Olivia Touzet

Alexandre Philips
email: alexandre.philips@inserm.fr

Resveratrol protects against protease inhibitor-induced ROS production, reticulum stress and lipid raft perturbation

Keywords: HIV, Protease Inhibitors, Antioxidant, membrane microdomain, muscle skeletal

Resveratrol protects against protease inhibitor-induced reactive oxygen species production, reticulum stress and lipid raft perturbation.

Introduction

HIV protease inhibitors (PIs) have been successfully used in highly active anti-retroviral therapy (HAART) of HIV-1 infection, the most effective treatment currently available.

Incorporation of PIs in HAART has significantly reduced the morbidity and mortality and prolonged the lifespan of patients with HIV infection. PI benefits are unfortunately compromised by a number of clinically important adverse side effects [START_REF] Hui | Effects of HIV protease inhibitor therapy on lipid metabolism[END_REF]. Most patients on HAART develop a metabolic syndrome associated with partial lipodystrophy, hyperlipidemia, insulin resistance, premature atherosclerosis and myocardial infarction [START_REF] Beregszaszi | Severe insulin resistance contrasting with mild anthropometric changes in the adipose tissue of HIV-infected children with lipohypertrophy[END_REF][START_REF] Bongiovanni | Lopinavir/ritonavir vs. indinavir/ritonavir in antiretroviral naive HIVinfected patients: immunovirological outcome and side effects[END_REF][START_REF] Carr | A syndrome of peripheral lipodystrophy, hyperlipidaemia and insulin resistance in patients receiving HIV protease inhibitors[END_REF][START_REF] Holmberg | Trends in rates of myocardial infarction among patients with HIV[END_REF][START_REF] Koster | HIV protease inhibitors acutely impair glucose-stimulated insulin release[END_REF]. Moreover, these patients face a growing number of other age-related comorbidities, such as neurodegeneration, osteopenia and malignancies [START_REF] Kohli | Aging and HIV infection[END_REF][START_REF] Senior | Growing old with HIV[END_REF]. The cellular and molecular mechanisms underlying PI-associated metabolic abnormalities remain elusive but they seem to be related to overproduction of reactive oxygen species (ROS) [START_REF] Mondal | HAART drugs induce oxidative stress in human endothelial cells and increase endothelial recruitment of mononuclear cells: exacerbation by inflammatory cytokines and amelioration by antioxidants[END_REF], induction of endoplasmic reticulum (ER) stress and activation of the unfolded protein response (UPR) [START_REF] Gills | Nelfinavir, A lead HIV protease inhibitor, is a broad-spectrum, anticancer agent that induces endoplasmic reticulum stress, autophagy, and apoptosis in vitro and in vivo[END_REF][START_REF] Pyrko | HIV-1 protease inhibitors nelfinavir and atazanavir induce malignant glioma death by triggering endoplasmic reticulum stress[END_REF].

In many cell models, such as pancreatic insulinoma cells, cultured vascular smooth muscle cells or fibroblasts, PIs induce intracellular ROS production that is prevented by co-treatment with the antioxidants thymoquinone [START_REF] Chandra | HIV-1 protease inhibitor induced oxidative stress suppresses glucose stimulated insulin release: protection with thymoquinone[END_REF] or N-acetyl cysteine [START_REF] Ben-Romano | Nelfinavir induces adipocyte insulin resistance through the induction of oxidative stress: differential protective effect of antioxidant agents[END_REF]. In rat pancreatic insulinoma cells, a direct role of ROS in the inhibition of insulin secretion by nelfinavir is suggested because thymoquinone reverses the suppressive effect of nelfinavir [START_REF] Chandra | HIV-1 protease inhibitor induced oxidative stress suppresses glucose stimulated insulin release: protection with thymoquinone[END_REF]. However, the clear implication of oxidative stress in PI-induced metabolic syndrome remains to be established. PIs also affect the overall functions of ER compartments in human fibrosarcoma [START_REF] Kraus | Ritonavir induces endoplasmic reticulum stress and sensitizes sarcoma cells toward bortezomib-induced apoptosis[END_REF] and myeloma cells [START_REF] Kraus | Ritonavir induces endoplasmic reticulum stress and sensitizes sarcoma cells toward bortezomib-induced apoptosis[END_REF] and in mouse macrophages [START_REF] Zhou | HIV protease inhibitors activate the unfolded protein response in macrophages: implication for atherosclerosis and cardiovascular disease[END_REF]. Specifically, they deplete ER calcium stores and induce ER stress leading to UPR activation and expression of the ER stress markers X-box Binding Protein 1 (XBP-1), activating transcription factor 4 (ATF-4) and C/EBP Homologous Protein (CHOP) [START_REF] Zhou | HIV protease inhibitors activate the unfolded protein response in macrophages: implication for atherosclerosis and cardiovascular disease[END_REF]. These transcription factors operate as downstream components of UPR to protect ER against accumulation of unfolded/misfolded proteins (reviewed in [START_REF] Kim | Cell death and endoplasmic reticulum stress: disease relevance and therapeutic opportunities[END_REF]). UPR hinders accumulation of misfolded proteins in ER by facilitating the expression of ER chaperones, inhibiting protein entry into ER through mRNA translation arrest and by increasing the efficiency of the ER-associated machinery for protein degradation (ERAD) [START_REF] Kim | Cell death and endoplasmic reticulum stress: disease relevance and therapeutic opportunities[END_REF]. Sarcoplasmic reticulum (SR) and ER are crucial for proper folding and post-transcriptional modifications of many cell surface proteins, including lipid raft microdomains, which are important cell signaling platforms (reviewed in [START_REF] Fantini | Lipid rafts: structure, function and role in HIV, Alzheimer's and prion diseases[END_REF]. In osteoclasts, recruitment of c-src and TNF receptor-associated factor 6 (TARF-6) to lipid rafts is inhibited by ritonavir treatment through an unknown mechanism, which could be linked to ER stress [START_REF] Wang | The HIV protease inhibitor ritonavir blocks osteoclastogenesis and function by impairing RANKL-induced signaling[END_REF].

Whether PIs inhibit protein localization at lipid rafts in other tissues and how PIs lead to accumulation of unfolded/misfolded proteins in ER lumen remain important questions to be elucidated as well as the precise implication of ROS production in these deleterious processes. To elucidate these questions, we treated cultures of primary human myoblasts after they fused to form mature differentiated myotubes with the PIs atazanavir, lopinavir, ritonavir and saquinavir. Such treatment increased ROS production, modified SR/ER morphology and induced SR/ER stress in myotubes.

Moreover, these PIs decreased the expression and localization at lipid rafts of Caveolin 3 and Flotillin 1, two major components of lipid raft microdomains [START_REF] Babuke | Dissecting the molecular function of reggie/flotillin proteins[END_REF][START_REF] Bickel | Flotillin and epidermal surface antigen define a new family of caveolae-associated integral membrane proteins[END_REF][START_REF] Morrow | Flotillin-1/reggie-2 traffics to surface raft domains via a novel golgi-independent pathway. Identification of a novel membrane targeting domain and a role for palmitoylation[END_REF][START_REF] Williams | The caveolin proteins[END_REF]. To evaluate the role of ROS production in these deleterious effects, we used the antioxidant resveratrol, a stilbene found naturally in various plants with the highest concentration in grape skin and peanuts [START_REF] Naylor | Cellular effects of resveratrol in skeletal muscle[END_REF][START_REF] Penumathsa | Resveratrol: a promising agent in promoting cardioprotection against coronary heart disease[END_REF]. Resveratrol supplementation protects against ischemia/reperfusion injury [START_REF] Das | Resveratrol in cardioprotection: a therapeutic promise of alternative medicine[END_REF][START_REF] Imamura | Pharmacological preconditioning with resveratrol: an insight with iNOS knockout mice[END_REF][START_REF] Ray | The red wine antioxidant resveratrol protects isolated rat hearts from ischemia reperfusion injury[END_REF] and attenuates many age-related illnesses [START_REF] Das | Resveratrol in cardioprotection: a therapeutic promise of alternative medicine[END_REF][START_REF] Bureau | Resveratrol and quercetin, two natural polyphenols, reduce apoptotic neuronal cell death induced by neuroinflammation[END_REF][START_REF] Howitz | Small molecule activators of sirtuins extend Saccharomyces cerevisiae lifespan[END_REF][START_REF] Sun | Anti-cancer effect of resveratrol is associated with induction of apoptosis via a mitochondrial pathway alignment[END_REF], such cardiovascular disease, various types of cancers, neurodegeneration and type II diabetes, making of it a particularly attractive molecule to attenuate PI-induced cellular damage. Following co-treatment with resveratrol, PI deleterious effects on myotubes were inhibited, suggesting that SR/ER stress and lipid raft perturbation depend on ROS overproduction.

