domain: info sdu.stu scco

text:
Claire Anjou
email: claire.anjou@espe-guadeloupe.fr

Thomas Forissier
email: tforissi@espe-guadeloupe.fr

Jacqueline Bourdeau
email: jacqueline.bourdeau@licef.ca

Yves Mazabraud
email: yves.mazabraud@espe-guadeloupe.fr

Roger Nkambou
email: nkambou.roger@uqam.cafournier.frederic@uqam.ca

Frédéric Fournier

Elaborating the Context Calculator: A Design Experiment in Geothermy

Keywords: Modelling, Context, Science Education, Learning, Context Effects, Geothermy 1

This paper presents a research project in science education that is positioned at the intersection of computer science and context in learning. The main objective is to improve learning process by creating a software tool that participates, from the inception to the achievement, in the design of leaning scenarios, based on context effects, and, to show that context effects' learning is an efficient method in the development of student's knowledge regarding a concept. The software will compute differences between twoor moreexternal contexts based on specific parameters, related to the phenomenon or object that students are expected to study. The elaboration of the calculator ("the MazCalc") is conducted using the design based research theory, meaning that several iterations of learning field experiments are conducted in order to collect relevant data which are used for the tool creation. In this paper, the design of the scenario involves students from North America and from the French West Indies and the concept studied is about geothermal energy. The instantiation of the context calculator is made with the geothermal object of study, and the differences are computed in the calculator between the two contexts mentioned. This example makes possible to validate the link between context effects predicted and observed, and also to study the impact of external context on the learning process. This study has been conducted thanks to the GEOTREF project support.

Introduction

Our study focuses on the contextualization of science education involving fieldwork [START_REF] Schwartz | Developing Views of Nature of Science in an Authentic Context: An Explicit Approach to Bridging the Gap between Nature of Science and Scientific inquiry[END_REF]. In general, context is defined as everything that surround an entity and that have connections with it [START_REF] Abowd | Toward a Better Understanding of Context and Context-Awareness[END_REF]. A more operational definition is given and discussed in 2.1 [see 2 .1].

One objective is to create a software tool, in order to support the design and development of context-sensitive tasks inside the learning scenario, by considering Context Effects [START_REF] Forissier | Computing the Context Effect for Science Learning[END_REF]. This software will be able to provide several external context instantiations [START_REF] Van Eijck | Towards a Chronotopic Theory of "Place" in Place-based Education[END_REF][START_REF] Van Wissen | A Constraint-based Approach to Context[END_REF], in relation with the scientific object of study, and to compare them to predict the emergence of Context Effects.

The other objective is to validate the hypotheses that scientific learning will occur thanks to the Context Effects predicted by this software tool. For that, an innovative learning project has been set up.

The project is built using the Design Based Research approach [START_REF] Sandoval | Design-Based Research Methods for Studying Learning in Context: Introduction[END_REF], meaning that we are developing theoretical methodology for the tool creation while experimenting it insitu. The design experiments are iterative, meaning that the results of one experiment are used to improve the design of the next one.

In this paper, the DBR iteration we are talking about is in the domain of geothermy. The software tool compares two geothermal learning contexts modelling, to highlight the points or concepts about geothermy that shows differences. After this comparison, the objective is to bring to light significant differences and take advantage of them to build a context-sensitive learning scenario about geothermy with the collaboration of two groups of students from those two contexts. The learning is expected to happen thanks to the gaps between contexts that may enrich learners' interactions within and between the two contexts.

Our position is that context gaps during student's collaboration are what create Context Effects [START_REF] Leurette | La Contextualisation dans l'Enseignement des Sciences et Techniques en Guadeloupe[END_REF]. The Context Effect learning is an innovative approach in science education, allowing authentic leaning where students will develop rich and complete conceptions, and will open their mind to very different world contexts.

Theoretical Framework

The complex subject area in which is involved this study makes necessary the explanation of all scales of theories that we are dealing with, from the science education one to the context effects one, including the DBR. The MazCalc is also introduced in this section as well as the concept of geothermy.

Context and Education

We adopted the operational definition of context by Bazire and Brézillon [START_REF] Bazire | Understanding Context Before Using It[END_REF], who considers that « context acts like a set of constraints that influence the behaviour of a system embedded in a given task. » In the teaching field, these constraints can be of different nature: epistemological, socio-economical or didactical [START_REF] Delcroix | Vers un Cadre d'Analyse Opérationnel des Phénomènes de Contextualisation Didactique[END_REF].

The spectrum of relations between context and education consist in inputs derived from several academic disciplines. The context is in most cases defined as an external factor to learning. In educational sociology, this factor shapes the relations between contextual and individual parameters in the pursuing of academic success [START_REF] Duru-Bellat | Le Déroulement de la Scolarité au Collège: le Contexte "Fait des Différences[END_REF]; in science education [START_REF] Sauvage Luntadi | La Compétence de Contextualisation au C ur de la situation d'Enseignement-Apprentissage[END_REF], it corresponds to institutional constraints weighting on the relationship between student and teacher; in language education [START_REF] Blanchet | Perspectives pour une Didactique des Langues Contextualisée[END_REF], it is coupled with the command of language for learners and with the teacher's ability to adapt their teaching. In science education, context-based approach [START_REF] King | New Perspectives on Context-based Chemistry Education: Using a Dialectical Sociocultural Approach to view Teaching and Learning[END_REF] deals with the natural context of student acting as a stimulating factor to providing authentic teaching [START_REF] Schwartz | Developing Views of Nature of Science in an Authentic Context: An Explicit Approach to Bridging the Gap between Nature of Science and Scientific inquiry[END_REF]. Here we use those ideas of authentic external context in view of the fact that they take part in the construction of the internal context thanks to Context Effects.

In the teaching situation studied, the internal context is situated at the level of students' conceptions. Like Giordan [START_REF] Giordan | Les Origines du savoir[END_REF], we define conceptions as coherent mental images. These conceptions are related to conjunctures and can be considered as situated conception [START_REF] Clément | Situated Conceptions and Obstacles. The Example of Digestion / Excretion[END_REF]. In this text, the dimension related to the contexts of these conceptions is denominated as internal context.

Context Effects

In this research work, we aim to explain that student conceptions about geothermy and geothermal context are connected. To this end, an experiment centred on a pedagogical project about geothermy has been set up between students from Canada and Guadeloupe.

Our purpose is to develop a tool that will predict the emergence of Context Effects. In the psychology field, Context Effects have been defined as the influence of an environmental factor on someone's perception [START_REF] Tourangeau | Cognitive Processes Underlying Context Effects in Attitude Measurement[END_REF]. In sciences education the approach is not the same, Context Effects are pedagogical events occurring when there is a clash between student's conceptions, coming from distinct environmental contexts, and about a shared topic being studied [START_REF] Forissier | Eléments de Conceptions des Etudiants de Première Année Scientifique de Guadeloupe sur les Saisons Climatiques et l'Orientation de la Lune[END_REF]. To study context effects, working collaborations on geothermy have been established between Canada and Guadeloupe students, so that they can formulate their conceptions and build together a more accurate understanding idea of geothermy.

In order to maximize the emergence of context effects, two key factors must be taken into account [START_REF] Forissier | Modeling Context Effects in Science Learning: The CLASH Model[END_REF] the amount of interactions between students, and the external context gap. The geothermal context in the Caribbean and North America shows many differences (see chap. 2.5) making it a good topic for the research.

Design Based Research (DBR)

Design-Based Research (DBR) is a recent research method that is used in educational research and more specifically in education science. It combines theoretical concerns and field considerations to develop solutions to produce successful learning [START_REF] Sandoval | Design-Based Research Methods for Studying Learning in Context: Introduction[END_REF].

The first specificity of DBR is the double purpose intended: the progress of theoretical knowledge related to the practical reality and the development of new practical solutions linked with the theory.

The second specificity is the "in situ" mode uses as much for the field actors collaboration than for the micro-experimental tests conducted during the different phases. The DBR engine is the different iterations that provide a wondering about the early or intermediate results and a testing of new ideas (theoretical and experimental) during the next iteration.

DBR distinguish itself from action research that studies problem coming from field actors, and solved jointly by researchers and practioners, and from developmental research, which has no theoretical claim, and can be perform in a laboratory setting [START_REF] Barrab | Design-Based Research: Putting Stake in the Ground[END_REF][START_REF] Bourdeau | DBR, une Méthodologie de Recherche pour le Design d'Environnements d'Apprentissage[END_REF].

The iteration we are talking about in this paper deals with geothermy and follows two other, the first one in biology was about frogs [START_REF] Fecil | Construire un Enseignement en tenant compte des Effets de Contexte[END_REF] and the second one in the environmental field was about water.

The Mazcalc : a Context Calculator

To better understand the context effects and their applications in DBR experiments, and to predict the probability of their occurrence, we have developed a computational model of a context-gap calculator. The idea is to propose a system able to capture and evaluate differences amongst contexts. Our hypothesis is that the more different the contexts under study are, the stronger the context effect is likely to be in the experiment. The very first step of our study was to characterize at best each context. To do so, one may have to study it at different scales (geographical, temporal, and political). The integration of the observations at each relevant scale will allow the modelling of the context (see CLASH model [START_REF] Forissier | Modeling Context Effects in Science Learning: The CLASH Model[END_REF]). The two, or more, contexts used in the experiment being well defined, it is then possible to study context effects.

