2k } 2 4k-2 λ 2k j n 2k E| λ j /λ j -1| 6k + 2 4k-2 E| λ j /λ j -1| 4k + 2 2k-1 E| λ j /λ j -1| 2k }.
Thus, (A.19) follows from (A.17) since X ∈ X 12k η 12k , which proves the lemma.

A.2 Proof of Proposition 3.3

Case [P-P] Since 2a + 2s + 1 > 0 it follows that the sequences δ, ∆ and M with δ m ≍ m 2a+2s+1 , ∆ m ≍ m (2a+2s) ∨0 and M n ≍ n 1/(2a+1+(2s) ∨0 ) , respectively, satisfy Assumption 3.1. Note that δ Mn /n 1, M n /n 1, min 1 j Mn λ j 2/n and ∀C > 0,

m △ m exp(-Cδ m /∆ m ) ≍ m m (2a+2s) ∨0 exp(-Cm (2a+2s+1)∧1 ) < +∞.
Therefore we can apply Theorem 3.1 and hence Corollary 3.2. In particular, by using m

⋄ n ≍ n 1/(2a+2p+1) , which satisfies γ m ⋄ n δ m ⋄ n /(nω m ⋄ n ) ≍ 1, it follows that the adaptive estimator β b m reaches the optimal rate ω m ⋄ n /γ m ⋄ n ≍ n -2(p-s)/(2p+2a+1) .
Case [E-P] The sequences δ, ∆, M are unchanged w.r.t. the previous case [P-P] and hence Assumption 3.1 is still satisfied. From Corollary 3.2 follows now again that the adaptive estimator β b m attains the optimal rate ω m

⋄ n /γ m ⋄ n ≍ n -1 (log n) (2a+1+2s)/(2p) since m ⋄ n ≍ {log[n(log n) -(2a+1)/(2p) ]} 1/(2p) satisfies γ m ⋄ n δ m ⋄ n /(nω m ⋄ n ) ≍ 1.
Case [P-E] Consider the sequences δ, ∆ and M with δ m = m 2a+1+(2s) ∨0 exp(m 2a ), ∆ m = m (2s) ∨0 exp(m 2a ) and M n = (log n/(log n) 2a+1+(2s) ∨0 )/(2a) ) 1/(2a) respectively. Then Assumption 3.1 is satisfied, that is δ Mn /n 1, M n /n 1, min 1 j Mn λ j 2/n and ∀C > 0, 2a) . Finally, due to Corollary 3.2 the adaptive estimator β b m attains again the optimal rate ω m ⋄ /γ m ⋄ ≍ (log n) -(p-s)/a , which completes the proof of Proposition 3.3.

m ∆ m exp(-Cδ m /∆ m ) m m (2s) ∨0 exp(m 2a ) exp(-Cm 2a+1 ) < +∞. Moreover, γ m ⋄ δ m ⋄ /(nω m ⋄ ) ≍ 1 implies m ⋄ n ≍ (log n/(log n) (2a+2p+1)/(2a) ) 1/(

A.3 Proof of Proposition 3.4

Let ∆ m := max 1 j m ω j /λ j , κ m := max 1 j m (ω j ) ∨1 /λ j and δ m := m∆ m log(κm∨(m+2)) log(m+2) as defined in (3.5). Note that | log(κ m ∨ (m + 2))/ log(m + 2)| 1 and hence δ m m j=1 ω j /λ j .

Case [P-P] and [E

-P]. Since a + s 0 it is easily verified that ∆ m ≍ m 2a+2s , κ m ≍ m 2a+(2s) ∨0 with | log(κ m ∨ (m + 2))/ log(m + 2)| ≍ (2a + (2s) ∨0 ) > 1 and hence, δ m ≍ m 1+2a+2s
. Therefore, the result follows from Proposition 3.3 case [P-P] and [E-P] since both sequences δ and ∆ are unchanged.

Case [P-E] We have ∆ m ≍ m 2s exp(m 2a ), κ m ≍ m (2s) ∨0 exp(m 2a ) with, for all m suf- ficiently large, log(κ m ∨ (m + 2))/ log(m + 2)| ≍ m 2a (1+(2s) ∨0 (log m)m -2a ) log(m+2) and hence δ m ≍ m 1+2a+2s exp(m 2a ) (1+(2s) ∨0 (log m)m -2a ) log m
. Then straightforward calculus shows that Assumption 3.1 (i) is fulfilled. Moreover, consider the sequence M given in Assumption 3.1 (ii), where M n ≍ (log

n (log log n)/(2a) (log n) (1+2a+(2s) ∨0 )/(2a) ) 1/(2a) = (log n) 1/(2a) 1 + o(1) , then also Assumption 3.1 (ii) is satisfied (as in the proof of case [P-E] in Proposition 3.3). Due to Corollary 3.2 it remains to balance n ≍ γ m ⋄ δ m ⋄ /ω m ⋄ ≍ (m ⋄ ) 1+2a+2p exp((m ⋄ ) 2a )/(log m ⋄ ) which implies m ⋄ n ≍ (log n (log log n)/(2a) (log n) (1+2a+2p)/(2a) ) 1/(2a) = (log n) 1/(2a) 1 + o(1) . Hence, ω m ⋄ /γ m ⋄ ≍ (log n) (p-s)/a
is the rate attained by the adaptive estimator β b m which is optimal and completes the proof of Proposition 3.4.