Methods

Reagents

Saquinavir, ritonavir, lopinavir and atazanavir were purchased from Sequoia Research Products (Pangbourne, United Kingdom). Thapsigargin and resveratrol were purchased from Sigma Aldrich (St.Quentin Fallavier, France), Nitro-Blue-Tetrazolium (NBT) from Merk (Darmstadt, Germany) and CM-H2DCFDA derivatives (5 (and 6) chloromethyl-2',7'dichlorodihydrofluorescein diacetate, acetyl ester) from Invitrogen (Cergy Pontoise, France).

Anti-α Tubulin monoclonal antibody (mAb) was from Sigma Aldrich (St Quentin Fallavier, France); anti-CHOP mAb was from Cell Signaling Technology (Ozyme, Montigny, France); anti-Bip/GRP78, -Flotillin-1, -Caveolin-3 were from BD Transduction Laboratories (Franklin Lakes, New Jersey, USA). The Alexa Fluor secondary antibodies were from Invitrogen (Cergy Pontoise, France).

Primary cell cultures

Primary human myoblasts were isolated as described previously [START_REF] Coisy-Quivy | TC10 controls human myofibril organization and is activated by the sarcomeric RhoGEF obscurin[END_REF] from quadriceps muscle biopsies obtained from the "AFM-BTR Banque de Tissus pour la Recherche" (Hôpital de la Pitié-Salpêtrière, 75013 Paris). 30 000 myoblasts/ml were grown in DMEM (Cambrex, Lonza, Verviers, Belgium) supplemented with 10% fetal calf serum (FCS) (Perbio Science, Brebieres, France), 1% UltroserG (Biosepra CI Pherben, Cergy St. Christophe, France) and Penicillin/Streptomycin (Cambrex, Lonza, Verviers, Bergium) for five days. At day 5, myoblasts started to fuse and cultures were switched to differentiation medium (DM) composed of DMEM supplemented with 10% FCS and Penicillin/Streptomycin for 4 days to obtain terminally differentiated myotubes that were then used for all the experiments. PIs were dissolved in dimethyl sulfoxide (DMSO) with 0.05% BSA and the final concentration of DMSO in DM was <0.1%; resveratrol was dissolved in ethanol and the final concentration of ethanol in DM was <0.1%. In control, cells were incubated with DMSO and/or ethanol. Cell viability was assessed by measuring cellular ATP levels using the ATPLite luminescence assay (PerkinElmer, Courtaboeuf, France).

ROS production

ROS production was detected by measuring NBT reduction. Cells were treated with PIs and 25 and 50 µM resveratrol, or not, for 18 hours before incubation with 0.2% NBT in DMEM for 90 minutes. Then cell extracts were prepared as described previously [START_REF] Causeret | N-cadherin association with lipid rafts regulates its dynamic assembly at cellcell junctions in C2C12 myoblasts[END_REF] and the amount of dark-blue reduced NBT in extracts in 50% acetic acid was assessed at 560 nm and the optical density values were normalized to protein concentration measured with the bicinchoninic acid assay. We also used CM-H2DCFDA as cellpermeant indicator for ROS. Myotubes were cultured in 6-well plates, treated with PIs and 25 µM resveratrol, or not, for 8 hours, before being incubated with 5 µM CM-H2DCFDA and DAPI in PBS at 37°C in the dark for 60 min. Cells were then washed with PBS three times, fixed in 4% paraformaldehyde and stored at 4°C for up to 16 h before image acquisition. Images were taken of at least 10 different regions with a Zeiss fluorescent microscope (AxioCamMRm equipped with a charged-coupled device camera Zeiss AxioCam MRm) and fluorescence quantification was performed using Image J (available at: rsb.nih.gov/ij/). All experiments were performed in triplicate.

Immunoblotting, immunofluorescence, electron microscopy

Cell extracts prepared as described previously [START_REF] Coisy-Quivy | TC10 controls human myofibril organization and is activated by the sarcomeric RhoGEF obscurin[END_REF] were subjected to SDS-PAGE, blotted onto nitrocellulose membranes and probed with the indicated antibodies. Bound antibodies were detected with a chemiluminescence detection kit (PerkinElmer, Courtaboeuf, France). Analysis and quantification were performed with Kodak 1D3.6 software. For immunofluorescence, cells were fixed in 4% paraformaldehyde in PBS and permeabilized with 0.1% Triton X-100 in PBS. Cells were incubated with primary antibodies for 2h, followed by Alexa Fluor secondary antibodies and DAPI staining.

Cell imaging was performed at the "Centre Regional d'Imageries Cellulaire de Montpellier" with a Leica DM6000 microscope (Wetlar, Germany) using a 40x/1.30 EC Plan-Neofluar grade oil objective. Images were captured as 16 TIFF files with a Micro Max 1300 CCD camera (RS-Princeton Instruments Inc.) driven by MetaMorph (version 7; Universal Imaging Corp.). Images were processed using ImageJ.

For ultrastructural evaluation, primary human myotubes were immersed in a solution of 3.5% glutaraldehyde in Sorensen's buffer (0.1M, pH 7.4) at 4°C overnight. They were then rinsed in Sorensen's buffer and post-fixed in 1% osmic acid and 0.8% potassium ferrocyanide in the dark and at room temperature for 2 h. After two rinses in Sorensen's buffer, cells were dehydrated in a graded series of ethanol solutions (30-100%) and then embedded in EmBed 812. Thin sections (85 nm; Leica-Reichert Ultracut E) were cut at different levels of each block, counterstained with uranyl acetate and observed using a Hitachi 7100 transmission electron microscope at the Centre de Ressources en Imagerie Cellulaire de Montpellier (France).

Isolation of detergent-resistant membrane (DRM) fractions

Ten 10cm dishes of myotubes were harvested, then centrifuged and cell pellets suspended in ice-cool lysis buffer containing 1% Triton X-100. After measurement of the total protein concentration with the bicinchoninic acid assay, cell extracts were fractionated on a 4 ml sucrose gradient [START_REF] Iwabuchi | Separation of "glycosphingolipid signaling domain" from caveolin-containing membrane fraction in mouse melanoma B16 cells and its role in cell adhesion coupled with signaling[END_REF]. Ten fractions were collected from the top.

Thirty microliters of each fraction was analyzed by immunoblotting for Caveolin 3, Flotillin 1, Tubulin α and Actin expression.

Statistical analysis

All quantitative data are presented as means ± s.e.m. Statistical analysis to determine significance was performed using the Student's t-test. Values of P ≤ 0.05 were considered significant.

Results

Resveratrol protects against PI-induced production of ROS in human myotubes.

As ROS induce cellular damage that could mediate PI-mediated metabolic syndrome, ROS production was investigated in cultures of primary human myoblasts that fused in vitro to form mature differentiated myotubes. Myotubes were treated with four PIs (20 µM atazanavir, 20 µM lopinavir, 40 µM ritonavir and 20 µM saquinavir) for 18 hours and ROS production was evaluated indirectly by measuring NBT reduction. ROS production was significantly increased in PI-treated cells in comparison to untreated control cells (Fig. 1A). Moreover, ROS production in PI-treated myotubes was inhibited by co-incubation with resveratrol in a dose-dependent manner (Fig. 1A), using noncytotoxic resveratrol concentrations (data not shown). These results were confirmed using the fluorescent probe CM-H2DCFDA. Dichloro-fluorescein oxidation increased twofold in myotubes treated with PIs for 8 hours compared to control cells (Fig. 1B) and again this effect was significantly inhibited by anti-oxidant treatment with 25 µM resveratrol (Fig. 1B), suggesting that resveratrol could protect against ROS-mediated iatrogenic effects of PIs.

Resveratrol protects against PI-induced sarco/endoplasmic reticulum stress in human myotubes.