Obviously, two contexts may appear significantly different in one aspect (weather, educational system, geography, economy, politics) and similar in another (language, content of teachings, economy). The context-gap, being the difference between both contexts, is then defined by the summation of all aspects, some very different, some not. Contexts effects may therefore also be studied at different scales. To undergo precise analyses of the context effects, we need to quantify the context gaps. Using a specific calculator [START_REF] Forissier | Computing the Context Effect for Science Learning[END_REF], we identify parameters, and then assign a numerical value to each context-defining parameter, then we measure the difference between the parameters and so, the overall difference between both contexts. It is then possible to study the link between the context-gaps and to predict the context effects. In the following section, we describe the prototype of this calculator and its testing in geothermy.

Geothermal Context

Geothermal energy is thermal energy generated and stored in the Earth. Geothermal context is something very complex. It can be define by all sorts of other concepts. For example, it depends on the geology, the industry, the climate, the environment, the resources and many others concepts.

In North America, the bedrock mainly consists of ancient Precambrian rocks from the Canadian Shield. The rocks are cold and the soil temperature in Quebec is constant: 12°C. The geothermal heat is used to regulate the temperature in buildings, to create spars, to grow crops and other purposes. It is called low temperature geothermy.

The Caribbean Islands are located in a subduction zone between the Atlantic plate and the Caribbean one. Those recent geological formations are creating a great potential for high temperature geothermal energy. In Guadeloupe, this potential has been harnessed and a geothermal power plant produces about 5% of the local demand for electricity.

Material and Methods

In this section, the material and methods selected and used for the research is described in two distinct parts: on the one hand for the context calculator; one the other, for the pedagogical experiment.

3.1

The Mazcalc

Mazcalc Ontology

The context, as we hear for the modelling, is something very complex to describe. There are many definitions and theories about this concept and it is choosey to select only one and to stick to it. Not all of the project participants have the same conceptions about how to describe and characterize context. This chapter suggests an ontology that we have clarified in order to formalize the concept of context and to facilitate its usage in the software creation and implementation.

Creation

 A context, in the context modelling tool, is always defined with respect to the object of study hereof : geothermy.  The object of study is specified by a set of Contextual Parameters to which a variable is assigned, and with a corresponding numerical value defined in natural numbers.  The parameters have Properties, which define the rules that will be applied for the calculation.  They can be composite, dependant or independent (if they are composite, the parameter variable choice gives rise to a new parameter depending on the first one, and that can give rise to another composite parameter, forming a tree where values by level are weighted).  They can be empirical (from the literature), estimated (by experts), calculated (from data) or measured (by learners).  They can have a qualitative or a quantitative nature.  They can have a scale of continuous or discrete variables.  They can have a list of unordered or ordered variables.  A measuring device (or for monitoring) can be associated to a parameter, providing an explanation for how to collect the appropriate data.  A parameter is part of one or several families.  A family is used to split the context: it can be assigned and used:

-For a investigative domains by student, -For an observation scale, -For a field of study...

Use

The software has to provide a calculation of context gap for each parameter (for the instructional designer).



Software Elaboration

Different scales of creation and use have been identified and specific actors participate to the software elaboration.

Level 1.

Actors: computer analyst. Software framework construction regardless from the object studied. The blank frame of the ontology was aforementioned.

As illustrated by the figure 1, in this step, only the software skeleton is created and none of the parameters is mentioned. The designer creates the possibility to add the specific parameter and plans the selection for the parameters properties implementation. Fig. 1 shows an example for one parameter property. Actors: Domain expert designer and specialists from several contexts. Modelling for every possible context, regarding the object of study. Specification of all contextual parameters and their properties (figure 2).

It is necessary that a number of experts or specialists from various contexts take part in this step, to provide a review of all possible parameters that can represent the domain, in every possible context.

Level 5

Actors: Teachers and students Uses of the various tools provided by the software (measurement and monitoring tools, communications tools, test questionnaires...)

Design Experiment in Geothermy Progression

Selected students for this iteration are teachers in training from UQAM (Université du Québec à Montréal) and from the UA (Université des Antilles) participating respectively in lessons about "Teaching and learning with project in sciences and technologies", and "Innovation in life and earth sciences". In Guadeloupe, students are between 21 and 27 years old and have grown up in the Caribbean. They are preparing a master degree (5 years after the bachelor degree). For Montreal, students have more diversified profiles, they are between 24 and 59 years old, and comes from various countries (Canada, North Africa, Lebanon...), they are also preparing a master degree but some of them already have PhD degree in others fields of study.

Students are committed in a common investigative process focussed on the implantation of a company willing to use geothermal energy as a complement for the cooling of server room and informatics equipments. They have to work together to find the best place in Guadeloupe and in Quebec form the implantation, related to the context characteristics.

The pedagogical experiment lasted four months (from February to May 2016) and is inspired with the Jigsaw method [START_REF] Aronson | The Jigsaw Classroom[END_REF]. The first session was organized in two parts: instructions and elaboration of 5 investigative topics with intragroup work, and then a videoconference between the two groups for the harmonization of the investigative topics (intergroup collaboration through a debate).

Fig. 1. Teams' organization

As explain the figure 4, the two groups (from the two distinct contexts) were split in 5 counterpart teams collaborating together about 5 investigative topics (identical ones in each group). Several means of communication were set available for students (Skype, Moodle platform, Chat, forums) so, the teams can easily collaborate.

After that, teams returned to their original group to pool their findings and build a unique result in the form of an oral presentation, shared during a last session videoconference.

Data Collection

A very detailed monitoring of all the project process have been collected in order to get all the information as possible for the documentation of the pedagogical iteration.

 Video recordings of all exchanges  Audio recordings of some interactions  Pre-test and post-test questionnaire  End of session questionnaires regarding students experience  Follow-up booklet filled by student  Statement form filled by teachers For now, only the video recordings of the exchanges and the pre-test and post-test questionnaires are studied and the work is still in progress.

Results

The Mazcalc

The MazCalc preliminary prototype has been elaborated as a brainstorming tool (using Excel) to grasp the potential difficulties that can be encountered in the tool programming and development, in the parameters ontology or in the general operation of the calculator.

Fig. 5. Small Excel portion of the MazCalculator prototype for geothermy

In the example shown in the figure 5, there are several parameters (B column). Variables are assigned to these parameters, related to the expected context model. Here, two contexts are specified (D and F columns). Specific properties related to each parameter provide the rules that will be applied for the gap calculation. For example, on line 7: "Type of geological formation" gives rise to a pick-list of options. Depending on the chosen variable (cell D7), the pick-list for "Type of rock" (cell D9) will be different. The "Type of geological formation" is so a composite parameter making "Type of rock" a dependent one. The parameter variables "Type of geological formation" are qualitative, discrete, and unordered. The gap calculation between the two contexts for this parameter is 1 or 0 depending on whether the variable is the same or not. For "Type of rock", values are weighted (J9 = 0.5 if the rock's types are different and 0 if they are identical) but only if the variance is zero for the parent parameter. Otherwise, if the variance is equal to 1 (I7) for "Type of geological formation" it is not taken into account for the gap calculation of "geology" family. For the parameter "Age of geological formation", the variables are ordered. That means that the gap value will not be the same depending on whether the ages are distant or not.

Fig. 6. Graph of the parameter's gaps shown in the MazCalc portion

The contexts gaps' values provided by the calculator are plotted graph (Figure 6). In this example, we can see that biggest gap is for "geodynamic" and "Type of geological formation" shows no differences. The identification of distant parameters is used to anticipate context's elements that will spearhead the Context Effects. This can give the teacher a line for the elaboration of a teaching, by informing the tasks in the learning scenario and on the themes that will allow the emergence of context effects.

4.2

The Pedagogical Project

The analysed data for the design experiment allows the identification of students' conceptions and context effects. Videos 95 minutes of video recordings between Guadeloupean and Quebec students have been meticulously examined mainly with the argumentative analysis method [START_REF] Baker | Rainbow: A Framework for Analysing Computer-Mediated Pedagogical Debates[END_REF]. Identified Context Effects are linked to different concepts. Here are some examples of context effect explained:  Climate : the soil is covered with snow in Montreal in winter;  Environment : there is hot water discharges in the sea in Guadeloupe;  Energy industry: in Montreal, mainly hydroelectric dams produce electricity;  Geology : in Guadeloupe the ground is 250°C at et 1000m whereas in Montreal it is around 100°C at 7000m;  Hydrology : there is no lake in Guadeloupe;  Volcanology : there is no active volcano in Quebec;  Pedagogy : in the light of these differences, the pedagogical approaches are perceived differently by students for Guadeloupe and Montreal;

This allows us to realize the large variety of concepts that can lead to the emergence of context effects, while at the same time being related to the geothermal context of the place. It is so possible to identify them by a video analysis: simultaneously with a content analysis and a gesture analysis.

Pre-test / Post-test

The questionnaire for initial and final conception of geothermy was submitted to the students just before the beginning of the first session and during the last minutes of the last session of the project. The analysis for this questionnaire was performed using statistics methods: Frequency distribution tables, cross-tabulations and Multiple Correspondence Analysis (MCA) with the Statistica software, and with Excel.

Students' conceptions before and after the experiment are synthesised on figure 8. The illustration shows the differences between pre-test and post-test for Guadeloupe and Montreal students for 5 questions:

 How do you explain the natural phenomenon of geothermy?  Draw it.  What is the product of a geothermal exploitation?  What deep can we drill for geothermal exploitation? Answers to those questions have been classified, and three categories have been identified:  General answer, gathering answers related to the general phenomena of geothermy: ground, soil or Earth heat, power or energy: the answers here describe geothermy in a simple or universal way.  Contextual answer, related to specific characteristics of geothermy in Quebec (for Canadian) and in the Caribbean (for Guadeloupeans).  Expert answers, with references to technical or scientific concept of geothermy and, answers describing geothermy in another geothermal context than the one from where the student is.

