A proof of stability of all‐pass PADé approximants TO e‐s

Abstract

This paper presents an explicit derivation of the all‐pass Routh approximants to e^‐s. It then shows that the stable all‐pass Routh approximants to e^‐s are exactly the same as the all‐pass Padé approximants. As a result, the derivation provides an alternative way of proving that the all‐pass Padé approximants to e^‐s are always asymptotically stable.     

Model simplification of stable discrete‐time systems by the squared magnitude Padé approximation

Abstract

A squared magnitude Padé approximation technique is presented for the model simplification of stable discrete‐time systems. The simplification is started from the squared magnitude function M(e^jTω ) =G(e^jTω )G(e^–jTω ), where G(z) is the z‐transfer function of a given high order discrete‐time system. The method is fully computer‐oriented and leads to a satisfactory approximation while preserving stability and minimum‐phase characteristics.     

Meletius Dutkiewicz (1836˜1897)

Abstract

Dutkiewicz, a fascinating and in some ways tragic character, has had no luck with biographers. Apart from one article in a photographic monthly ¹ Polish Biographical Dictionary, Cracow (1948), Vol. 6, pp. 15-16. This is in fact a résumé of an article by Aleksander Karoli, ‘In memoriam of Meletius Dutkiewicz’, published in a photographic monthly of Warsaw, Swiat?o (Light), No. 10 (1899), pp. 448-452. , and a brief entry in the Polish Biographical Dictionary, no published information concerning him has been traced. However, much that is of interest can be gleaned from surviving correspondence between the eminent Polish photographers Karol Beyer, Joseph Kordysz and Michal Greim ² Only a part of this correspondence was published by Juliusz Garztecki in his book on Michal Greim Mistrz zapominiany (Forgotten Master), Wydawnictwo Literackie, Cracow (1972). .     

Computing Parametric Geometric Resolutions

 Given a polynomial system of n equations in n unknowns that depends on some parameters, we define the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d ⁿ , where d is a bound on the degrees of the input system. Then we present a probabilistic algorithm to compute a parametric resolution. Its complexity is polynomial in the size of the output and in the complexity of evaluation of the input system. The probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry. Received: July 5, 2001; revised version: September 5, 2002 Key words: Polynomial systems with parameters, Complexity, Theory of elimination, Symbolic Newton operator.     

Jules Itier,Daguerreotypist

Abstract

Jules Itier's work with daguerreotypes is a recent discovery¹. A group of 40 examples has been acquired by the Musée de Bièvres (Essonne), and about 80 plates have been traced to a private collection. This body of work conveys the idea of travel photography as it was practised during the 1840s, although only a few examples of the genre survive². E. Lacan, a critic for La Lumière, commented at the time: ‘the application of daguerreotypy to travel offered more than one practical difficulty, and did not respond to the great need of our century, which was/or popuiarization’³. In the first place, the transportation of a considerable number of silvered plates during a long journey was cumbersome and costly and, above all, the images that were finally brought hack ⁴ were unique. It was possible to make an admirable private collection, but there was no practical way to share with the general public the fascinating views that had been captured in faraway places, often at great expense and with great effort.     

Etude expérimentale du retrait du béton

Résumé Pour évaluer les déformations dues au retrait, on utilise couramment les valeurs indiquées par les normes existantes; toutefois, lorsque l'on effectue des mesures pendant une longue période, on constate que les déformations sont plus importantes que celles préconisées. Dans cet article, nous présentons des résultats expérimentaux pouvant être utilisés pour contr?ler la validité des modèles existants ou pour en développer de nouveaux, car nous avons pris en considération plusieurs paramètres. Pour décrire le retrait en fonction du temps des dessiccation, nous avons admis la fonction ɛ=a(t/b+t) ⁿ . Les coefficientsa, b etn ont été déterminés par la méthode des moindres carrés. Des mesures effectuées on peut tirer les conclusions suivantes: (i) le retrait final diminue lorsque le diamètre des échantillons augmente; (ii) le temps nécessaire pour atteindre la moitié du retrait final est, en première approximation, de la formet _1/2=A _ϕ+B _ϕ ²; (iii) l'exposantn augmente avec le diamètre et tend vers 1; la fonctionn=1−C e^−dϕ semble convenir; (iv) la différence entre le retrait des éprouvettes sciées ou moulées est négligeable; (v) pour un même volume de pate de ciment, le retrait est peu influencé par le rapport E/C; (vi) le superfluidifiant utilisé augmente faiblement le retrait mais cette augmentation est compensée par la diminution que l'on peut opérer sur le rapport E/C; (vii) le dosage en ciment n'influence pas le retrait lorsque l'ouvrabilité du béton est maintenue constante; (viii) la variabilité du retrait diminue fortement si des précautions sont prises pour mesurer les déformations pendant les premières minutes de dessiccation.