Accumulation of Prelamin A following PI-dependent inhibition of the ER metalloproteinase ZMPSTE24 was proposed to mediate PI-associated iatrogenic effects in fibroblasts. Thus, maturation of Lamin A was assessed in proliferating myoblasts and differentiated myotubes. Treatment with 20 µM lopinavir, 40 µM ritonavir and 20 µM saquinavir for 18 hours induced Prelamin A accumulation in myoblasts (Fig. 2A), like in fibroblasts, but not in myotubes (Fig. 2A), suggesting that PI-induced iatrogenic effects are independent of Prelamin A accumulation in human myotubes. Next, we analyzed PI effects on SR/ER integrity because ROS can alter cellular redox reactions that interfere with protein disulphide bonding and result in protein misfolding leading to SR stress and UPR activation (reviewed in [START_REF] Kim | Cell death and endoplasmic reticulum stress: disease relevance and therapeutic opportunities[END_REF]). Following incubation with atazanavir, lopinavir, ritonavir and saquinavir (same concentrations as before) and 25µM resveratrol, or not, SR/ER distribution was monitored by immunodetection of BiP, a major ER chaperone protein and by high-resolution electron microscopy. In untreated myotubes, BiP was detected in the entire cell confirming the broad distribution of SR/ER (Fig. 2B) and electron microscopy showed the typical SR/ER morphology as flattened sacs (Fig. 2C). In myotubes treated with PIs for 18 hours or with 100 nM of thapsigargin (TG), an irreversible inhibitor of the ER Ca2+-ATPase that induces SR/ER stress, for 4 hours (Fig. 2B and2C) the broad expression of BiP was lost. Moreover, SR/ER vacuolization, observed in the immunofluorescence images (arrows in Fig. 2B: addition of arrows will facilitate reading), was confirmed by electron microscopy (arrows in Fig. 2C). On the other hand, co-incubation with resveratrol protected against PI-induced SR/ER morphological alterations (Fig. 2B and2C), suggesting that they, at least in part, are due to ROS overproduction.

We next tested whether PIs could induce SR/ER stress and activate UPR in human myotubes. To this aim, myotubes were incubated with PIs for 18 hours and with TG for 4 hours. Expression of CHOP, which is a key downstream transcription factor of UPR activated during SR/ER stress (Fig. 3A), strongly increased in TG-and in PI-treated myotubes. Co-incubation with 25 µM resveratrol (Fig. 3B), however, inhibited CHOP upregulation in PI-treated myotubes (from 80-90% of CHOP-positive nuclei to 9-13%).

Thus, induction of SR/ER stress by PIs is hindered by resveratrol.

Resveratrol protects against PI-dependent inhibition of Caveolin 3 and Flotillin expression and localization to lipid rafts.

In an initial step to reestablish homeostasis and normal SR/ER function, UPR leads to the export of unfolded or misfolded SR/ER proteins to the cytosol for degradation. As SR/ER is crucial for proper folding and post-transcriptional modifications of many cell surface proteins, including lipid raft microdomains, which are important cell signaling platforms, we assessed the effect of PIs on the expression of Caveolin 3 and Flotillin 1, two lipid raft components. Treatment with different concentrations (from 5 to 20 µM) of PIs for 18 and 24 hours reduced expression of Caveolin 3 and Flotillin 1 in myotubes (Fig. 4A for atazanavir and data not shown for the other PIs). Moreover, this decrease was dose-and time-dependent in human myotubes and was obtained at therapeutic concentrations (Fig. 4B). As for the previously described PI effects, co-treatment with different concentrations (from 5 to 25 µM) of resveratrol for 18 hours inhibited reduction of Caveolin 3 expression in myotubes in a dose-dependent manner (Fig. 4C and 4D).

To determine whether PIs also affected Caveolin 3 and Flotillin 1 localization at lipid raft microdomains, Triton X-100 lysates of human myotubes, which had been treated with 20 µM of atazanavir and 25 µM of resveratrol (or not), were fractionated on sucrose density gradients. Buoyant density and resistance to solubilization by Triton X-100 at 4°C are widely used to isolate raft membranes. In these preparations, Caveolin and Flotillin 1 retain their association with lipids and they are frequently used as lipid raft markers. Indeed they strongly accumulated in fractions 3-5 (DMR fractions) of control cells (DMSO alone) (Fig. 5A), but less in those of atazanavir-treated myotubes (Fig. 5A). Again, resveratrol reduced this PI effect (Fig. 5A). Western blot analysis of pooled fractions 3-5 (LR: lipid raft fractions) and pooled fractions 8-10 (NR: no raft fractions) (Fig. 5B) confirmed that localization of Caveolin 3 and Flotillin 1 at lipid rafts was impaired in myotubes treated with atazanavir, lopinavir, ritonavir and saquinavir and that this PI effect was hindered by resveratrol (Fig. 5B). These results indicate that PIs decrease the expression and membrane localization of two important components of lipid raft microdomains. Moreover, resveratrol protects against PIinduced loss of these proteins in lipid raft microdomains, suggesting that this antioxidant could attenuate PI-mediated iatrogenic effects.

Discussion

In this work we show that PI treatment induces ROS production, SR/ER damage, SR stress and reduces Caveolin 3 and Flotillin 1 localization at lipid rafts. Co-treatment with the anti-oxidant resveratrol protects against these effects.

Our results suggest that, at least in myotubes, PI side effects are not due to defects in Prelamin A conversion. A role for Lamin A/C metabolism in PI-associated metabolic/lipodystrophy syndrome and premature ageing was proposed because PIs block the zinc metalloproteinase ZMPSTE24, an ER protein localized at the cytosolic surface of ER membranes, thus causing accumulation of farnesylated Prelamin A [START_REF] Caron | Human lipodystrophies linked to mutations in A-type lamins and to HIV protease inhibitor therapy are both associated with prelamin A accumulation, oxidative stress and premature cellular senescence[END_REF][START_REF] Coffinier | HIV protease inhibitors block the zinc metalloproteinase ZMPSTE24 and lead to an accumulation of prelamin A in cells[END_REF][START_REF] Coffinier | A potent HIV protease inhibitor, darunavir, does not inhibit ZMPSTE24 or lead to an accumulation of farnesyl-prelamin A in cells[END_REF]. Retention of the farnesylated residue confers toxic properties to the partially processed Prelamin A [START_REF] Fong | Prelamin A and lamin A appear to be dispensable in the nuclear lamina[END_REF][START_REF] Fong | Heterozygosity for Lmna deficiency eliminates the progeria-like phenotypes in Zmpste24-deficient mice[END_REF][START_REF] Young | Prelamin A, Zmpste24, misshapen cell nuclei, and progeria--new evidence suggesting that protein farnesylation could be important for disease pathogenesis[END_REF] and genetic defects in the conversion of Prelamin A to mature Lamin A (e.g., ZMPSTE24 deficiency) cause lipodystrophy and premature ageing syndromes, including the Hutchinson-Gilford progeria syndrome [START_REF] Agarwal | Zinc metalloproteinase, ZMPSTE24, is mutated in mandibuloacral dysplasia[END_REF][START_REF] Moulson | Homozygous and compound heterozygous mutations in ZMPSTE24 cause the laminopathy restrictive dermopathy[END_REF][START_REF] Navarro | Loss of ZMPSTE24 (FACE-1) causes autosomal recessive restrictive dermopathy and accumulation of Lamin A precursors[END_REF]. However, our results suggest that, at least in myotubes, PI side effects are due to another mechanism.