According to the graph (figure 7), conceptions have changed between the pre-test and post-test. This analysis reveals that in Guadeloupe the conceptions are mainly contextual before and after the experiment with a small evolution of the expert conceptions, whereas in Montreal, student's conception evolved in a very significantly way, from general (in pre-test) to expert (in post-test). On the basis of the questionnaire analysis, students' conceptions regarding an object can be various; the local environment plays an important role in the building of those conceptions. In Guadeloupe, the basic understanding of geothermy is very contextual due to the specificities of the geothermal context, (the only French geothermal plant), well known by Guadeloupeans. The conceptions of geothermy are consequently tightly linked to this particular environment.

By contrast, there is no big geothermal plant in Quebec, but low temperature geothermy is something fairly common, it is found in certain buildings (school, museum) or in private homes. The common understanding is therefore linked to this low temperature geothermal context, and the well-known general principle of geothermy is to extract a small amount of degrees from the soil to produce heat.

The video analysis is still in progress and the methodology is being studied to enrich the data treatment. The Rainbow analysis method, [START_REF] Baker | Rainbow: A Framework for Analysing Computer-Mediated Pedagogical Debates[END_REF] for argumentation-based collaborative learning and computer-mediated argumentative interactions can help in understanding how argumentation relates to attempting to solve students' problems of a learning scenario, using support of external representations and knowledge elaboration.

Conclusion

The Mazcalc tool has been elaborated and its preliminary prototype implemented and tested on two contexts (Quebec and Guadeloupe) with geothermy as the learning subject. Our context modelling approach and the formal ontology we have designed for it allows us to describe each context and then apply the Mazcalc prototype. The Mazcalc clearly highlighting context gaps between the two distinct geothermal contexts.

The results confirm that the MazCalc was effective in informing the scenario and predicting the probability of context effects. The results of the questionnaire show an improvement of the quality of the of the experts' representations on both sites. By combining the two analyses (Excel and Statistica adapted depending of the question type), we can say that Guadeloupean have evolved from Contextual to Expert-Contextual conceptions, and that Canadian from General-Unclassifiable to Expert-Contextual conception. This, thanks to context effects created by the social interactions planned in the learning scenario and brought to light by the analysis of the video recordings.

The design experiment showed us that there is a strong link between external context and students' mental models. It is possible to change them for more accurate ones via context effects induced by exchanges between students with different mental models.

The recommendations for the next DBR iteration that we are that the lesson objectives, the project instructions and the speech held by the teachers in both contexts have to be similar or at least completely homogeneous. Students must have the same objectives to maximize collaborations and interactions. It is also important to find a good reason in the pedagogical scenario for interactions, so students can find an interest for the project outcomes in the collaborations with the students from the other environment.

Fig. 1 .

 1 Fig. 1. Wireframe of the first level of creation (add a parameter)

Fig. 2 .

 2 Fig. 2. Wireframe of the second level of creation (specifying the parameter properties)

Fig. 3 .

 3 Fig. 3. Wireframe of the third level (implementing variables for the parameters)

Fig. 7 .

 7 Fig. 7. Graphs of conception's evolution in Guadeloupe and in Montreal between the pre-test and the post-test (population size is on the y-axis).

  It can compute the global context gap for each family (for the teacher).

	 It must contain several pedagogical tools such as:
	-	Communication tools.
	-	Activities book.
	-	Follow-up booklet.
	-	Support for test establishment
	-	Tools for the identification of context effects.
	-	Sensors to capture the observation data.
	-	Tools for video analysis (emotion sensor).


J M Thiriet

M J Martins

H Yahoui
email: yahoui@cegely.univ-lyon1.fr

M Robert

H Fremont
email: fremont@ixl.fr

EIE-Surveyor: Observatory of the Bologna-process in EIE: an aid for the enhancement of mobility

An accurate survey of the present status of the situation of the implementation of the Bologna-process in the fields of Electrical and Information Engineering in Europe is one of the objectives of the EIE-Surveyor project (www.eie-surveyor.org) ERASMUS thematic network. To attain this goal, a survey is proposed, identifying the curricula available in Europe in these disciplines, at the levels of bachelor, master and doctoral studies.

The results obtained will be presented in a dedicated website and also a paper monograph in order to allow free access of students or colleagues to this information.

Introduction

Today, in various European countries, work is in progress to put the curricula in conformity with the Bologna-process scheme. Some of them are actually running it and others are in a reflection or implementation stage. At the moment the result of the European policy relative to ERASMUS mobility has strongly encouraged the mobility of students. Some works remain to be done in order to enhance the mobility thanks to the use of some standardised descriptors in order to describe curricula and competences, and also some works and reflections in order to emphasise the understanding of accreditation methodologies and procedures and the recognition of diploma, which are a huge aspect if we want to encourage actually the mobility of students and also of workers. Task 3 of the EIE-Surveyor thematic network (www.eie-surveyor.org) aims at emphasise on having a better understanding about the existing curricula available in Europe in Electrical and Information Engineering.

Objectives

The objectives of the task are to provide information and synthesis on the existing curricula in EIE in Europe, and also to get a snapshot of the implementation of the Bologna-process in Europe in our disciplines. The main purpose is to have a better understanding of the present and evolving situation and to provide this information to other universities in Europe and also outside Europe: it will be possible so to enhance the attractiveness of the European HEI and curricula landscape to extra-European students.

In the curricula we are interested in, two kinds of difficulties could be identified:

-The first are structural difficulties due to the fact that higher education systems are different from one country to another one, and also because the diploma accreditation procedures are also different, even if the situation is a little bit clearer thanks to the Bologna-process implementation. This set of difficulties has to be identified in order to well understand the various national practices.

-The second concerns some fundamental aspects of the field or pedagogical approach: it is interesting for us as "specialists" to identify what are the difficulties at the moment to think about or elaborate common or shared curricula.

This study is made in 2 stages. The first stage is to identify the present structures of institutions, curricula, diploma and gateways in the various European systems, in order to have a common global view.

The second point consists in characterising with more details the contents of the curricula/diploma in Electrical and Information Engineering thanks to a set of disciplines/keywords (i.e. a common glossary).

Various target groups have been identified: professional engineering associations, life long learning institutions, students, teachers and accreditation boards.

Activities

For that, the main activities presently running on concerning that activity are the following ones:

-Completion and update of the study (monograph) achieved during the THEIERE project, integration of international curricula. -Definition of a tool, for the research of curricula abroad.

A sub-task on international degrees available in Europe at the master level is also proposed in this project. The purpose is to analyse these international courses: languages used, organisation (actual or virtual mobility...), accreditation and recognition.

Partnership

Within the project, some national contacts have been identified. These national contacts have the responsibility to provide and check the information relative to their country. In some countries, several partners work together, in some countries (France) the French group lays on the strong partnership with the Club EEA.

Background

The background of the project consists in the results of the THEIERE (Thematic Harmonisation in Electrical and Information EngineeRing in Europe) project (www.eaeeie.org/theiere) during which was set up an observatory on the implementation of the Bologna process in Europe, in Electrical and Information Engineering. This observatory resulted in a monograph [START_REF] Thiriet | Monograph: Towards the harmonisation of Electrical and Information Engineering Education in Europe[END_REF] which was published and sent all around Europe (700 copies).

Methodology

Starting from the monographs developed during the THEIERE TN (2000TN ( -2003)), and THEIERE-DISS TN (2004-05) projects, the aim is to complete the missing information as well as to update the existing one, because the situation is always evolving.

PhD studies are completed, together with the identification of the existing links between masters and PhDs, taking into account the master courses which will prepare the students for PhD studies. A census of the existing international curricula in EIE will also be integrated in the monographs.

This task also includes a survey about mobility in order to build-up a portal that underlines the key issues to improve mobility in our fields.

As shown in Figure 2, we want to point out that core activities deal with competencies (Tuning activity) which are an important element for the implementation of accreditation procedures, the mean to reach this purpose being the Bologna process approach.

-"Overview per country" : This part, the most important of the monograph, consists in several files ; one per country, with a description of the respective Higher Education system in our fields, the delimitation of disciplinary fields, an explanation of the processes for the definition, the validation, and the accreditation of the contents of the curricula and the degree systems. Some statistics will be provided, allowing interested people to get an idea on the weight of our disciplines in the various countries.

Bologna Process

Tuning activity

Quality assurance Accreditation

Figure 2: The role of competencies in the Bologna process -Two appendices give the monograph a non negligible interest: Appendix A proposes a list of the institutions providing curricula in our fields, with their internet sites. Appendix B proposes, generally as tables, the list of degrees in our fields in each European university. Because of the present reflections about Bologna process, some of these tables will need to be updated quite soon, but the general interest to show which are the specificities of the universities as far as Electrical and Information Engineering is concerned, remains important.

Planned outputs

The main planned outputs for this task of this three-year project are:

• Guidelines for a proposition for an accreditation methodology, together with a census on accreditation boards and methodologies (this work is in fact done mainly in a dedicated task [START_REF] Barbosa | Accreditation of higher education in EIE in Europe[END_REF] but some results will be used in these outcomes). • Update of the monograph on the degrees and international degrees available in EIE in Europe, available both as a book and as website.

• Update of the monograph on the Doctoral studies in Europe.

The described outcomes will be disseminated to various universities, inside and outside the project partnership, as well as institutions at the national levels (ministries of Higher Education, Professional bodies, accreditation bodies) and European level (European Commission, Thematic networks, European societies…). Some other partners outside Europe will be also targeted, but may be not at a wider level at first. These outcomes will contribute at showing the important work which is done in these directions at the European level, and also to contribute at rendering the EIE-Surveyor thematic network project as well as the EAEEIE association to act as a Reference Point for Electrical and Information Engineering in Europe.