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Summary The shrinkage of concrete is generally determined using the standards. When shrinkage is measured over a long period of time, deformation is found to be larger than predicted by the standards. Therefore it seems to be interesting to measure shrinkage on different samples for a relatively long period of time. In this paper, we do not suggest a new model, but present the experimental measurements that can be used to check or develop any model. Our samples are divided into five series according to the examined parameters: (I) cylinders with different diameters; (II) prism with different dimensions; (III) different quality of cement paste; (IV) concrete with same workability but different cement content; (V) variability. The following empirical Equation 1 was used to link shrinkage and time of drying: ɛ=a (t/b+t) ⁿ . The coefficientsa, b andn were fitted using the least-square method. This allows us to see the influence of some factors on shrinkage. All samples were cast using the same aggregates and the same cement type (CPN). In some cases we also used a superplasticizer. Both axial and superficial shrinkage have been measured. As the data seem to fit correctly Equation 1 the following conclusions can be drawn: the final shrinkage,a, is decreasing linearly with increasing diameter. The slope depends on the concrete mixture. It is obvious that for a large diameter final shrinkage will reach an asymptotic line; the timet _1/2 needed to obtain half of the final shrinkage is increasing with the diameter ϕ and, as a first approximation, we gett _1/2=Aϕ+Bϕ²; therefore the theory of diffusion does not apply for small diameters; the coefficientn increases also with the diameter, it reaches asymptotically the value of 1. It varies between 0.45 and 0.8 as seen by our measurements, and the functionn=1−C e^−dϕ fits correctly. If we assume that Equation 1 describes realistically the behaviour of concrete during drying, the coefficientsa andn should not vary with the sample dimension. We should not forget that shrinkage is a complex phenomenon and therefore cannot be described by such a simple equation. From our measurement we can draw the following conclusions: (i) there is no significant difference between the shrinkage of samples cast or sawed; (ii) when the volume of cement paste is constant, shrinkage is almost uninfluenced by the W/C ratio; (iii) the superplasticizer increases shrinkage slightly, but this increase is compensated by the decrease in W/C ratio; (iv) the cement content does not influence the shrinkage, if the workability is constant; (v) variability can be drastically reduced when precautions are taken to measure deformation during the first minutes of drying.

     

Explanation of size effect in concrete fracture using non-uniform energy distribution

A local fracture energy model originally proposed to explain the influence of fracture process zone (FPZ) on fracture energy of cementitious materials is further developed in this study. By assuming a bilinear distribution for the fracture energy distribution, the ligament-dependent fracture energyG _f is obtained. The analytical expression ofG _f contains two important prameters: the intrinsic size-independent fracture energyG _F and a reference ligament sizea _l ^* which determines the intersection of the two linear fracture energy functions. It is shown that the ligament-dependentG _f approaches the size-independentG _F asymptotically. As a result,G _F can be determined from the ligament-dependentG _f results. It is also found that while the reference ligament sizea _l ^* is influenced by the specimen geometry, size and loading conditions, the derived fracture energyG _F is virtually constant. The present local fracture energy distribution model is also discussed and compared with the original local fracture energy model.

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Résumé Cette étude présente un développement plus approfondi d'un modèle de l'énergie locale de rupture, originairement proposé pour expliquer l'effet de la taille de la zone d'endommagement sur l'énergie de rupture des matériaux à base de ciment. En supposant une distribution bilinéaire de l'énergie locale de rupture, on peut obtenir l'énergie de rupture G_f dépendante de la longueur du ligament. L'expression analytique de l'énergie de rupture G_F contient deux paramètres importants: l'énergie de rupture G_F, indépendante de la taille, et une valeur de référence de la longueur du ligament, a _l ^* , qui détermine l'intersection de la distribution bilinéaire de l'énergie locale de rupture. On montre que l'énergie de rupture dépendante du ligament G_f tend asymptotiquement vers la valeur de l'énergie de rupture non-dépendante de la taille, G_F. Ainsi, G_F peut être déduite de l'énergie dépendante du ligament, G_f. On a aussi remarqué que l'énergie de rupture G_F, ainsi déduite, demeure virtuellement constante bien que la logueur de référence du ligament, a _l ^* , puisse être influencée par la géométrie et la taille de l'éprouvette et les conditions de sollicitations. Ce modèle de distribution d'énergie locale de rupture est discuté en détail dans cette contribution et est comparé avec le modèle original de l'énergie locale de rupture.