Our work indicates that ROS overproduction, SR/ER stress and lipid raft perturbation were co-induced by PI treatment in human primary myotubes. The deleterious effects of HIV PIs had been previously associated with ROS overproduction and ER stress but the hierarchy between these two events was not known. In human aortic endothelial cells, PIs alone and as part of HAART regimens increase ROS production [START_REF] Mondal | HAART drugs induce oxidative stress in human endothelial cells and increase endothelial recruitment of mononuclear cells: exacerbation by inflammatory cytokines and amelioration by antioxidants[END_REF]. Nelfinavir induces oxidative stress and can also lead to adipocyte necrosis [START_REF] Vincent | Nelfinavir induces necrosis of 3T3F44-2A adipocytes by oxidative stress[END_REF] and premature senescence in human fibroblasts [START_REF] Caron | Human lipodystrophies linked to mutations in A-type lamins and to HIV protease inhibitor therapy are both associated with prelamin A accumulation, oxidative stress and premature cellular senescence[END_REF]. The clear implication of oxidative stress in PIinduced cellular damages is demonstrated by co-treatment with antioxidants. In 3T3-L1 adipocytes, the superoxide dismutase-mimetic antioxidant MnTBAP protects against nelfinavir-induced insulin resistance and cleavage of poly ADP ribose polymerase, a marker of apoptosis [START_REF] Ben-Romano | Nelfinavir induces adipocyte insulin resistance through the induction of oxidative stress: differential protective effect of antioxidant agents[END_REF]. In rat pancreatic insulinoma cells, nelfinavir-dependent inhibition of glucose-stimulated insulin secretion is prevented by thymoquinone [START_REF] Chandra | HIV-1 protease inhibitor induced oxidative stress suppresses glucose stimulated insulin release: protection with thymoquinone[END_REF], a component of black seed oil with potent antioxidant properties. Similarly, we show that atazanavir, lopinavir, ritonavir and saquinavir induce ROS overproduction and SR/ER stress (SR/ER swelling, CHOP expression and loss of Caveolin 3 and Flotillin 1 expression) in differentiated myotubes. Induction of ER stress by PIs was already shown in cancer cell lines [START_REF] Gills | Nelfinavir, A lead HIV protease inhibitor, is a broad-spectrum, anticancer agent that induces endoplasmic reticulum stress, autophagy, and apoptosis in vitro and in vivo[END_REF][START_REF] Pyrko | HIV-1 protease inhibitors nelfinavir and atazanavir induce malignant glioma death by triggering endoplasmic reticulum stress[END_REF][START_REF] Kraus | Ritonavir induces endoplasmic reticulum stress and sensitizes sarcoma cells toward bortezomib-induced apoptosis[END_REF] and in macrophages [START_REF] Zhou | HIV protease inhibitors activate the unfolded protein response in macrophages: implication for atherosclerosis and cardiovascular disease[END_REF], but the potential implication of ROS production on the induction of ER stress was not investigated in these studies. Our results clearly suggest that PI-induced SR/ER stress and UPR are mediated by ROS overproduction because the antioxidant resveratrol protects against SR/ER swelling, CHOP upregulation and loss of Caveolin 3 and Flotillin 1 expression. PI-induced oxidative stress could induce SR/ER stress through different mechanisms.

Nitric oxide (NO) and other ROS may modify oxidizable residues (cysteine, tyrosine) of SR-associated calcium channels, such as RyR (S-nitrosylation), and SR/ER Ca 2+ ATPases (by tyrosine nitration) leading to SR/ER calcium depletion, which is a cause of protein misfolding [START_REF] Viner | Protein modification during biological aging: selective tyrosine nitration of the SERCA2a isoform of the sarcoplasmic reticulum Ca2+-ATPase in skeletal muscle[END_REF][START_REF] Xu | Nitric oxide synthase in cardiac sarcoplasmic reticulum[END_REF][START_REF] Xu | Activation of the cardiac calcium release channel (ryanodine receptor) by poly-S-nitrosylation[END_REF] and consequently of SR/ER stress. Such mechanism could explain how ritonavir, atazanavir and indinavir decreased ER calcium content in a dosedependent manner in mouse macrophages [START_REF] Zhou | HIV protease inhibitors activate the unfolded protein response in macrophages: implication for atherosclerosis and cardiovascular disease[END_REF]. In addition, production of NO, a mediator of protein nitrosylation, and of other ROS that alter cellular redox-dependent reactions interferes with protein disulphide bonding and results in misfolding of proteins which accumulate in ER lumen [START_REF] Gotoh | Nitric oxide and endoplasmic reticulum stress[END_REF]. Interestingly, our results show that PIs inhibit localization of Caveolin 3 and Flotillin 1 at lipid raft microdomains. These two proteins are palmitoylated on cysteine residues in association with the ER and this post-translational modification is needed for their localization at lipid rafts microdomains (reviewed in [START_REF] Babuke | Dissecting the molecular function of reggie/flotillin proteins[END_REF][START_REF] Fruhbeck | Role of caveolins in body weight and insulin resistance regulation[END_REF]. Caveolin 3 and Flotillin 1 are two major components of lipid rafts and are implicated in insulin signaling to trigger glucose transporter redistribution in adipocytes and myotubes [START_REF] Fecchi | Spatial and temporal regulation of GLUT4 translocation by flotillin-1 and caveolin-3 in skeletal muscle cells[END_REF]. Thus, inhibition of their recruitment to lipid rafts by PIs could play an important role in PI-induced insulin resistance. The PI effect on lipid rafts is not restricted to myotubes and it has been described also in osteoclast precursors [START_REF] Wang | The HIV protease inhibitor ritonavir blocks osteoclastogenesis and function by impairing RANKL-induced signaling[END_REF].

Ritonavir blocks osteoclastogenesis and osteoclast function by disrupting the recruitment of TARF6/c-Src complex to lipid rafts by an unidentified mechanism [START_REF] Wang | The HIV protease inhibitor ritonavir blocks osteoclastogenesis and function by impairing RANKL-induced signaling[END_REF].

In myotubes, resveratrol protects against degradation and mislocalization of Caveolin 3 and Flotillin 1 induced by PIs, suggesting that these cellular damages are mediated by SR/ER stress and oxidative stress. Resveratrol holds a number of reported health benefits (cardioprotective, antiaging, neuroprotective, antiviral and anticancer) [START_REF] Das | Resveratrol in cardioprotection: a therapeutic promise of alternative medicine[END_REF][START_REF] Imamura | Pharmacological preconditioning with resveratrol: an insight with iNOS knockout mice[END_REF][START_REF] Ray | The red wine antioxidant resveratrol protects isolated rat hearts from ischemia reperfusion injury[END_REF][START_REF] Bureau | Resveratrol and quercetin, two natural polyphenols, reduce apoptotic neuronal cell death induced by neuroinflammation[END_REF][START_REF] Howitz | Small molecule activators of sirtuins extend Saccharomyces cerevisiae lifespan[END_REF][START_REF] Sun | Anti-cancer effect of resveratrol is associated with induction of apoptosis via a mitochondrial pathway alignment[END_REF] that make of it a particularly attractive molecule to attenuate PI-induced side effects.

Using the antioxidant resveratrol, we show that inhibition of PI-induced ROS overproduction protects against PI-induced SR/ER stress and mislocalization of lipid raft proteins in human myotubes. These results strongly suggest that resveratrol treatment may be used as a potential therapeutic agent to attenuate PI-induced side effects. (C) Human myotubes were treated as described in (B) and after 18 hours, cells were fixed and analyzed by electron microscopy (X 10,000). Scale bar: 500 nm.

Legends to figures

Figure 1 :

 1 Figure 1: Increased production of ROS in myotubes treated with PIs.

Figure 2 :

 2 Figure 2: Resveratrol protects against PI-induced SR/ER vacuolization.

Figure 3 :

 3 Figure 3: Resveratrol protects against PI-induced CHOP expression.

Figure 4 :

 4 Figure 4: Resveratrol protects against PI-induced inhibition of Caveolin 3

Acknowledgements

The authors are grateful to Chantal Cazevieille and Cécile Sanchez for their technical assistance and interpretation of data concerning ultrastructural evaluation. We would like to thank the Tissue Bank for Research (MYOBANK-AFM) of the Association Française contre les Myopathies (AFM) for generously providing human muscle biopsies. This work was supported by grants to A.P. from AFM and the Agence Nationale de Recherche Contre le Sida. O.T. was supported by the ANRS. A.P. was responsible for the overall conception and design of the study, drafting the article, and obtaining funding. O.T. collected and assembled data and performed data analysis. All coauthors were responsible for interpretation of the data, critical revision of the article, and all provided final approval of the article.