Conclusion

The TN purpose is to participate in the setting of the academic world for tomorrow in the fields of Electrical and Information Engineering. The task presented in this paper is dedicated to the analysis of existing curricula in EIE in Europe and to the understanding of the implementation of the Bologna-process in the various European countries.

The results of the tasks, both as a website and a paper monograph, will be widely disseminated and very useful both for students interested by mobility and by academics interested in developing a new common multinational curriculum.

These various aspects will be also very useful in order to have a better understanding of the functioning of the accreditation methods, since these procedures are presently very different from one country to another, and these differences are an obstacle to the recognition of diploma all around Europe.

Figure 1 :

 1 Figure 1: Cover of the THEIERE monograph

Figure 3 :

 3 Figure 3: Example of a table

Acknowledgments

The authors would like to thank the European Commission for its supports and grants.

The authors would like to thank all the EIE-Surveyor partners for their contribution.


L Baratchart

J Leblond

F Seyfert

André Boivin

Constrained L 2 -approximation by polynomials on subsets of the circle

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.

Introduction

This paper deals with best approximation to a square summable function, on a finite union I of arcs of the unit circle T, by a polynomial of fixed degree which is bounded by 1 in modulus on the complementary system of arcs J = T \ I. This we call, for short, the polynomial problem. We are also concerned with the natural limiting version when the degree goes large, namely best approximation in L 2 (I) by a Hardy function of class H 2 which is bounded by 1 on J. To distinguish this issue from the polynomial problem, we term it the analytic problem. The latter is a variant, involving mixed norms, of constrained extremal problems for analytic functions considered in [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF][START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF][START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of toeplitz operators[END_REF][START_REF] Leblond | Constrained approximation and interpolation in hilbert function spaces[END_REF][START_REF] Schneck | Constrained optimization in hardy spaces[END_REF]. As we shall see, solutions to the polynomial problem converge to those of the analytic problem as the degree tends to infinity, in a sense to be made precise below. This is why solving for high degree the polynomial problem (which is finite-dimensional) is an interesting way to regularize and approximately solve the analytic problem (which is infinite-dimensional). This is the gist of the present work.

Constrained extremal problems for analytic functions, in particular the analytic problem defined above, can be set up more generally in the context of weighted approximation, i.e. seeking best approximation in L 2 (I, w) where w is a weight on I. In fact, that kind of generalization is useful for applications as we shall see. As soon as w is invertible in L ∞ (I), though, such a weighted problem turns out to be equivalent to another one with unit weight, hence the present formulation warrants most practical situations. This property allows one to carry the analytic problem over to more general curves than the circle. In particular, in view of the isomorphism between Hardy spaces of the disk and the half-plane arising by composition with a Möbius transform [10, ch. 10], best approximation in L 2 (I) from H 2 of the disk can be converted to weighted best approximation in L 2 (I, w) from the Hardy space h 2 of a half-plane with I a finite union of bounded intervals on the line and w a weight arising from the derivative of the Möbius transform. Since this weight is boundedly invertible on I, it follows that the analytic problem on the circle and its analog on the line are equivalent. One may also define another Hardy space H 2 , say of the right half-plane as the space of analytic functions whose L 2 -means over vertical lines are uniformly bounded. Then, best approximation in L 2 (I) from H 2 is equivalent to best approximation from H 2 in L 2 (I), i.e. weight is no longer needed. Of course, such considerations hold for many other domains and boundary curves than the half-plane and the line, but the latter are of special significance to us as we now explain.

Indeed, on the line, constrained extremal problems for analytic functions naturally arise in Engineering when studying deconvolution issues, in particular those pertaining to system identification and design. This motivation is stressed in [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF][START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF][START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF][START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF][START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of toeplitz operators[END_REF], whose results are effectively used today to identify microwave devices [START_REF] Baratchart | Identification d'un filtre hyperfréquences par approximation dans le domaine complexe[END_REF][START_REF] Olivi | Identification of microwave filters by analytic and rational h 2 approximation[END_REF]. More precisely, recall that a linear time-invariant dynamical system is just a convolution operator, hence the Fourier-Laplace transform of its output is that of its input times the Fourier-Laplace transform of its kernel. The latter is called the transfer-function. Now, by feeding periodic inputs to a stable system, one can essentially recover the transfer function pointwise on the line, but typically in a restricted range of frequencies only, corresponding to the passband of the system, say I [START_REF] Doyle | Feedback Control Theory[END_REF]. Here, the type of stability under consideration impinges on the smoothness of the transfer function as well as on the precise kind of recovery that can be achieved, and we refer the reader to [6, Appendix 2] for a more thorough analysis. For the present discussion, it suffices to assume that the system is stable in the L 2 sense, i.e. that it maps square summable inputs to square summable outputs. Then, its transfer function lies in H ∞ of the half-plane [START_REF] Partington | Linear operators and linear systems[END_REF], and to identify it we are led to approximate the measurements on I by a Hardy function with a bound on its modulus. Still, on I, a natural criterion from the stochastic viewpoint is L 2 (I, w), where the weight w is the reciprocal of the pointwise covariance of the noise assumed to be additive [START_REF] Pintelon | System Identification: A Frequency Domain Approach[END_REF]. Since this covariance is boundedly invertible on I, we face an analytic problem on the line upon normalizing the bound on the transfer function to be 1. This stresses how the analytic problem on the line, which can be mapped back to the circle, connects to system identification. Now, this analytic problem is convex but infinite-dimensional. Moreover, as Hardy functions have no discontinuity of the first kind on the boundary [START_REF] Garnett | Bounded analytic functions[END_REF]ch. II,ex. 7] and since the solution to an analytic problem generically has exact modulus 1 on J, as we prove later on, it will typically oscillate at the endpoints of I, J which is unsuited. One way around these difficulties is to solve the polynomial problem for sufficiently high degree, as a means to regularize and approximately solve the analytic one. This was an initial motivation by the authors to write the present paper, and we provide the reader in Section 6 with a numerical example on real data from a hyperfrequency filter. It must be said that the polynomial problem itself has numerical issues: though it is convex in finitely many variables, bounding the modulus on J involves infinitely many convex constraints which makes it of so-called semi-infinite programming type. A popular technique to handle such problems is through linear matrix inequalities, but we found it easier to approximate from below the polynomial problem by a finite-dimensional one with finitely many constraints, in a demonstrably convergent manner as the number of these constraints gets large.

The organization of the paper is as follows. In section 3 we set some notation and we recall standard properties of Hardy spaces. We state the polynomial and analytic problems in Section 4, where we also show they are well-posed. Section 5 deals with the critical point equations characterizing the solutions, and with convergence of the polynomial problem to the analytic one. Finally, we report on some numerical experiment in Section 6.

Notations and preliminaries

Throughout we let T be the unit circle and I ⊂ T a finite union of nonempty open arcs whose complement J = T \ I has nonempty interior. If h 1 (resp. h 2 ) is a function defined on a set containing I (resp. J), we put h 1 ∨ h 2 for the concatenated function, defined on the whole of T, which is h 1 on I and h 2 on J.

For E ⊂ T, we let ∂E and • E denote respectively the boundary and the interior of E when viewed as a subset of T; we also let χ E for the characteristic function of E and h | E for the restriction of h to E. Lebesgue measure on T is just the image of Lebesgue measure on [0, 2π) under the parametrization θ → e iθ . We denote by |E| the measure of a measurable subset E ⊂ T, and if 1 ≤ p ≤ ∞ we write L p (E) for the familiar Lebesgue space of (equivalence classes of a.e. coinciding) complex-valued measurable functions on E with norm

f L p (E) = 1 2π E |f (e iθ )| p dθ 1/p < ∞ if 1 ≤ p < ∞, f L ∞ (E) = ess. sup θ∈E |f (e iθ )| < ∞.
We sometimes indicate by L p R (E) the real subspace of real-valued functions. We also set

f, g E = 1 2π E f (e iθ )g(e iθ ) dθ (1)
whenever f ∈ L p (E) and g ∈ L q (E) with 1/p + 1/q = 1. If f and g are defined on a set containing E, we write for simplicity f, g E to mean

< f | E , g | E > and f L p (E) to mean f | E L p (E) . Hereafter C(E)
stands for the space of bounded complex-valued continuous functions on E endowed with the sup norm, while C R (E) indicates real-valued continuous functions.