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Editorial Note Prof. Folker H. Wittmann is a RILEM Senior Member.     

Joseph A. Maybin and Two Steamboat Wrecks

Abstract

Just after 10 a.m. on Monday 17th August 1885, as the steamboat S. M. Felton left its dock in Philadelphia headed for Wilmington, Delaware, an explosion ripped through its forward deck structure. Miraculously, no one was killed, although several people were injured. The ship returned to the dock and, after the passengers were disembarked, was towed to the Pusey & Jones Company in Wilmington for repairs. The ship's captain and others were convinced that the explosion had not been an accident, and that for some unknown reason a ‘veritable infernal device’ had been ‘smuggled’ on board and set to go off to make it look as if the boiler had exploded. No theory was advanced as to who would have done such a thing, although it was pointed out that the Felton was the archrival of the ship Brandywine. Since the two ships frequently left Philadelphia at the same time, races between them were common¹. Another good cause for speculation was the report in the Wilmington Every Evening less than two weeks before the explosion that the Felton charged a fare ‘so low that it is intended to drive all other boats from the river’². In the long run nothing came of these suspicions, and no one was ever convicted in the case, despite the work of a Captain Linden of the Pinkerton Detective Agency.     

Influence des charges sur le comportement mécanique d'un joint collé cisaillé en traction

Résumé Le comportement mécanique d'un joint collé cisaillé en traction dépend de nombreux facteurs tant mécaniques que géométriques. Parmi tous ces paramètres, nous nous sommes proposés d'étudier, tout particulièrement, l'influence des charges sur la tenue d'un collage ainsi que leur r?le sur la microfissuration d'un joint. Après avoir caractérisé la nature minéralogique des charges vulgaires et renfor?antes, nous déterminons les contraintes radiales et tangentielles dans la matrice et dans l'inclusion aux points les plus sollicités mécaniquement. La comparaison avec l'état des contraintes autour d'un trou circulaire nous permet, alors, de déterminer un diamètre limite en-de?à duquel les charges ne jouent plus aucun r?le dans la rupture du film d'adhésif.

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Summary In the course of study of the influence of fillers on the mechanical behaviour of a glued joint, the author was first led to characterize them by spectroscopic analysis and scanning microscopy. These fillers of essentially mineral origin can be classified as common fillers (chalk, calcite, kaolin ...) and reinforcing fillers (aluminia, titanium oxide, ferrous oxide). Roughly spherical in shape, they may reach a diameter of 15 μm, whereas the maximum pore size is close to 20 μm. In a second phase of research, the author determined the radial and tangential stresses in the matrix and in the inclusion at places in the joint under the greatest mechanical stress by a uniform plane field of compressive stresses. The calculation is based on the Mauskhelisvili formulae and shows that the radial and tangential stresses (σ _r andσ _θ respectively) in the matrix are fairly equal and maximum at points M₂ and M ₂ ^′ -extremities of the diameter of the circular load, in the direction of monoaxial loading p. The values ofσ _r andσ _θ are 1.45 and 0.45 respectively whatever the type of reinforcing filler. At M₁ and M ₁ ^′ these stresses are nil (M₁ and M ₁ ^′ are the extremities of the diameter perpendicular to M₂M ₂ ^′ ). In the circular inclusion for the same type of filler, the stress tensor is reduced to a compressive stress of 1,45 p at M₁, M ₁ ^′ , M₂ and M ₂ ^′ . Using these results as a basis and applying the Isha? fracture criterion, the authors show that the onset of microcraking at points M₂ and M ₂ ^′ in the joints occurs for a value of p equal to two thirds of the tensile elastic limit of the resin. By approaching this value of limit load of a circular hole, it is possible to determine a 7μm limit diameter for an inclusion beyond which the filler plays no role in the microcracking of the glued joint. Finally, the authors show that a common fillter fractures well before the onset of microcracking in the matrix. The singularity then behaves like a circular hole under increasing uniform compressive stress p.