A.4 Proof of Theorem 4.1

We begin by defining additional notations to be used in the proof. Consider sequences δ, △, M and m ⋄ satisfying Assumption 4.1 and the random upper bound M defined in (4.1). Denote by Ω := Ω I ∩ Ω II the event given by

Ω I := ∀j ∈ {1, . . . , M n }, 1 λ j - 1 λ j < 1 2λ j and λ j 1/n , Ω II := {m ⋄ n M n M n }.
It is easily seen that on Ω I we have for all 1 m M n Below we show that there exist a numerical constant C ′ > 0 and a constant K ′ = K ′ (Σ, η, ξ, δ 1 ) only depending on Σ, η, ξ and δ 1 such that for all n 1 we have (ω m ⋄ n /γ m ⋄ n )ρ. Moreover, we have σ 2 Y ρE X 2 + σ 2 . From these properties by combining the decomposition (A.22) and the estimates (A.23) -(A.25) we conclude that there exists a numerical constant C > 0 and a constant K = K(Σ, η, ξ, δ 1 ) only depending on Σ, η, ξ and δ 1 such that for all n 1

(1/2)∆ m ∆ m (3/2)∆ m and (1/2)κ m κ m (3/2)κ m and hence (1/2)[κ m ∨ (m + 2)] [ κ m ∨ (m + 2)] (3/2)[κ m ∨ (m + 2)] which implies (1/2)m∆ m log[κ m ∨ (m + 2)] log(m + 2) 1 - log 2 log(m + 2) log(m + 2) log(κ m ∨ [m + 2]) δ m (3/2)m∆ m log(κ m ∨ [m + 2]) log(m + 2) 1 + log 3/2 log(m + 2) log(m + 2) log(κ m ∨ [m + 2]) , together with log(κ m ∨ [m + 2])/log(m + 2) 1 we get δ m /10 (log 3/2)/(2 log 3)δ m (1/2)δ m [1 -(log 2)/ log(m + 2)] δ m (3/2)δ m [1 + (log 3/2)/ log(m + 2)] 3δ m . Since pen(m) = 192σ 2 Y ηδ m n -1 and pen(m) = 1920σ 2 Y η δ m n -1 it
E β b m -β 2 ω ½ Ω C ′ β -β m ⋄ n 2 ω + δ m ⋄ n n σ 2 Y η + K ′ n σ 2 Y [δ 1 + β 2 ω ][1 + (E X 2 ) 2 ] (A.23) E β b m -β 2 ω ½ Ω c I ∩Ω II C ′ β -β m ⋄ n 2 ω + K ′ n σ 2 Y [δ 1 + β 2 ω ][1 + (E X 2 ) 2 ] (A.24) E β b m -β 2 ω ½ Ω c II C ′ ξ n [σ 2 Y + β 2 ω ][1 + E X 2 ].
E β b m -β 2 ω C ω m ⋄ n γ m ⋄ n ρ+ δ m ⋄ n n [ρE X 2 +σ 2 ]η+ K n [ρE X 2 +σ 2 ] [1+δ 1 +ρ][1+(E X 2 ) 2 ] .
The result follows now from the definition of m ⋄ n , that is, γ m ⋄ n δ m ⋄ n /(n ω m ⋄ n ) c. Proof of (A.23). Observe that on Ω we have m ⋄ n M n M n . Thus, following line by line the proof of (A.4) it is easily seen that +4 pen(m ⋄ n ), where the last inequality follows from (A.20). Combining the last bound with (A.5) in Lemma A.2, (A.9) and (A.10) in Lemma A.3 we conclude that there exists a numerical constant C ′ > 0 and a constant K ′ = K ′ (Σ, η, ξ, δ 1 ) depending on Σ, η, ξ, δ 1 only such that (A.23) for all n 1 holds true.