Recall that the Hardy space H p is the closed subspace of L p (T) consisting of functions whose Fourier coefficients of strictly negative index do vanish. We refer the reader to [START_REF] Garnett | Bounded analytic functions[END_REF] for standard facts on Hardy spaces, in particular those recorded hereafter. Hardy functions are the nontangential limits a.e. on T of functions holomorphic in the unit disk D having uniformly bounded L p means over all circles centered at 0 of radius less than 1:

f H p = sup 0≤r<1 1 2π 2π 0 |f (re iθ )| p dθ 1/p if 1 ≤ p < ∞, f H ∞ = sup z∈D |f (z)|. (2)
The correspondence between such a holomorphic function f and its non tangential limit f is one-to-one and even isometric, namely the supremum in ( 2) is equal to f p , thereby allowing us to identify f and f and to drop the superscript . Under this identification, we regard members of H p both as functions in L p (T) and as holomorphic functions in the variable z ∈ D, but the argument (which belongs to T in the former case and to D in the latter) helps preventing confusion. It holds in fact that f r (e iθ ) = f (re iθ ) converges as r → 1 -to f (e iθ ) in L p (T) when f ∈ H p and 1 ≤ p < ∞. It follows immediately from (2) and Hölder's inequality that, whenever g 1 ∈ H p1 and g 2 ∈ H p2 , we have

g 1 g 2 ∈ H p3 if 1/p 1 + 1/p 2 = 1/p 3 .
Given f ∈ H p , its values on D are obtained from its values on T through a Cauchy as well as a Poisson integral [17, ch. 17, thm 11], namely:

f (z) = 1 2 i π T f (ξ) ξ -z dξ , and also f (z) = 1 2 π T Re e iθ + z e iθ -z f (e iθ ) dθ , z ∈ D, (3)
where the right hand side of the first equality in (3) is a line integral. The latter immediately implies that the Fourier coefficients of a Hardy function on the circle are the Taylor coefficients of its power series expansion at 0 when viewed as a holomorphic function on D. In this connection, the space H 2 is especially simple to describe: it consists of those holomorphic functions g in D whose Taylor coefficients at 0 are square summable, namely

g(z) = ∞ k=0 a k z k : g 2 H 2 := ∞ k=0 |a k | 2 < +∞, g(e iθ ) = ∞ k=0 a k e ikθ , (4)
where the convergence of the last Fourier series holds in L 2 (T) by Parseval's theorem (and also pointwise a.e. by Carleson's theorem but we do not need this deep result). Incidentally, let us mention that for no other value of p is it known how to characterize H p in terms of the size of its Fourier coefficients. By the Poisson representation (i.e. the second integral in (3)), a Hardy function g is also uniquely represented, up to a purely imaginary constant, by its real part h on T according to:

g(z) = iImg(0) + 1 2 π T e iθ + z e iθ -z h(e iθ ) dθ , z ∈ D. (5)
The integral in ( 5) is called the Riesz-Herglotz transform of h and, whenever h ∈ L 1 R (T), it defines a holomorphic function in D which is real at 0 and whose nontangential limit exists a.e. on T with real part equal to h. Hence the Riesz-Herglotz transform (5) assumes the form h(e iθ ) + i h(e iθ ) a.e. on T, where the real-valued function h is said to be conjugate to h. It is a theorem of M. Riesz [11,

chap. III, thm 2.3] that if 1 < p < ∞, then h ∈ L p R (T) when h ∈ L p R (T)
. This neither holds for p = 1 nor for p = ∞. A nonzero f ∈ H p can be uniquely factored as f = jw where

w(z) = exp 1 2π 2π 0 e iθ + z e iθ -z log |f (e iθ )| dθ (6)
belongs to H p and is called the outer factor of f , while j ∈ H ∞ has modulus 1 a.e. on T and is called the inner factor of f . That w(z) in ( 6) is well-defined rests on the fact that log

|f | ∈ L 1 if f ∈ H 1 \ {0};
it entails that a H p function cannot vanish on a subset of strictly positive Lebesgue measure on T unless it is identically zero. For simplicity, we often say that a function is outer (resp. inner) if it is equal, up to a unimodular multiplicative constant, to its outer (resp. inner) factor. Closely connected to Hardy spaces is the Nevanlinna class N + , consisting of holomorphic functions in D that can be factored as jE, where j is an inner function and E an outer function of the form

E(z) = exp 1 2π 2π 0 e iθ + z e iθ -z log ρ(e iθ ) dθ , (7)
with ρ a positive function such that log ρ ∈ L 1 (T) (though ρ itself may not be summable). Such a function has nontangential limits of modulus ρ a.e. on T. The Nevanlinna class is instrumental in that [10, thm 2.11] or [11, 5.8, ch.II]. Thus, formula [START_REF] Borwein | Convex Analysis and Nonlinear Optimization[END_REF] defines a H p -function if and only if ρ ∈ L p (T). Let Ĉ = C ∪ {∞} be the Riemann sphere. The Hardy space Hp of Ĉ \ D can be given a treatment parallel to H p upon changing z into 1/z. Specifically, Hp consists of functions in L p (T) whose Fourier coefficients of strictly positive index do vanish; these are, a.e. on T, the complex conjugates of H pfunctions, and they can also be viewed as nontangential limits of functions analytic in Ĉ \ D having uniformly bounded L p means over all circles centered at 0 of radius bigger than 1. We further single out the subspace Hp 0 of Hp , consisting of functions vanishing at infinity or, equivalently, having vanishing mean on T. Thus, a function belongs to Hp 0 if, and only if it is of the form e -iθ g(e iθ ) for some g ∈ H p . For G ∈ Hp 0 , the Cauchy formula assumes the form :

N + ∩ L p (T) = H p , see
G(z) = 1 2 i π T G(ξ) z -ξ dξ , z ∈ Ĉ \ D. (8)
It follows at once from the Cauchy formula that the duality product , T makes H p and Hq 0 orthogonal to each other, and it reduces to the familiar scalar product when p = q = 2. In particular, we have the orthogonal decomposition :

L 2 (T) = H 2 ⊕ H2 0 . (9)
For f ∈ C(T) and ν ∈ M, the space of complex Borel measures on T, we set

ν.f = T f (e iθ ) dν(θ) (10)
and this pairing induces an isometric isomorphism between M (endowed with the norm of the total variation) and the dual of C(T) [17, thm 6.19]. If we let A ⊂ H ∞ designate the disk algebra of functions analytic in D and continuous on D, and if A 0 indicates those functions in A vanishing at zero, it is easy to see that A 0 is the orthogonal space under [START_REF] Duren | Theory of H p spaces[END_REF] to those measures whose Fourier coefficients of strictly negative index do vanish. Now, it is a fundamental theorem of F. and M. Riesz that such measures are absolutely continuous, that is have the form dν(θ) = g(e iθ ) dθ with g ∈ H 1 . The Hahn-Banach theorem implies that H 1 is dual via (10) to the quotient space C(T)/A 0 [11, chap. IV, sec. 1]. Equivalently, H1 0 is dual to C(T)/A under the pairing arising from the line integral :

( ḟ , F ) = 1 2iπ T f (ξ)F (ξ) dξ , (11)
where F belongs to H1 0 and ḟ indicates the equivalence class of f ∈ C(T) modulo A. Therefore, contrary to L 1 (T), the spaces H 1 and H1 0 enjoy a weak-* compactness property of their unit ball. We define the analytic and anti-analytic projections P + and P -on Fourier series by :

P + ∞ n=-∞ a n e inθ = ∞ n=0 a n e inθ , P - ∞ n=-∞ a n e inθ = -1 n=-∞ a n e inθ .
It is a theorem of M. Riesz theorem [11, ch. III, sec, 1] that P + : L p → H p and P -: L p → Hp 0 are bounded for 1 < p < ∞, in which case they coincide with the Cauchy projections:

P + (h)(z) = 1 2iπ T h(ξ) ξ -z dξ, z ∈ D, P -(h)(s) = 1 2iπ T h(ξ) s -ξ dξ, s ∈ Ĉ \ D. (12)
When restricted to L 2 (T), the projections P + and P -are just the orthogonal projections onto H 2 and H2 0 respectively. Although P ± (h) needs not be the Fourier series of a function when h is merely in L 1 (T), it is Abel summable almost everywhere to a function lying in L s (T) for 0 < s < 1 and it can be interpreted as a function in the Hardy space of exponent s that we did not introduce [10, cor. to thm 3.2]. To us it will be sufficient, when h ∈ L 1 , to regard P ± (f ) as the Fourier series of a distribution.

Finally, we let P n denote throughout the space of complex algebraic polynomials of degree at most n. Clearly, P n ⊂ H p for all p.

Two extremal problems

We first state the polynomial problem discussed in Section 2. We call it P BEP (n) for "Polynomial Bounded Extremal Problem":

PBEP(n)

For f ∈ L 2 (I), find k n ∈ P n such that |k n (e iθ )| ≤ 1 for a.e. e iθ ∈ J and

f -k n L 2 (I) = inf g∈Pn |g|≤1 a.e. on J f -g L 2 (I) . (13)
Next, we state the analytic problem from Section 2 that we call ABEP for "Analytic Bounded Extremal Problem":

ABEP Given f ∈ L 2 (I), find g 0 ∈ H 2 such that |g 0 (e iθ )| ≤ 1 a.e. on J and f -g 0 L 2 (I) = inf g∈H 2 |g|≤1 a.e. on J f -g L 2 (I) . (14)
Note that, in ABEP , the constraint |g| ≤ 1 on J could be replaced by |g| ≤ ρ where ρ is a positive function in L 2 (J). For if log ρ ∈ L 1 (J) then, denoting by w 1∨(1/ρ) the outer factor having modulus 1 on I and 1/ρ on J, we find that g ∈ H

| I = |g| | I while |h| | J = |g| | J /ρ. If, however, log ρ / ∈ L 1 (J)
, then we must have J log ρ = -∞ because ρ ∈ L 2 (J), consequently the set of candidate approximants reduces to {0} anyway because a nonzero Hardy function has summable log-modulus. Altogether, it is thus equivalent to consider ABEP for the product f times (w 1∨ρ -1 ) | I . A similar argument shows that we could replace the error criterion . L 2 (I) by a weighted norm . L 2 (I,w) for some weight w which is non-negative and invertible in L ∞ (I).

Then, the problem reduces to ABEP for f (w ρ 1/2 ∨0 ) I . Such equivalences do not hold for P BEP (n) because the polynomial character of k n is not preserved under multiplication by outer factors. Still, the results to come continue to hold if we replace in P BEP (n) the constraint |k n | ≤ 1 by |k n | ≤ ρ on J and the criterion . L 2 (I) by . L 2 (I,w) , provided that ρ ∈ C(J) and that w is invertible in L ∞ (I). Indeed, we leave it to the reader to check that proofs go through with obvious modifications.

After these preliminaries, we are ready to state a basic existence and uniqueness result.

Theorem 1 . Problems P BEP (n) and ABEP have a unique solution. Moreover, the solution g 0 to ABEP satisfies |g 0 | = 1 almost everywhere on J, unless f = g | I for some g ∈ H 2 such that g L ∞ (J) ≤ 1.