Quantification of the expression level of the proteins obtained in (A) after


François Chaplais
email: chaplais@cas.ensmp.fr

Keywords:

Generalization of the interaction between the Haar approximation and polynomial operators to higher order methods

Generalization of the interaction between the Haar approximation and polynomial operators to higher order methods

Introduction

The theory of averaging [START_REF] Sanders | Averaging Methods in Nonlinear Dynamical Systems[END_REF] approximates the solutions of a differential systems dx/dt = f (x, t, t/), where is a small parameter, by the solution of an "averaged" system dx/dt = f (x, t). In practice, f is not given as a function of two time scales t and t/, but rather as a function f (x, u) where u is an input which is a function of time. The question is to give a practical sense to the "average" of f (x, u) as a function of the input u.

The answer is relatively simple when the averaging of f (and u) is obtained by computing its averages on a sequence of intervals [kδ, (k + 1)δ). In this paper, we assume that f is a polynomial function. If E denotes the averaging operator, it can be easily shown that the following identity holds:

E(f (u)) = k=∞ k=0 f (k) (e) k! E(w k ) (1)
where e = E(u) and w = ue. We can see that f is expanded around the average of u, and the powers of w are replaced in the expansion by their averages. If one interprets the averaging operator E as an empirical expectation, then [START_REF] Sanders | Averaging Methods in Nonlinear Dynamical Systems[END_REF] shows that the expectation of f (u) is computed by using the derivatives of f at the expectation e of u and the moments of the "noise" w. While it is easy to derive, equation [START_REF] Sanders | Averaging Methods in Nonlinear Dynamical Systems[END_REF] appears to be new or at least not a classic. This previous method of averaging is also an approximation procedure which is related to the Haar multiresolution analysis. There are other methods of approximation which are more efficient and yet give a sense of local average. Wavelets [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Meyer | Wavelets and Operators[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] provide such methods. The order of approximation are then given by the classic Strang and Fix conditions [START_REF] Strang | A Fourier analysis of the finite element variational method[END_REF] which states that the order of approximation of smooth functions is characterized by the ability of the approximation method to reproduce polynomials up to a certain order. This condition is related to the number of vanishing moments of the wavelet. The approximation of a function is obtained by projection on a resolution space, which is itself generated by the translates of a so called scaling function. The previous averaging is a particular case of approximation on a resolution space where the scaling function is the characteristic function of an interval. The corresponding multiresolution analysis is called the Haar system. Among all wavelet methods, it yields the lowest order of approximation, since its scaling function reproduces only polynomials of degree zero.

The following question then arises: is it possible to have the relation (1) for higher order approximation methods (wavelets or not)? This will be the main subject of this article. It turns out that projectors which satisfy equation ( 1) can be simply characterized by two conditions on their image space and their kernel:

Proposition 1 Let V the image space of the projector E and W the image of Id -E. Equation ( 1) is verified if and only if (P1) the product of two elements of V belongs to V (P2) the product of a element of V with a element of W is a element of W

The sufficient part is proved by expanding f around e and applying E. The necessary condition is derived from (1) using f (x) = x 2 with x = e + b and x = eb, e ∈ V and successively b ∈ V and b ∈ W.

Finding approximation projectors which satisfy equation ( 1) if thus equivalent to satisfying properties (P1) and (P2). As we shall see in the next section, property (P1) implies that the function whose translates generates V is the characteristic function of an interval. It appears then that high order approximation methods cannot satisfy property (P1).

This last limitation is due to the use of the classical product on functions. Since we are dealing with approximations of functions, we can replace the classical product by an approximate product without degrading the order of approximation of the result. A general characterization of such approximate products is given in section 2.2.2. Like the Strang and Fix conditions, it is based on the reproduction of polynomials.

There remains to construct such approximate products. The difficulty is to verify the associativity of the product operator. In the case where the functions are piecewise polynomial, we give in section 3.2 a constructive characterization of approximate products in this functional space. These products are defined by a Hermite interpolation. In the simple case of a two point interpolation, we exhibit in section 3.4 an approximation projector which satisfies properties (P1) and (P2), and thus equation (1).

Product invariance and approximation

Two results on product invariance

The property (P1) states that the space V should be product invariant. We consider here the case where

V = V δ ⊂ L 2 (R) has a Riesz basis (φ k ) k∈Z with φ k (t) = φ(t/δ -k)/
(δ), and φ is compactly supported. In the wavelet context, V δ is a resolution space and φ is a scaling function. Given a function e ∈ V δ , C k (e) will denote its coordinate on the vector φ k . The following general lemma characterizes the product operators for which (P1) holds:

Lemma 1 Let * a (nonzero) product operator such that (V δ , +, * ) be a commutative ring. It is assumed that that * commutes with the shift of length δ, and that the product of two compactly supported functions is zero beyond some fixed distance. Then there exist a constant A = 0 such that, for any x and y in V δ , and any n ∈ N,

C n (x * y) = AC n (x)C n (y). ( 2
)
If * is the usual product on functions, then the intersection of the supports of φ i and φ j is of measure zero if i = j.

This lemma is proved in appendix A. Let us apply it to the case where an approximation operator is associated to the space V δ :

Theorem 1 Let E δ a projector on V δ defined by a rescaled kernel: E δ x(t) = 1/δ K(t/δ, s/δ)x(s)ds. We assume that E δ is an approximation operator, e.g.,for any f ∈ L 2 (R),

E δ f -f L 2 (R) → 0 when δ → 0. ( 3
)
If V 1 is product invariant for the usual product, then φ is proportional to the characteristic function of an interval of length 1.

Proof. Thanks to the projector E δ , the linear form C k is extended to L 2 (R).

Under mild assumptions, it can be represented by an L 2 (R) function φ * k ; one can easily see that φ * (t) = φ * (t/δk)/ (δ) with φ * compactly supported and that K(t, s) = φ(t)φ * (s). The Strang and Fix conditions [START_REF] Strang | A Fourier analysis of the finite element variational method[END_REF] imply that E(1) is defined and is equal to 1. Since the φ k 's have disjoint support and have the same shape, they must be proportional to the characteristic function of an interval of length 1. Theorem 1 implies that, if E δ yields an order of approximation strictly greater than the minimum provided by the Haar basis, then V δ cannot satisfy (P1). Indeed, φ cannot be a piecewise constant function anymore, and the only assumption that may be invalid in lemma 1 with the usual product is the product invariance of V δ . The product operator * δ is defined on V δ by rescaling: f (t/δ) * δ g(t/δ) = (f * g)(t/δ). It is assumed that E δ satisfies the Strang and Fix conditions [START_REF] Strang | A Fourier analysis of the finite element variational method[END_REF] at order N , e.g.

Characterization

E δ (t i ) = t i if i ≤ N . Then Theorem 2
The following two conditions are equivalent

• There exists K such that, for any f and g of class C N +1 and any δ ≤ 1,

|fg(t) -[(E δ f ) * δ (E δ g)] (t)| ≤ Kδ N +1 sup (k,l)∈I N ,|s-t|≤(µ+M )δ f (k) (s) g (l) (s) (4)
where

I N denotes the set of integers (k, l) ∈ N × N such that 0 ≤ k ≤ N + 1, 0 ≤ l ≤ N + 1 and k + l ≥ N + 1. • t i * t j = t i+j if i + j ≤ N (5)
The proof of this theorem is in appendix B. Observe its similarity with the classical Strang and Fix conditions for linear operators [START_REF] Strang | A Fourier analysis of the finite element variational method[END_REF]. Note that Beylkin [START_REF] Beylkin | On the fast algorithm for multiplication of functions in the wavelet bases[END_REF] has designed a recursive algorithm to compute the wavelet coefficients on the product of two functions. It converges fast if the functions are regular. The computation of the exact product involves an infinite number of computations; an approximation is computed by stopping the recursion before the infinity. The result presented here is different in the sense that it characterizes approximate product operators which are compatible with the scaling operation. In section 3 we effectively build such operators for piecewise polynomial functions.

3 Working with piecewise polynomial functions

Introduction

Since the mapping x → a × x is linear, building a product operator appears at a first glance as the construction of a collection of linear operators. The extra requirement that the operator be associative shows that things are more complicate; indeed, checking the associativity is a nonlinear problem. This eliminates Fourier methods for finding approximate products.