Proof of (A.24). Note that on Ω c I ∩ Ω II we have still m ⋄ n M n M n . Thus, by using (A.21) rather than (A.20) + 1920σ 2 Y η δ Mn /n + M n ( max

1 j Mn ω j )log(n + 2) ½ Ω c I ∩Ω II .
From the last bound together with (A.5) in Lemma A.2, (A.9) and (A.10) in Lemma A.3 we conclude that there exist a numerical constant C > 0 and a constant K = K(Σ, η, ξ, δ 1 ) depending on Σ, η, ξ and δ 1 only such that for all n 1 we have

E β b m -β 2 ω ½ Ω c I ∩Ω II C β -β m ⋄ n 2 ω + K n σ 2 Y [δ 1 + β 2 ω ][1 + (E X 2 ) 2 ]
+ σ 2 Y η n -1 δ Mn + n -2 M n ( max 1 j Mn ω j ) n 2 log(n + 2)P (Ω c I ∩ Ω II ) . (A.26)

  .3) Define S m := span{ϕ 1 , . . . , ϕ m }. Obviously for all t ∈ S m it follows that t, Φ b g ω = t, β m ω and hence Υ(t) = tβ m 2 ωβ m 2 ω . Therefore, we have for all m 1 arg min t∈Sm Υ(t) = β m . Let X ∈ X 4 η and E|Y /σ Y | 4 η with σ 2 Y := Var(Y ). Under Assumption 3.1, we consider the penalty function pen(m) := 192σ 2 Y η δ m n . The adaptive estimator β b m is obtained from (1.4) by choosing the dimension parameter m := arg min 1 m Mn Υ( β m ) + pen(m) .

. 5 )

 5 Then Assumption 3.1 is fulfilled and, under the additional assumptions of Theorem 3.1, the adaptive estimator β b m reaches the optimal rate.

Corollary 4. 2 .

 2 In cases[P-P] and [E-P] with a + s 0 or in case [P-E], Assumption 4.1 is fulfilled and, under the additional assumptions of Theorem 4.1, the fully adaptive estimator β b m with m given by (4.2) reaches the optimal rate.

  follows on Ω I that pen(m) pen(m) 30 pen(m) for all 1 m M n , and hencepen(m ⋄ n ∨ m)+ pen(m ⋄ n )-pen( m) ½ Ω pen(m ⋄ n )+pen( m)+ pen(m ⋄ n )-pen( m) ½ Ω 31 pen(m ⋄ n )by using 1 m M n and m ⋄ n M n M n . On the other hand, it is not hard to see that on Ω c I we have ∆ m n max 1 j m ω j and κ m n for all m 1. From these properties we conclude that for all 1 m M n δ m mn( max1 j m ω j ) log(n ∨ (m + 2)) log(m + 2) mn( max 1 j m ω j ) log(n + 2), (A.20) which implies pen(m ⋄ n ) 1920σ 2 Y ηM n (max 1 j Mn ω j )log(n + 2) and hence pen(m ⋄ n ∨ m) + pen(m ⋄ n )pen( m) ½ Ω c I ∩Ω II pen(M n ) + 1920σ 2 Y ηM n ( max 1 j Mn ω j )log(n + 2) ½ Ω c I ∩Ω II 1920σ 2 Y η δ Mn /n + M n ( max 1 j Mn ω j )log(n + 2) ½ Ω c I ∩Ω II (A.21)We shall prove in the end of this section the technical Lemma A.5 which is used in the following steps of the proof together with the technical Lemmas A.2 -A.4 above.Consider now the decompositionE β b mβ 2 ω = E β b mβ 2 ω ½ Ω + E β b mβ 2 ω ½ Ω c I ∩Ω II + E β b mβ 2 ω ½ Ω c II . (A.22)

  γ) is monotonically non increasing we obtain in case β ∈ F ρ γ that β 2 ω ρ and ββ m ⋄ n 2 ω

(

  Φ b g -Φ b g ω | 2 + pen(m ⋄ n ∨ m) + pen(m ⋄ n )pen( m) ½ Ω (7/4) ββ m ⋄ n | t, Φ b h ω | 2 -6σ 2 Y ηδ m /n + +32 sup t∈B Mn | t, Φ b f ω | 2 + 32 sup t∈B Mn | t, Φ b g -Φ b g ω | 2

+

  it follows in analogy to(A.22Φ b h ω | 2 -6σ 2 Y ηδ m /n + +32 sup t∈B Mn | t, Φ b f ω | 2 +32 sup t∈B Mn | t, Φ b g -Φ b g ω | 2 + pen(m ⋄ n ∨ m)+ pen(m ⋄ n )-pen( m) ½ Ω c I ∩Ω II (7/4) ββ m ⋄ n Φ b h ω | 2 -6σ 2 Y ηδ m /n Φ b g -Φ b g ω | 2

Since X ∈ X 24 ξ and Ω c I ∩ Ω II ⊂ {∃j ∈ {1, . . . , M n } : |λ j / λ j -1| > 1/2 or λ j < 1/n} it follows from (A.29) in Lemma A.29 that P (Ω c I ∩ Ω II ) Cξ M n n -6 for some numerical constant C > 0. Moreover, due to Assumption 4.1 we have δ Mn /n δ 1 , M n /n 1 and max 1 j Mn ω j max 1 j Nn ω j n. Combining the last estimates and (A.26) implies now (A.24).