Proof. Consider the sets

E n = {g | I : g ∈ P n , g L ∞ (J) ≤ 1}, F = {g | I : g ∈ H 2 , g L ∞ (J) ≤ 1}.
Clearly E n ⊂ F are convex and nonempty subsets of L 2 (I), as they contain 0. To prove existence and uniqueness, it is therefore enough to show they are closed, for we can appeal then to well-known properties of the projection on a closed convex set in a Hilbert space. Since E n = P n ∩F , it is enough in fact to show that F is closed. For this, let g m be a sequence in

H 2 with |g m | | J ≤ 1 and such that (g m ) | I converges in L 2 (I).
Obviously g m is a bounded sequence in L 2 (T), some subsequence of which converges weakly to h ∈ H 2 . We continue to denote this subsequence with g m . The restrictions (g m ) | I a fortiori converge weakly to h | I in L 2 (I), and since the strong and the weak limit must coincide when both exist we find that (g m )

| I converges to h | I in L 2 (I). Besides, (g m ) | J is contained in the unit ball of L ∞ (J) which is dual to L 1 (J), hence some subsequence (again denoted by (g m ) | J ) converges weak-* to some h 1 ∈ L ∞ (J) with h 1 L ∞ (J) ≤ 1. But since (g m ) | J also converges weakly to h | J in L 2 (J), we have that h 1 , ϕ J = lim m→∞ g m , ϕ J = h | J , ϕ J for all ϕ ∈ L 2 (J) which is dense in L 1 (J). Consequently h 1 = h | J , thereby showing that h L ∞ (J) ≤ 1,
which proves that F is closed. Assume now that f is not the trace on I of an H 2 -function which is less than 1 in modulus on I. To prove that |g 0 | = 1 a.e. on J, we argue by contradiction. If not, there is a compact set K of positive measure, lying interior to J, such that g 0 L ∞ (K) ≤ 1 -δ for some 0 < δ < 1; it is so because, by hypothesis, J must consist of finitely many closed arcs, of which one at least has nonempty interior. For K an arbitrary subset of K, consider the Riesz-Herglotz transform of its characteristic function:

h K (z) = 1 2 π K e iθ + z e iθ -z dθ , z ∈ D, (15)
and put w t = exp(th K ) for t ∈ R, which is the outer function with modulus exp t on K and 1 elsewhere. By construction, g 0 w t is a candidate approximant in ABEP for all t < -log(1 -δ). Thus, the map t → f -g 0 w t 2 L 2 (I) attains a minimum at t = 0. Because K is at strictly positive distance from I, we may differentiate this expression with respect to t under the integral sign and equate the derivative at t = 0 to zero which gives us 2Re f -g 0 , h K g 0 I = 0. Replacing g 0 w t by ig 0 w t , which is a candidate approximant as well, we get a similar equation for the imaginary part so that

0 = f -g 0 , h K g 0 I = (f -g 0 )ḡ 0 , h K I . ( 16
)
Let e it0 be a density point of K and I l the arc centered at e it0 of length l, so that Applying now ( 16) with K = I l ∩K and taking into account that (e it0 +e iθ )/(e it0 -e iθ ) is pure imaginary on I, we find in the limit, as l → 0 that

|I l ∩ K)|/l → 1 as l → 0. Since e it + e iθ e it -e iθ - e it0 + e iθ e it0 -e iθ ≤ 2l dist 2 (K, I) for e it ∈ I l ∩ K, e iθ ∈ I, (17)
1 2π I e it0 + e iθ e it0 -e iθ (f -g 0 )ḡ 0 (e iθ ) dθ = 0. (18)
Next, let us consider the function

F (z) = 1 2π I e iθ + z e iθ -z (f -g 0 )ḡ 0 (e iθ ) dθ = - 1 2π I (f -g 0 )ḡ 0 (e iθ )dθ + 1 iπ I (f -g 0 )ḡ 0 (ξ)dξ ξ -z
which is the sum of a constant and of twice the Cauchy integral of (f -(g 0 ) | I )(ḡ 0 ) | I ∈ L 1 (I), hence is analytic in Ĉ\I. Equation [START_REF] Schneck | Constrained optimization in hardy spaces[END_REF] means that F vanishes at every density point of K, and since a.e. point in K is a density point F must vanish identically because its zeros accumulate in the interior of J. Denoting by F + and F -the nontangential limits of F from sequences of points in D or C \ D respectively, we now get from the Plemelj-Sokhotski formulas [11, ch. III] that

0 = F + (ξ) -F -(ξ) = (f -g 0 )(ξ)g 0 (ξ), a.e. ξ ∈ I.
Thus, either g 0 is nonzero a.e. on I, in which case f = (g 0 ) | I and we reach the desired contradiction, or else g 0 ≡ 0. In the latter case, if we put id for the identity map on T, we find that t → f -t id k 2 L 2 (I) has a minimum at t = 0 for each integer k ≥ 0, since e iθ → te ikθ is a candidate approximant for t ∈ [-1, 1]. Differentiating with respect to t and expressing that the derivative at t = 0 is zero, we deduce that all Fourier coefficients of non-negative index of (f -(g 0 ) | I ) ∨ 0 do vanish. This means this last function lies in H2 , but as it vanishes on J it is identically zero, therefore f = (g 0 ) | I in all cases. Remark: the theorem shows that the constraint |g 0 | ≤ 1 on J is saturated in a very strong sense for problem ABEP , namely |g 0 | = 1 a.e. on J unless f is already the trace of the solution on I. In contrast, it is not true that k n L ∞ (J) = 1 unless f = g | I for some g ∈ P n such that g L ∞ (J) < 1. To see this, observe that the set E n is not only closed but compact. Indeed, if we pick distinct points ξ 1 , • • • , ξ n+1 in J and form the Lagrange interpolation polynomials L j ∈ P n such that L j (ξ j ) = 1 and L j (ξ ) = 0 if = j, we get a basis of P n in which the coordinates of every g ∈ P n meeting g L ∞ (J) ≤ 1 are bounded by 1 in modulus. Hence E n is bounded in P n , and since it is closed by the proof of Theorem 1 it is compact. Thus, each f ∈ L 2 (I) has a best approximant from E n , and if (p n ) I is a best approximant to f with p n ∈ P n , then for λ > p n L ∞ (J) we find that p n /λ is a best approximant to f /λ in L 2 (I) which is strictly less than 1 on J. This justifies the remark.

Critical point equations and convergence of approximants

At this point, it is worth recalling informally some basic principles from convex optimization, for which the reader may consult [START_REF] Borwein | Convex Analysis and Nonlinear Optimization[END_REF]. The solution to a strictly convex minimization problem is characterized by a variational inequality expressing that the criterium increases under admissible increments of the variable. If the problem is smooth enough, such increments admit a tangent space at the point under consideration (i.e. the solution) in the variable space. We term it the tangent space to the constraints, and its orthogonal in the dual space to the variable space is called the orthogonal space to the constraints (at the point under consideration). The variation of the objective function must vanish on the tangent space to the constraints to the first order, thereby giving rise to the so-called critical point equation. It says that the gradient of the objective function, viewed as an element of the dual space to the variable space, lies in the orthogonal space to the constraints. If a basis of the latter is chosen, the coordinates of the gradient in this basis are known as the Lagrange parameters. More generally, one can form the Lagrangian which is a function of the variable and of the Lagrange parameters, not necessarily optimal ones. It is obtained by adding the gradient of the criterion, at the considered value of the variable, with the member of the orthogonal space to the constraints defined by the chosen Lagrange parameters. By what precedes, the Lagrangian must vanish at the solution for appropriate values of the Lagrange parameters. One can further define a function of the Lagrange parameters only, by minimizing the Lagrangian with respect to the variable. This results in a concave function which gets maximized at the optimal value of the Lagrange parameters for the original problem. This way, one reduces the original constrained convex minimization problem to an unsconstrained concave maximization problem, called the dual problem. In an infinite-dimensional context, the arguments needed to put this program to work may be quite subtle.

Below we derive the critical point equation for P BEP (n) described in [START_REF] Leblond | Constrained approximation and interpolation in hilbert function spaces[END_REF]. For g ∈ P n define

E(g) = {x ∈ J, |g(x)| = ||g|| L ∞ (J) },
which is the set of extremal points of g on J.

Theorem 2 A polynomial g ∈ P n is the solution to P BEP (n) iff the following two conditions hold:

• ||g|| L ∞ (J) ≤ 1,
• there exists a set of r distinct points x 1 , • • • , x r ∈ E(g) and non-negative real numbers

λ 1 , • • • , λ r , with 0 ≤ r ≤ 2n + 2, such that g -f, h I + r j=1 λ j g(x j )h(x j ) = 0, ∀h ∈ P n . ( 19
)
Moreover the λ j 's meet the following bound

r j=1 λ j ≤ 2||f || 2 L 2 (I) . ( 20
)
We emphasize that the set of extremal points {x j , j = 1, . . . , r} is possibly empty (i.e r = 0).

Proof. Suppose g verifies the two conditions and differs from the solution k n . Set h = k n -g ∈ P n and observe that Re g(

x i )h(x i ) = Re g(x i )k n (x i ) -1 ≤ 0, i = 1 . . . r. ( 21
)
From the uniqueness and optimality of k n we deduce that

||k n -f || 2 L 2 (I) = ||g -f + h|| 2 L 2 (I) = ||g -f || 2 L 2 (I) + ||h|| 2 L 2 (I) + 2Re g -f, h I < ||g -f || 2 L 2 (I) .
Consequently Re g -f, h L 2 (I) < 0 which, combined with (21), contradicts [START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF]. Conversely, suppose that g is the solution to P BEP (n) and let φ 0 be the R-linear forms on P n given by φ 0 (h) = Re g -f, h I , h ∈ P n .