A case where approximate products are easy to find is when the coordinate of the approximation of a function is a sample (interpolating scaling functions [START_REF] Deslauriers | Symmetric iterative interpolation[END_REF][START_REF] Donoho | Interpolating wavelet transforms[END_REF]) or an approximation of a sample (Coiflets). The approximate product is defined by having the coordinate of the product to be the product of the coordinates, as in lemma 1. The functional space on which the general product is defined is a resolution space at a fine scale. There remains to define an approximation subspace which is product invariant. Since we are dealing with multiresolution analysis, the natural idea is to consider a coarser resolution space. In general, it is not product invariant under the action of the product which is defined on the fine scale. Take for instance the Schauder basis, where approximations are defined by linear approximations between samples. The scaling function is the hat function, which is the autocorrelation of the Haar function 1 [0,1) . Then the coarser resolution spaces are not product invariant. Indeed, let us consider a coarse scaling function, e.g. a scaled hat function; if the resolution space was product invariant, the square of the scaling function would be a linear interpolation over the coarse grid. This is not the case, because the approximate square is computed by performing an interpolation of the square over the fine grid, and the result is a linear interpolation over samples of a parabola. It is not a linear interpolation over points of the coarse grid. So, even in this simple case, the matter of finding product invariant approximation subspaces is not obvious.

The nature of approximating product invariant subspaces is linked to the choice of the approximate product operator. To better understand the nature of this link, we have chosen to restrict our attention to the special case where functions are piecewise polynomial ones. This will allow us to characterize the relationship between the choice of the operator and the approximation subalgebras.

Observe that wavelets can detect singularities in piecewise regular functions (provided the singularities are not too close) and analyze them ( [START_REF] Jaffard | Exposants de hölder en des points donnés et coefficients d'ondelettes[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]). Then these function pieces can be approximated by piecewise polynomial functions. We can therefore assume to have an approximation of piecewise regular functions as piecewise polynomials.

We now assume assume that the function has been preprocessed to be represented as an equally spaced piecewise polynomial function of degree smaller than some N ∈ N. The study of conditions (P 1) and (P 2) will not bear on general functions but on piecewise polynomial ones.

This section is organized as follows. Section 3.2 characterizes the approximate product on piecewise polynomial functions by showing that they must be defined by a Hermite interpolation. To recover property (P1), section 3.3 studies the approximation subspaces of piecewise polynomial functions which are product invariant. In particular, all of these subspaces include a minimal one which consists in so called regular functions. Finally, section 3.4 studies the case where the approximate product is defined by a Hermite interpolation between two points and exhibits an approximation projector which satisfies properties (P1) and (P2), and hence equation (1).

Characterization of approximate products for piece-

wise polynomial approximations 3.2.1 The algebraic Strang and Fix conditions for piecewise polynomial approximations Denote by S N,δ (t) the space of piecewise polynomial functions over successive intervals of length δ and with a degree smaller or equal to N . To maintain consistency with the linear and nonlinear approximation conditions previously mentioned, we assume there exists a linear approximation operator P δ which transforms functions into elements of S N,δ (t). P δ is defined by scaling a kernel K which is assumed to have the structure

K(t, s) = k∈Z l=N l=0 (t -k) l 1 [k,k+1) (t)ϕ l (s -k) (6)
and to satisfy the assumptions 1 of the result of Strang and Fix with the order N (two such operators are presented in [START_REF] Chaplais | Product invariant piecewise polynomial approximations of signals[END_REF]). Theorem 2 is transposed to S N,δ (t) and relates it to an algebraic condition:

Proposition 2 Denote by R N [t]
the space of real valued polynomials with degree smaller or equal to N , and by F δ the operator

[F δ p] (t) = k∈Z p k (t/δ -k) 1 [k,k+1) (t/δ) which identifies polynomial sequences in R N [t] to functions in S N,δ (t). If × is a product operator over R N [t],
it is transposed to an operator * δ on functions in S N,δ (t) as follows:

(f * δ g)(t) = F δ (F -1 δ f ) k × (F -1 δ g) k k∈Z (7)
Then * δ approximates the product on the image of P δ like in condition (4) of theorem 2 if and only if × satisfies:

t i × t j = t i+j ∈ R N [t] if i + j ≤ N ( 8
)
This result is proved in [START_REF] Chaplais | Product invariant piecewise polynomial approximations of signals[END_REF].

Comment: the degree of the product is restricted to be smaller or equal to N in order to have S N,δ (t) product invariant. This, in turns, makes sure that the complexity of the function representation does not increase with every product operation. It also makes sense numerically: since the elements of S N,δ (t) are at best approximations of regular functions at the order N , there is no need for a greater precision on the computation of the product.

Description of all approximate products

Proposition 2 reduces the search for approximate products over S N,δ (t) to algebraic products on R N [t] which satisfy [START_REF] Deslauriers | Symmetric iterative interpolation[END_REF]. These are characterized in the following lemma:

1

In particular, it is defined by a shift invariant scaled kernel.

Lemma 2 (Hermite interpolation product) Let × an associative, commutative product over R N [t] which satisfies [START_REF] Deslauriers | Symmetric iterative interpolation[END_REF], and define T N +1 ∈ R N [t] by T N +1 = t N ×t. Then, for any p and q in R N [t], p×q is the Hermite interpolation of pq at the (possibly multiple) zeros of t N +1 -T N +1 (t) in C. Conversely, any such product is associative, commutative, and satisfies [START_REF] Deslauriers | Symmetric iterative interpolation[END_REF]

in R N [t].
This lemma is proved in appendix C.

Interpretation: it is convenient to understand the product operators × of lemma 2 as follows:

• at every interpolation point z of order o, a Taylor expansion at order o of the polynomials x and y is computed;

• at every interpolation point z of order o, the product of the expansions is computed and the coefficients of degree greater than o are set to zero; denote by p z the result;

• the product x × y is then the Hermite interpolation of the various p z .

This leads to the following constructive characterization of approximate product operators on S N,δ (t)

Theorem 3 Let * δ a product operator on S N,δ (t) related to an algebraic product operator × by [START_REF] Beylkin | On the fast algorithm for multiplication of functions in the wavelet bases[END_REF]. Then * δ approximates the product on the image of P δ like in condition (4) of theorem 2 if and only p × q is defined by the Hermite interpolation of pq in R N [t] on a set of interpolation points.

Examples Assume that N is odd, N = 2p + 1. Then the product can be defined as the Hermite interpolation of the product at the points 0 and 1 and orders p. For p = 0, the product is the linear interpolation of product of the samples. For continuous functions, the resulting product is the same as the product previously exhibited on the Schauder basis.

Product invariant approximation subspaces

Our working space is now S N,δ (t). If we remember the decomposition x = e + w, e should belong to an approximation space which satisfies condition (P1), e.g. it is product invariant. This section shows that all such approximation spaces include a minimal algebra which is defined by "regular" functions. In particular, the number of parameters required per time step to describe a regular function has a non zero lower bound. This is a contrast to the situation of multiresolution approximations where a nested structure of approximation spaces exist such that the number of parameters per time unit required to describe functions in a resolution space tends to zero as the scale tends to the infinity.

A variation on the Strang and Fix condition

All of the previous results require the existence of an approximation operator for regular functions. Consistently, we characterize the linear operators Q δ on S N,δ (t) which approximate regular functions at the order N . Consistently again, we assume that

Q δ = F δ Q F -1 δ
, where Q applies a unique operator Q on blocks of q successive polynomials. In other words, Q δ is defined by q-shifting and scaling an operator Q on (R N [t]) q . Proposition 3 Let P δ the scaled operator which codes functions into piecewise polynomials. Then Q δ P δ approximates functions of class C N +1 with an error of order δ N +1 if and only if

Q ((t + k) p ) k∈Z = ((t + k) p ) k∈Z for 0 ≤ p ≤ N (9)
The proof essentially consists in showing that Q δ P δ has the structure required by the result of Strang and Fix with a shift length qδ. Details of the proof are in [START_REF] Chaplais | Product invariant piecewise polynomial approximations of signals[END_REF]. The choice of q > 1 is to allow more general operations on piecewise polynomials like combined interpolation and subsampling.

In looking for subspaces of S N,δ (t), we will require that they be defined as the image of an approximation operator Q δ . An important consequence of the previous proposition is that these subspaces must contain the functions t i , i ≤ N ; if they are product invariant, they must contain all the monomials defined by the approximate product operator.