Proof of (A.25). Let βm := m j=1 [β] j ½{ λ j 1/n}ϕ j . Then it is not hard to see that

ω for all m m ′ and βmβ 2 ω β 2 ω . By using these properties together with 1 m M n N n we conclude .30) and (A.31) in Lemma A.5 that P (Ω c II ) Cξn -6 for some numerical constant C > 0 and hence

Moreover, from (A.16) and (A.17) in Lemma A.4 together with X ∈ X 28 ξ and E|Y /σ Y | 28 ξ it follows that there exists a numerical constant C > 0 such that

By combination of (A.27), (A.28) and E X 2 = j 1 λ j max j 1 λ j we obtain

for some numerical constant C ′ > 0. The estimate (A.25) follows now from max 1 j Nn ω j n (Assumption 4.1), which completes the proof of Theorem 4.1.

Technical assertions.

The following lemma gathers technical results used in the proof of Theorem 4.1.

Lemma A.5. Suppose X ∈ X 4k η 4k , k 1, with associated sequence λ of eigenvalues. Let M and m ⋄ be sequences satisfying Assumption 4.1. Then there exist a numerical constant C k > 0 only depending on k such that for all n 1 we have

Proof. Proof of (A.29). We start our proof with the observation that the event {|λ j / λ j -1| > 1/2} can equivalently be written as {1λ j /λ j > 1/3 or λ j /λ j -1 > 1}, and hence is a subset of {| λ j /λ j -1| > 1/3}. Moreover, since λ j 2/n for all 1 j M n it follows that { λ j < 1/n} ⊂ {| λ j /λ j -1| > 1/2}. Combining both estimates we conclude

Thus applying Markov's inequality together with (A.17) in Lemma A.4 implies (A.29). Proof of (A.30). Due to the definition of M n given in (4.1) the event

2(log n)/n (Assumption 4.1 (iii)). Thereby, (A.30) follows from the second bound in (A.18) in Lemma A.17.

Proof of (A.31). Due to the definition (4.1) of

. Thereby, the first bound in (A.18) in Lemma A.17 together with N n /n 1 (Assumption 4.1 (iv)) implies (A.31), which completes the proof of Lemma A.5.

A.5 Proof of Corollary 4.2

First, note that in all three cases, the sequences δ, ∆, M and m ⋄ have been calculated in the proof of Proposition 3.4. If in addition Assumption 4.1 holds true, then from Theorem 4.1 follows that the fully adaptive estimator attains the rate ω m ⋄ n /γ m ⋄ n , which in the proof of Proposition 3.4 has been confirmed to be optimal in all three cases. Therefore it only remains to check (i)-(iii) of Assumption 4.1.

Case [P-P]

In this case, we have M n ≍ n 1/(2a+1+(2s) ∨0 ) and m ⋄ n ≍ n 1/(2a+2p+1) . Then (i) of Assumption 4.1 holds true, since min 1 j Mn λ j ≍ M -2a n ≍ n -2a/(2a+1+(2s) ∨0 ) 2/n and max m Mn

Moreover (ii) of Assumption 4.1 is satisfied by using that for all p > s min

Finally, consider (iii) of Assumption 4.1. It is easily verified that N n ≍ n 1/(1+(2s) ∨0 ) which satisfies max 1 m Nn ω m N (2s) ∨0 n

≍ n (2s) ∨0 /(1+(2s) ∨0 ) n and M n ≍ n 1/(2a+1+(2s) ∨0 ) N n n. Thereby also (iii) of Assumption 4.1 holds true.

Case [E-P]. We have 2p) and N n ≍ n 1/(1+(2s) ∨0 ) . Then as in case [P-P] (i) and (iii) of Assumption 4.1 hold true since M n and N n are unchanged. Furthermore, for all s ∈ R we have min

which shows (ii) of Assumption 4.1.

Case [P-E].

Here we have M n ≍ (log

) and N n ≍ n 1/(1+(2s) ∨0 ) . It is easily seen that (iii) of Assumption 4.1 is satisfied. Moreover, (i) of Assumption 4.1 holds true, since min 1 j Mn λ j ≍ exp(-M 2a n ) ≍ (log n) (1+2a+(2s) ∨0 )/( 2a Finally, consider (ii) of Assumption 4.1 which is satisfied by using that for all p > s min