     

Interaction of radionuclides with some functional derivatives of chitosan in solutions

New derivatives of low- and high-molecular-weight chitosan (LMWC and HMWC), containing various functional groups, were prepared. The solubility of the compounds in aqueous solutions at pH 1–10 and the sorption and complexation of radionuclides with these compounds were studied. Water-insoluble preparations based on LMWC and HMWC efficiently sorbed ⁹⁰Y and ²³³UO₂²⁺ with the distribution coefficients K _d > 10³ ml g⁻¹ at a phase contact time of 1 h and V/m = 100. The water-soluble preparations based on LMWC showed high complexing power toward ²³³UO₂²⁺ and virtually completely retained hydrolyzed U(VI) species in solutions in a wide pH range. In the presence of N-sulfosuccinyl-LMWC (N-SSLMWC), ²³³UO₂SO₄ solutions with pH 8–9 and U(VI)/N-SSLMWC weight ratio of 0.5 remained stable for 200 days, with no precipitation of hydrolyzed ²³³UO₂²⁺ species.     

Mechanical properties of reactive powder concretes

Reactive Powder Concretes (RPC) are a set of ultrahigh-strength concretes reinforced with steel fibers. Their compressive strength is between 200 and 800MPa, and their flexural strength can reach 140MPa.RPC200 has been studied with respect to compressive strength and two-point loading strength to define its mechanical behavior.RPC800, which has been mostly studied from the point of view of compressive strength, displays hardening elastic non-linear behavior at low stress. This behavior is similar to that of some natural rocks. The critical stress intensity factorK _Ic , and the average fracture energy, , ofRPC200 andRPC800 have been studied experimentally by applying the theory of linear fracture mechanics (compliance method). The fracture energy, which is a measurement of ductility, can reach 40,000 J/m² forRPC200, as compared to 100 to 150 J/m² for ordinary concretes. Fracture energy depends on the volume of fibers added to the concrete. The optimum content is between 2 and 3% by volume.

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Résumé Les Bétons de Poudres Réactives (BPR) constituent une famille de bétons à ultra hautes performances renforcés de fibres métalliques. Leur résistance en compression se situe entre 200 et 800 MPa et leur résistance en flexion peut atteindre 140 MPa. Le BPR200 a été étudié en compression et en flexion 4-points pour définir son comportement mécanique. Le BPR800, étudié principalement en compression, présente un comportement non linéaire élastique à faible contrainte. Ce comportement s'apparente à celui de certaines roches naturelles. Le facteur d'intensité de contrainte K_Ic et l'énergie moyenne de fracturation du BPR20 et du BPR800 ont été étudiés expérimentalement en appliquant la théorie de la mécanique linéaire de la rupture (méthode de la complaisance). L'énergie de fracturation qui caractérise la ductilité peut atteindre 40 000 J/m² pour le BPR200 à comparer à des valeurs de 100 à 150 J/m² pour les bétons traditionnels. L'énergie de fracturation dépend du volume de fibres ajouté au béton. Le dosage optimal se situe entre 2 et 3 vol-%.

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Editorial note Gérard Bernier is working at the Laboratoire de Mécanique et de Technologie, école Normale Supérieure, Cachan (France). This laboratory is a RILEM Associate Member.     

Early scottish photographic exhibitions

Abstract

There is that nerve-shredding time after the finalization of a text before, and immediately after, publication, when the author wonders what new photographs, correspondence, or other invaluable materials will rise up from unknown sources to mock his or her temporary certainties. So far, no one has yet come forward with another print of Camille Silvy's River Scene, France, or La Vallée de l'Huisne, from 1858 to disturb the (admittedly open) conclusions of my book on the photograph.¹ There is, however, new material to add. Pointed in the right direction by Sara Stevenson of the Scottish National Portrait Gallery, I recently read some fascinating documents in the Scottish Record Office. Deposited on indefinite loan there by the Edinburgh Photo graphic Society in 1979 are the papers of the Photographic Society of Scotland (founded 1856, dissolved 1873). Bundies of hundreds of letters, invoices, and other papers document some of the early Edinburgh exhibitions more amply and significantly than any other photographic exhibitions of the time, so far as I am aware.² I concentrate here on the exhibition held in the Winter of 1858/59, at which Silvy's River Scene, France (1858) was first shown.     