Product invariant subspaces and Hermite interpolation points

We have characterized approximate product operators over S N,δ (t). To define an approximation on this space which satisfies property (P1), we must study the subspaces of S N,δ (t) which are product invariant and include polynomials of degree smaller or equal to N , a condition which is necessary for them to qualify as an approximation spaces. It turns out that all of these various spaces include a minimal one, which is the space of so-called regular functions of S N .

Notations Let × an associative, commutative product over R N [t] which is defined by a Hermite interpolation as in theorem 2.

• Z is the set of interpolation points which defines the product ×, without counting the order of multiplicity.

• if z ∈ Z, o(z) denotes the order of interpolation used by the product at the point z.

• I denotes an arbitrary finite subset of I and S N,I [t] is the space of sequences with elements in R N [t] and indices in I.

Regular functions

Definition 1 (Regular functions) The subspace SN,I [t] of "regular functions" in S N,I [t] is the set of sequences p ∈ S N,I [t] such that p

(j) k (z) = p (j) k (z) if z + k = z + k and j < min(o(z), o(z)).
Interpretation: sampling points z and z such that z + k = z + k with k, k ∈ Z are called overlapping sampling points. The redundancy introduced by the regularity can be removed by identifying any regular function to a unique polynomial, while preserving the product. At every sampling point, for all indices, the maximum order of interpolation is computed for all sampling points which may overlap here. The global polynomial is defined by Hermite interpolation at these maximum orders. The polynomial product is defined by the same interpolation. This establishes an algebraic isomorphism between regular functions and a single polynomial algebra. The dimension of the latter is Ω(I), which is the sum of all the orders of interpolation increased by one2 . An important property of Ω(I) is that it is greater or equal to N plus the number of elements of I, if I is finite. Indeed, Ω(I) is an increasing function of this number of elements, since each shift of the interpolation points adds at least one node which does not overlap the previous ones. If I has one element, then Ω(I) = N + 1.

Minimum approximation subalgebras Regular functions are the minimum approximation product invariant subspace of piecewise polynomials:

Theorem 4 (Minimum subalgebra)

• SN,I [t] is a product invariant subspace of S N,I [t] which includes the sequences ((t + k) p ) k∈I for 0 ≤ p ≤ N .

• If I is finite and if Σ I is a subalgebra of S N,I [t] which includes the sequences ((t + k) p ) k∈I for p = 0 and p = 1, then SN,I [t] ⊂ Σ I .

Proof.

Let us prove the first statement. From the definition of SN,I [t] and the definition of the product, it is easy to verify that SN,I [t] is product invariant. If z and z are two interpolation points, and k and k two integers

such that z + k = z + k, then ((t + k) p ) (j) (z) = (t + k) p (j) (

Recovering properties (P1) and (P2) in the case of a two point interpolation product

There are as many spaces of regular functions as there are interpolation points. To simplify, we consider here the case where the product is defined by a Hermite interpolation between two points of equal order. We shall assume, without loss of generality, that these two points are 0 and 1. This implies that the piecewise polynomials are defined by an even number of parameters, e.g., their degree N is odd, e.g. say, N = 2n + 1.

Hermite function representation and interpolation

After a suitable piecewise polynomial approximation of degree 2n + 1, functions are represented as sequences of pairs (l k , r k ) k∈Z of polynomials of degree n. The corresponding piecewise polynomial function can be written

k∈Z i=n i=0 l (i) k (0)H i t δ -k + r (i) k (0)(-1) i H i 1 -t δ -k 1 [k,k+1) t δ
where H i (t) is the unique polynomial of degree 2n + 1 which satisfies

H (j) i (0) = δ i,j and
H (j)
i (1) = 0 for 0 ≤ j ≤ n A method for constructing H i , 0 ≤ i ≤ n can be found in [START_REF] Stoer | Introduction to Numerical Analysis[END_REF]. Repeated applications of Rolle's theorem prove that H i (t) > 0 if 0 < t < 1.

p 1 (t)=t 3 L 1 (t)=0 R 2 (t)=0 R 1 (t)= δ 3 +3δ 2 t L 2 (t)= δ 3 -3δ 2 t p 2 (t)=(δ-t) 3 δ Figure 1: A sequence (p 1 (t) = t 3 , p 2 (t) = (δ -t) 3
) of two cubic polynomials can be viewed as a sequence ((L

1 (t) = 0, R 1 (t) = δ 3 + 3δ 2 t) , (L 2 (t) = δ 3 -3δ 2 t, R 2 (t) = (δ -t) 3
))) of couples of polynomials with degree 1. L i polynomials are Taylor expansions at the left point; R i are expansions at the right point. "Regular functions" are defined by R 1 = L 2 .

Product definition

The product

(l k , r k )××(λ k , ρ k ) of two sequences (l k , r k ) and (λ k , ρ k ) is defined by (l k , r k ) × ×(λ k , ρ k ) = (l k × λ k , r k × ρ k ) with (p × q)(t) = i+j≤n p i q j t i+j ∈ R n [t]
3.4.3 Recovering properties (P1) and (P2)

The results of section 3.3 are used to build product invariant subspaces using a regularization operator which removes singularities; (P1) is thus recovered. The symmetric structure of this operator makes it possible to recover property (P2).

Regularization operator Subalgebras are defined by removing singularities. Define the regularization operator R k by

R k (l, r) k = l k , r k + l k+1 2 R k (l, r) k+1 = r k + l k+1 2 , r k+1 R k (l, r) j = (l j , r j ) otherwise
All of the R k 's commute, and, given an index set I, the operator R I is defined as the composition of the R k 's for

k ∈ I. If S is a subalgebra of S N [t], then R I S is also a subalgebra of S N [t]
. Moreover, if S includes regular functions, then R I S also includes regular functions. Finally, R I reproduces polynomials like in the Strang and Fix conditions (9) because polynomials sequences are regular functions which are unchanged by R I . It is thus a high order approximation projector when the interval between the polynomial samples tends to 0.

Singularity operator

The singularity operator S k is defined by

S k = Id -R k , e.g. S k (l, r) k = 0, r k -l k+1 2 S k (l, r) k+1 = l k -r k+1 2 , 0 S k (l, r) j = (0, 0) otherwise If I ⊂ Z, S I is defined as S I = Id -R I .
Recovering properties (P1) and (P2) These two operators are symmetry based, like the decomposition in the Haar basis. For this reason, properties (P1) and (P2) are recovered:

L 1 (t)=0 L 1 (t)=0 R 1 (t) = δ 3 = L 2 (t) R 2 (t)=0 R 2 (t)=0 R 1 (t) = 3δ 2 t = -L 2 (t)
with e = R I x and w = S I x, and P (x) is defined with the product operator 3 * .

3

Actually, R is identified to the constants of S N [t], e.g., the coefficients of the polynomial P are constant functions.

Application

We give a very simple application. We wish to estimate the commutation defect between the polynomial and the regularization (approximation) operator. To do so, we estimate R I (P (x)) -P (R I x). This calls for a norm on the elements of S N,δ (t) which is compatible with the product, e.g. x * y ≤ xy. Let us represent the elements σ of S N,δ (t) as sequences (l k , r k ) of polynomial pairs of degree less or equal to n and renumber it as a sequence of polynomials p k . A * product compatible norm σ * of σ is sup k p k n , where p n is the product compatible polynomial norm i=n i=0 p i t i n = i=n i=0 |p i |δ i . Assuming that the norms of the derivatives P (2k) (e) are uniformly bounded by K, we get the estimate from equation [START_REF] Jaffard | Exposants de hölder en des points donnés et coefficients d'ondelettes[END_REF]:

R I (P (x)) -P (R I x) * ≤ K(cosh S I x * -1)
This estimate can be used to select on which index set I the regularization operator R I must bear in order to have a commutation defect which is smaller than a given threshold .

Conclusion

From a simple averaging scheme we have drawn equation (1) which relates the average of f (u) to the average E(u) of u and the averages of the powers of the remainder u -E(u). A projector satisfies [START_REF] Sanders | Averaging Methods in Nonlinear Dynamical Systems[END_REF] if and only if it satisfies properties (P1) and (P2). This implies that the product of two "averaged" functions be itself an "averaged" function. This property is very strong; in fact, we show that, when the space of "averaged" functions is generated by the shifts of a single waveform, this property implies that the waveform is the characteristic function of an interval. In terms of approximation theory, this means that the order of approximation provided by the projection on the space of "averaged" functions is the lowest possible.