For each extremal point x ∈ E(g), define further a R-linear form φ x by φ x (h) = Re g(x)h(x) , h ∈ P n .

Put K for the union of these forms:

K = {φ 0 } ∪ {φ x , x ∈ E(g)}.
If we let P R n indicate P n viewed as a real vector space, K is a subset of the dual (P R n ) * . As J is closed by definition, simple inspection shows that K is closed and bounded in (P R n ) * (it is in fact finite unless g is a constant), hence it is compact and so is its convex hull K as (P R n ) * is finite-dimensional. Suppose for a contradiction that 0 ∈ K. Then, since (P R n ) * * = P R n because P R n is finite-dimensional, there exists by the Hahn-Banach theorem an h 0 ∈ P n such that,

φ(h 0 ) ≥ τ > 0, ∀φ ∈ K.
The latter and the continuity of g and h 0 ensure the existence of a neighborhood V of E(g) on T such that for x in U = J ∩ V we have Re g(x)h 0 (x) ≥ τ 2 > 0, whereas for x in J\U it holds that |g(x)| ≤ 1 -δ for some δ > 0. Clearly, for > 0 with ||h 0 || L ∞ (J) < δ, we get that sup

J\U |g(x) -h 0 (x)| ≤ 1. ( 22
)
Moreover, assuming without loss of generality that < 1, it holds for x ∈ U that

|g(x) -h 0 (x)| 2 = |g(x)| 2 -2Re g(x)h 0 (x) + 2 |h 0 (x)| 2 ≤ |g(x)| 2 -2Re g(x)h 0 (x) + 2 |h 0 (x)| 2 ≤ 1 -τ + 2 ||h 0 || 2 L ∞ (J) .
The latter combined with (22) shows that, for sufficiently small, we have

||g -h 0 || L ∞ (J) ≤ 1. (23)
However, since

||f -g -h 0 || 2 L 2 (J) = ||f -g|| 2 L 2 (J) -2 φ 0 (h 0 ) + 2 ||h 0 || 2 L 2 (J) ≤ ||f -g|| 2 L 2 (J) -2 τ + 2 ||h 0 || 2 L 2 (J) , (24)
we deduce in view of (23) that for small enough the polynomial g -h 0 performs better than g in F BEP , thereby contradicting optimality. Hence 0 ∈ K, therefore by Carathedory's theorem [8, ch. 1, sec. 5] there are r elements γ j of K, with 1 ≤ r ≤ 2(n + 1) + 1 (the real dimension of P R n plus one), such that r j=1

α j γ j = 0 (25)
for some positive α j satisfying α j = 1. Of necessity φ 0 is a γ j , otherwise evaluating (25) at g yields the absurd conclusion that

0 = r j=1 α j γ j (g) = r j=1 α j |g(x j )| 2 = 1.
Equation (25) can therefore be rewritten as

α 1 Re f -g, h I + r j=2 α j Re(g(x j )h(x j )) = 0 ∀h ∈ P n , α 1 = 0.
Dividing by α 1 and noting that the last equation is also true with ih instead of h yields [START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF] with r = r -1. Finally, replacing h by g in ( 19) we obtain r j=1

|λ j | = r j=1 λ j | = f -g, g I ≤ f -g, f -g I + | f -g, f I | ≤ ||f -g|| 2 L 2 (I) + ||f -g|| L 2 (I) ||f || L 2 (I) ≤ 2||f || 2 L 2 (I)
where the next to last majorization uses the Schwarz inequality and the last that 0 is a candidate approximant for P BEP (n) whereas g is the optimum.

The next result describes the behavior of k n when n goes to infinity, in connection with the solution g 0 to ABEP . Theorem 3 Let k n be the solution to P BEP (n) defined in [START_REF] Leblond | Constrained approximation and interpolation in hilbert function spaces[END_REF], and g 0 the solution to ABEP described in [START_REF] Olivi | Identification of microwave filters by analytic and rational h 2 approximation[END_REF]. When n → ∞, the sequence (k n ) | I converges to (g 0 ) | I in L 2 (I) , and the sequence (k n ) | J converges to (g 0 ) | J in the weak-* topology of L ∞ (J), as well as in L p (J)-norm for 1 ≤ p < ∞ if f is not the race on I of a H 2 -function which is at most 1 in modulus on J. Altogether this amounts to:

lim n→∞ ||g 0 -k n || L p (T) = 0, 1 ≤ p ≤ 2, (26)
lim n→∞ k n , h J = g 0 , h J ∀h ∈ L 1 (J), ( 27
) if f = g 0 on I, lim n→∞ ||g 0 -k n || L p (J) = 0, 1 ≤ p < ∞. (28)
Proof. Our first objective is to show that g 0 can be approximated arbitrary close in L 2 (I) by polynomials that remain bounded by 1 in modulus on J. By hypothesis I is the finite union of N ≥ 1 open disjoint sub-arcs of T. Without loss of generality, it can thus be written as

I = N i=1 (e iai , e ibi ), 0 = a 1 ≤ b 1 ≤ a 2 • • • ≤ b N ≤ 2π.
Let ( n ) be a sequence of positive real numbers decreasing to 0. We define a sequence (v n ) in H 2 by

v n (z) = g 0 (z) exp - 1 2π N i=1 ai+ n ai e it + z e it -z log |g 0 |dt + bi bi-n e it + z e it -z log |g 0 |dt
Note that indeed v n ∈ H 2 for n large enough because then it has the same modulus as g 0 except over the arcs (a i , a i + n ) and (b in , b i ) where it has modulus 1. We claim that (v n ) | I converges to g 0 in L 2 (I) as n → ∞. To see this, observe that v n converges a.e. on I to g 0 , for each z ∈ I remains at some distance from the sub-arcs (a i , a i + n ) and (b i , b i + n ) for all n sufficiently large, hence the argument of the exponential in (29) converges to zero as n → ∞ by absolute continuity of log |g 0 |dt. Now, we remark that by construction |v n | ≤ |g 0 | + 1, hence by dominated convergence, we get that

lim n→∞ ||g 0 -v n || L 2 (I) = 0.
This proves the claim. Now, let > 0 and 0 < α < 1 such that ||g 0 -αg 0 || L 2 (I) ≤ 4 . Let also n 0 be so large that ||v n0 -g 0 || L 2 (I) ≤ 4 . For 0 < r < 1 define u r ∈ A (the disk algebra) by u r (z) = v n0 (rz) so that, by Poisson representation,

u r (e iθ ) = T P r (θ -t)v n0 (re it )dt,
where P r is the Poisson kernel. Whenever e iφ ∈ J, we note by construction that |v n | = 1 a.e on the sub-arc (e i(φ-n 0 ) , e i(φ+ n 0 )). This is to the effect that

|u r (e iφ )| ≤ T P r (φ -t)|v n0 (re it )|dt ≤ P r ( n0 ) T |v n0 (re it )|dt + + n 0 -n 0 P r (t)dt ≤ P r ( n0 )||v n0 || L 1 (T) + 1 ≤ P r ( n0 )||v n0 || L 2 (T) + 1
by Hölder's inequality. Hence, for r sufficiently close to 1, we certainly have that |u r | ≤ 1/α 2 on J and otherwise that ||u r -v n0 || 2 L 2 (I) ≤ 4 since u r → v n0 in H 2 . Finally, call q the truncated Taylor expansion of u r (which converges uniformly to the latter on T), where the order of truncation has been chosen large enough to ensure that |q| ≤ 1/α on J and that ||q -u r || 2 L 2 (I) ≤ 4 . Then, we have that

||αq -g 0 || L 2 (I) ≤ α ||q -u r || L 2 (I) + ||u r -v n0 || L 2 (I) + ||v n0 -g 0 || L 2 (I) + ||g 0 -αg 0 || L 2 (I)
≤ .

Thus, we have found a polynomial (namely αq) which is bounded by 1 in modulus on J and close by to g 0 in L 2 (I). By comparison, this immediately implies that

lim n→∞ ||f -k n || L 2 (I) = ||f -g 0 || L 2 (I) , (29)
from which (26) follows by Hölder's inequality. Moreover, being bounded in H 2 , the sequence (k n ) has a weakly convergent sub-sequence. The traces on J of this subsequence are in fact bounded by 1 in L ∞ (J)norm, hence up to another subsequence we obtain (k nm ) converging also in the weak-* sense on J. Let g be the weak limit (H 2 sense) of k nm , and observe that g | J is necessarily the weak-* limit of (k nm ) | J in L ∞ (J), as follows by integrating against functions from L 2 (J) which is dense in L 1 (J). Since balls are weak-* closed in L ∞ (J), we have that ||g|| L ∞ (J) ≤ 1, and it follows from (29) that ||f -g|| L 2 (I) = ||f -g 0 || L 2 (I) . Thus, g = g 0 by the uniqueness part of Theorem 1. Finally, if f = g 0 on J, then we know from Theorem 1 that |g 0 | = 1 a.e. on J. In this case, (29) implies that lim sup ||k nm || L 2 (T) ≤ ||g 0 || L 2 (T) , and since the norm of the weak limit is no less than the limit of the norms it follows that (k nm ) | J converges strongly to (g 0 ) | J in the strictly convex space L 2 (J). The same reasoning applies in L p (J) for 1 < p < ∞. Finally we remark that the preceding arguments hold true when k n is replaced by any subsequence of itself; hence k n contains no subsequence not converging to g 0 in the sense stated before, which achieves the proof.