To work around this limitation, we observe that the space of "averaged" functions is an approximation space and that an approximate product may be substituted to the actual one without degrading the order of approximation. We have exhibited in section 2.2.2 a new result on the characterization of such approximate product operators by means of a Strang and Fix like condition.

However, it seems difficult to construct such approximate products in a general framework. This is why we have restricted ourselves to piecewise polynomial functions. For this space of functions we have exhibited a complete parameterization of approximate product operators which is based on Hermite interpolation.

Using such an approximate product implies the existence of a minimal product invariance subspace; it excludes the construction of embedded approximation spaces obtained by rescaling, in the manner of multiresolution approximations. By characterizing the minimum approximation subalgebra, we can observe that the "smooth" part of the function should be obtained by removing the singularities at the overlapping interpolation points.

In the case of a two point interpolation scheme, this idea is explicited in section 3.4.3 and a suitable decomposition of the function into a regular and a singular part is exhibited which satisfies conditions (P1) and (P2). The regularization operator is an approximation projector which satisfies a simplified version of equation ( 1), which was the goal of the article. This operator is based on the averaging of the left and right derivatives of the function at the interpolation points. As such, it gives sense to the "average" of a nonlinear function f (u) as a function of the input u.

This simple decomposition works because it is based on the symmetry properties of its arguments, a structure which is sufficiently strong to be preserved through nonlinearities.

Overall, it is possible to retrieve properties (P1) and (P2) by using approximate product operators on piecewise polynomial approximations of functions. Equation ( 10) is very simple and can be used, for instance, to detect which singularities are negligible in the commutation between the regularization and the polynomial operator.

A Proof of lemma 1

Proof. Let r the maximum integer n for which φ 0 * φ n = 0; because the product is commutative, r ≥ 0. Let M r = max {k ∈ Z s.t. C k (ϕ 0 * ϕ r ) = 0}. Consider now (φ 0 * φ r ) * φ Mr+r ; there exist a non zero constant a such that it is equal to aφ Mr * φ Mr+r , which is itself non zero. On the other hand, it is also equal to (φ 0 * φ Mr+r ) * φ r , which is zero if M r > 0; hence, M r ≤ O. By a symmetric argument, one verifies that m r = min {k ∈ Z/C k (ϕ 0 * ϕ r ) = 0} satisfies m r ≥ 0; hence, m r = M r = 0. This implies that φ 0 * φ r = Aφ 0 . Consider now φ -r * (φ 0 * φ r ) = A 2 φ -r ; it is also equal to (φ -r * φ r ) * φ 0 , which is non zero if and only if r = 0. If * is the usual product, taking x = φ i and y = φ j with i = j in (2) implies C n (φ i φ j ) = 0 for all n, e.g. φ i φ j = 0 almost everywhere. The domain argument used in the proof is illustrated in figure 3.

B Proof of theorem 2

Proof. Necessary condition: if i + j ≤ N and (k, l) ∈ I N , then k ≥ i + 1 or l ≥ j + 1. Inserting f (t) = t i and g(t) = t j with i + j ≤ N in (4) proves, with the Strang and Fix conditions, that (5) holds. Similarly, (13) is bounded by the sum of two terms with k = N + 1 and l = 1 and l = N +1. Finally, the Strang and Fix condition E δ (t i ) = t i and condition [START_REF] Strang | A Fourier analysis of the finite element variational method[END_REF] imply that ( 14) is equal to [T(fg) -Tf * δ Tg] (t) = (k,l)∈In f (k) (t)k! × g (l) (t)/l! × ((st) i * δ (st) j ) (t) Together with the continuity of the product, this proves that (14) is bounded by a term like in (4).

C Proof of lemma 2

Proof. Let T N +1 = t N × t. There exists a unique family (a 0 , . . . , a N ) such that

T N +1 = i=N i=0 a i t i def = Q(t) (15)
Let (z 0 , . . . , z N ) the complex roots of the (usual) polynomial t N +1 -Q(t).

Then T N +1 is the Hermite interpolation of t N +1 at these points. We are going to show that × is obtained by Hermite interpolation of the usual product at (z 0 , . . . , z N ). To do so, we only have to prove it is true for monomials generated by ×.

The latter are defined recursively by T N +p = T N +p-1 × t. We assume that T N +p-1 is the interpolation of t N +p-1 at (z 0 , . . . , z N ). There also exist a polynomial q(t) of degree < N and a real b such that T N +p-1 = q + bt N . Then T N +p = qt + bT N +1

By assumption, q(z i ) + b(z i ) N = z N +p-1 i ; hence,

T N +p (z i ) = z N +p-1 i -b(z i ) N z i + bT N +1 (z i ) = z N +p i -bz N +1 i -bz N +1 i = z N +p i
which proves the interpolation result at the order 0. For higher orders k, the induction becomes:

T (k) N +p (z i ) = (qt) (k) (z i ) + bT (k) N +1 (z i ) = T N +p-1 t -bt N +1 (k) (z i ) + bT (k) N +1 (z i ) = (T N +p-1 t) (k) (z i ) = T (k) N +p-1 (z i )z i + T (k-1) N +p-1 (z i ) = t N +p-1 (k) (z i )z i + t N +p-1 (k-1) (z i ) = t N +p (k) (z i )
Conversely, one verifies that the Hermite interpolation is indeed associative and that, by definition, it preserves the polynomials of degree ≤ N .

  z). This proves that the sequences F -1 1 (t p ) = ((t + k) p ) k∈Z belong to SN,I [t]. Let us prove now the inclusion property. Denote by T l the sequence ((t + k) l ) k∈I , l = 0, 1. By assumption, T 0 ∈ Σ I ∩ SN,I [t] andT 1 ∈ Σ I ∩ SN,I [t]; hence the powers T n def = T n-1 * T 1 belong to Σ I ∩ SN,I [t]. One can prove that any element p of SN,I [t] can be uniquely represented by a polynomial H -1 p of R Ω I [t],where Ω I is an integer determined by the overlapping sampling points, and that H commutes with the product. Hence, if n < Ω I , T n has the unique representationH -1 T n = t n in R Ω I [t].As a consequence, (T 0 , T 1 , . . . , T Ω I -1 ) is a basis of SN,I [t] because it is the image of the basis (1, t, . . . , t Ω I -1 ) of R Ω I [t]. Since (T 0 , T 1 , . . . , T Ω I -1 ) is also in Σ I , necessarily Σ I includes the algebra generated by this family, e.g. SN,I [t]. Theorem 4 indicates that the only degrees of freedom which are left in the definition of an algebra Σ I lie in the singularities at the overlapping sampling points.

Theorem 5 (

 5 Product and decomposition) Let I ⊂ Z, V I the image of R I , W I the image of S I . Then S N [t] = V I ⊕ W I , and (P1) the product of two elements of V I is an element of V I (P2) the product of an element of V I with an element of W I is an element of W I (P3) moreover, the product of two elements of W I is an element of V IProof. The decomposition of x between R I x and S I x is the decomposition of a pair of vectors into the symmetric and their antisymmetric parts. Theorem 5 yields a polynomial formula which is similar to (1):

Figure 2 :Corollary 1

 21 Figure 2: The piecewise polynomial sequence of figure 1 is decomposed by symmetry into its regular part and its singular part.

Figure 3 :

 3 Figure 3: The diffusion of the analysis and reconstruction processes implies a diffusion of the product operator.

  Sufficient condition: Denote by Tx the Taylor expansion of a function x at point t with degree N . The left handside of (4) is decomposed as|fg(t) -[(E δ f ) * δ (E δ g)] (t)| ≤ |fg(t) -T(fg)(t)| (11) + |[E δ f * δ E δ (g -Tg)] (t)| (12) + |[E δ (f -Tf ) * δ E δ Tg] (t)| (13) + |T(fg)(t) -[E δ (Tf ) * δ E δ (Tg)] (t)| (14)The definition of Tx implies that (11) is zero. A localized version of the Strang and Fix approximation conditions and the localization of the product implies that (12) is bounded by a term like (4) with k = 0 and l = N + 1.

At an interpolation point, the number of parameters describing the function is equal to the order plus 1