We come now to an analog of theorem 2 in the infinite dimensional case. We define H 2,∞ J and H 2,1 I to be the following vector spaces:

H 2,∞ J = {h ∈ H 2 , ||h|| L ∞ (J) < ∞}, H 2,1 I = {h ∈ H 1 , ||h|| L 2 (I) < ∞},
endowed with the natural norms. We begin with an elementary lemma.

Lemma 1 Let v ∈ L 1 (J) such that P + (0 ∨ v) ∈ H 2,1 I . Then:

∀h ∈ H 2,∞ J , P + (0 ∨ v), h T = v, h J .
Proof. Let u be the function defined on T by

u = (0 ∧ v) -P + (0 ∨ v).
By assumption u ∈ L 1 (T), and by its very definition all Fourier coefficients of u of non-negative index vanish. Hence u ∈ H1 0 , and since it is L 2 integrable on I where it coincides with -P + (0 ∨ v), .we conclude that u ∈ H 2,1 I and that u(0) = 0 Now, for h ∈ H 2,∞ J we have that

vχ J , h T = u, h T + P + (0 ∨ v), h T = u(0)h(0) + P + (0 ∨ v), h T = P + (0 ∨ v), h T (30)
where the second equality follows from the Cauchy formula because (uh) ∈ H 1 .

Theorem 4 Suppose that f ∈ L 2 (I) is not the trace on I of a H 2 -function of modulus less or equal to 1 a.e on J. Then, g ∈ H 2 is the solution to ABEP iff the following two conditions hold.

• |g(e iθ )| = 1 for a.e. e iθ ∈ J,

• there exists a nonnegative real function λ

∈ L 1 R (J) such that, ∀h ∈ H 2,∞ J , g -f, h I + λg, h J = 0. (31)
Proof. Suppose g verifies the two conditions and differs from g 0 . Set h = (g 0 -g) ∈ H2,∞ J and observe that Re λg, h J = 1 2π J λ(Re(gg 0 ) -1) ≤ 0.

In another connection, since -h is an admissible increment from g 0 , the variational inequality characterizing the projection onto a closed convex set gives us (cf. Theorem 1) Re g 0 -f, h I ≤ 0, whence Re g -f, h I = Re g 0 -f, h I -h, h I < 0 which, combined with (32), contradicts (31). Suppose now that g is the solution of ABEP . The property that |g| = 1 on J has been proven in Theorem 1. In order to let n tend to infinity, we rewrite [START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF] with self-explaining notations as

k n -f, e imθ I + r(n) j=1 λ n j k n (e iθ n j )e imθ n j = 0, ∀m ∈ {0 . . . n}, . (33)
We define (Λ n ), n ∈ N, to be a family of linear forms on C(J) defined as

Λ n (u) = r(n) j=1 λ n j k n (e iθ n j )u(e θ n j ), ∀u ∈ C(J).
Equation (20) shows that (Λ n ) is a bounded sequence in the dual C(J) * which by the Banach-Alaoglu theorem admits a weak-* converging subsequence whose limit we call Λ. Moreover, the Riesz representation theorem ensures the existence of a complex measure µ to represent Λ so that, appealing to Theorem 3 and taking the limit in (33), we obtain g 0 -f, e imθ I + J e imθ dµ = 0, ∀m ∈ N..

Now, the F. and M. Riesz theorem asserts that the measure which is µ on J and (g 0 -f )dθ on I is absolutely continuous with respect to Lebesgue measure, because its Fourier coefficients of nonnegative index do vanish, by (34). Therefore there is v ∈ L 1 (J) such that,

g 0 -f, e imθ I + v, e imθ J = 0, ∀m ∈ N , which is equivalent to g 0 -f, e imθ I + λg 0 , e imθ J = 0, ∀m ∈ N, (35)
where we have set λ(z) = v(z)g 0 (z) ∀z ∈ J. Equation (35) means that

P + ((g 0 -f )χ I ) = -P + (0 ∨ λg 0 ),
which indicates that P + (0 ∨ λg 0 ) lies in H 2 . Thus, thanks to Lemma 1, we get that

g 0 -f, u I + λg 0 , u J = 0, ∀u ∈ H 2,∞ J . (36)
In order to prove the realness as well as the nonnegativity of λ, we pick h ∈ C ∞ c,R (I), the space of smooth real-valued functions with compact support on I, and we consider its Riesz-Herglotz transform b

(z) = 1 2 π I e it + z e it -z h(e it ) dt = 1 2 π T e it + z e it -z χ I (e it ) h(e it ) dt . (37)
It is standard that b is continuous on D [11, ch. III, thm. 1.3]. For t ∈ R, define ω t = exp(tb) which is the outer function whose modulus is equal to exp th on I and 1 on J. The function g 0 ω λ is a candidate approximant in problem ABEP , hence t → f -g 0 ω t the boundedness of b, we may differentiate this function with respect to t under the integral sign, and equating the derivative to 0 at t = 0 yields 0 = Re (f -g 0 )g 0 , b I = Re (f -g 0 ), bg 0 I .

In view of (36), it implies that 0 = Re λg 0 , bg 0 J = Re λ, b J , where we used that |g 0 | ≡ 1 on J. Remarking that b is pure imaginary on J, this means

Im(λ), b L 2 (J) = 0, ∀h ∈ C ∞ c,R (

I).

Letting h = h m range over a sequence of smooth positive functions which are approximate identies, namely of unit L 1 (I)-norm and supported on the arc [θ -1/m, θ + 1/m] with e iθ ∈ I, we get in the limit, as m → ∞, that Im(λ), (e iθ + .)/(e iθ -.) J = 0, e iθ ∈ I.

Then, appealing to he Plemelj-Sokhotski formulas as in the proof of Theorem 1, this time on J, we obtain that Im(λ) = 0 which proves that λ is real-valued. Note that the argument based on the Plemelj-Sokhotski formulas and the Hahn-Banach theorem together imply that the space generated by ξ → (e iθ +ξ)/(e iθ -ξ), as e iθ ranges over an infinite compact subset lying interior to J, is dense in L p (I) for 1 < p < ∞. In fact using the F. and M. Riesz theorem and the Plemelj-Sokhoski formulas, it is easy to see that such functions are also uniformly dense in C(I). Then, using that ABEP is a convex problem, we obtain upon differentiating once more that Re (g 0 -f )ḡ 0 , b 2 I ≥ 0, which leads us by (36) to Re λ, ((e iθ + .)/(e iθ -.)) 2 J = Re λg 0 , g 0 ((e iθ + .)/(e iθ -.)) 2 J ≤ 0, e iθ ∈ I.

By the density property just mentioned this implies that ((e iθ + .)/(e iθ -.)) 2 | Ī is dense in the set of nonpositive continuous functions on I, therefore λ ≥ 0. Note also that (35) implies (f -g 0 ) ∨ λg 0 ∈ H1 , hence it cannot vanish on a subset of T of positive measure unless it is the zero function. But this would imply f = g a.e on I which contradicts the hypothesis. This yields λ > 0 a.e on J.

A numerical example

For practical applications the continuous constraint of P BEP on the arc J is discretized in m + 1 points. Suppose that J = {e it , t ∈ [-θ, θ]}, for some θ ∈ [0, π]. Call J m the discrete version of the arc J defined by

For the discretized problem DBEP(n,m), the following holds.

where λ ≥ 0 means that each component of λ is non negative.

• For a fixed n, lim m→∞ k n,m = k n in P n .

The proof of Theorem 5 follows from standard convex optimization theory, using in addition that the sup-norm of the derivative of a polynomial of degree n on T is controlled by the values it assumes at a set of n+1 points. This depends on Bernstein's inequality and on the argument using Lagrange interpolation polynomials used in the Remark after Theorem 1.

In the minmax problem (39) , the minimization is a quadratic convex problem. It can be tackled efficiently by solving the critical point equation which is a linear system of equations similar to [START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF]. Eventually, an explicit expression of the gradient and of the hessian of the concave maximization problem (39) allows us for a fast converging computational procedure to estimate k n,m .

Figure [START_REF] Baratchart | Identification d'un filtre hyperfréquences par approximation dans le domaine complexe[END_REF] represents a solution to problem DBEP(n,m), where f is obtained from partial measurement of the scattering reflexion parameter of a wave-guide microwave filter by the CNES (French Space Agency). The problem is solved for n = 400 and m = 800, while the constraint on J has been renormalized to 0.96 (instead of 1). The modulus of k 400,800 is plotted as a blue continuous line while the measurements |f | appear as red dots. As the reader can see, the fit is extremely good.

it follows by dominated convergence that lim l→0 1

 1 |I l ∩ K| I l ∩K e it + e iθ e it -e iθ -e it0 + e iθ e it0 -e iθ dt = 0, uniformly w.r. to e iθ ∈ I, and therefore that lim l→0 h I l ∩K (e iθ ) |I l ∩ K| = e it0 + e iθ e it0 -e iθ uniformly w.r. to e iθ ∈ I.

J

  m = {e it , t ∈ {-θ + 2kθ m , k ∈ {0 . . . m}}we define following auxiliary extremal problem:DBEP(n,m) For f ∈ L 2 (I), find k n,m ∈ P n such that ∀t ∈ J m |k n,m(t)| ≤ 1 and f -k n,m L 2 (I) = min g∈Pn |g|≤1 a.e. on Jm f -g L 2 (I) .

Theorem 5 Figure 1 :

 51 Figure 1: Solution of DBEP at hand of partial scattering measurements of a microwave filter

  2 satisfies |g| ≤ ρ on J if and only if h = gw 1∨(1/ρ) lies in H 2 and satisfies |h| ≤ 1 on J. It is so because, for g as indicated, h lies in the Nevanlinna class by construction and |h|

L 2 (I) reaches a minimum at t = 0. By