Post-cracking behaviour of steel fibre reinforced concrete

Recently, RILEM TC 162-TDF has proposed equivalent,f _eq , and residual,f _R , flexural tensile strength parameters to characterize and simulate the post-cracking behaviour of steel fibre reinforced concrete (SFRC) structures. In the current work, more than two hundred flexural tests are carried out according to the RILEM TC 162-TDF recommendations and the corresponding values off _eq andf _R parameters are evaluated. In series of specimens reinforced with fibres of a distinct length/diameter ratio, similar values off _eq andf _R parameters were obtained in these series. Although a strong correlation betweenf _eq andf _R was determined, a larger scatter off _R values was observed thereby revealingf _eq to be more appropriate for design purposes. A numerical strategy involving a cross sectional layered model and an inverse analysis was developed to evaluate the post-cracking stress-strain and the stress-crack opening diagrams for the tested SFRC. This strategy was also used to determine a relation between the post-cracking strain, ɛ ^pcr , and the crack opening displacement,w, (ɛ ^pcr =w/L _p ) which is useful for evaluating the crack opening when numerical strategies based on a stress-strain approach are used. The obtainedL _p values range from half the specimen cross section height to half the distance between the tip of the notch and the top of the cross section.

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Résumé Récemment, pour caractériser et simuler le comportement post-fissuration en traction du béton renforcé des fibres d'acier, la Commission Technique 162-TDF de la RILEM a proposé des paramètres désignés par résistance équivalente, f_eq, et résistance résiduelle, f_R, à la contrainte en flexion. Dans le travail présent, des valeurs de ces paramètres sont obtenues sur plus de deux cents essais de flexion effectués en accord avec les recommandations du TC 162-TDF de la RILEM. Des valeurs semblables de f_eq et f_R ont été obtenues dans des séries d'éprouvettes renforcées avec des fibres d'un rapport longueur/diamètre distinct. Bien qu'une forte corrélation entre f_eq et f_R ait été déterminée, une plus grande dispersion de valeurs du f_R a été observée, en démontrant que f_eq est plus approprié pour les buts du projet. Pour évaluer les diagrammes contrainte-déformation et contrainte-ouverture après fissuration, une stratégie numérique a été développée, en utilisant un modèle de section et en effectuant une analyse inverse. Cette stratégie a aussi été utilisée pour déterminer une relation entre la contrainte après fissuration, ɛ ^pcr , et l'ouverture de fissure, w, (ɛ ^pcr =w/L _p ) utile pour évaluer l'ouverture de la fissure quand les stratégies numériques sont basées sur une approche contrainte-déformation. Les valeurs de L_p obtenues ont varié entre la demi-hauteur de la section de l'éprouvette et la demi-distance entre l'extrémité de l'entaille et le sommet de la section.

     

Some observations on unsymmetric radial basis function collocation methods for convection–diffusion problems

In this paper, we re‐investigate the unsymmetric radial basis function (RBF) collocation method for solving convection–diffusion problems with high Péclet number as in Power and Barraco (Computers and Mathematics with Applications 2002; 43: 551). By testing different RBFs and different numbers of nodes, we found that the unsymmetric method can still solve high Péclet number problems reasonably well by using more nodes and domain decomposition techniques. Compared to solving one large problem, the domain decomposition method is shown to be very efficient and can improve the accuracy especially when the Péclet number is not that high. From our tests, it seems that the RBFs r⁴ ln r and r⁸ ln r are not very stable, while r⁶ ln r, r⁵ and r⁷ perform very well. Copyright © 2003 John Wiley & Sons, Ltd.     

Early Serbian photography

Abstract

At the onset of the industrial revolution, at the end of the 18th and beginning of the 19th centuries, while all civilized nations of the world were building up their tcchnological resources, Serbia was making its first attempts to free itself from Turkish occupation. When photography was announced in 1839, Serbia was a small country, still partly governed by Turkey. A newly formed class of city dwellers tried eagerly to make up, at an accelerated pace, for all that it had missed in culture and civilization while dominated by the Turks. In their efforts they were aided by numerous émigré Serbs from Vojvodina¹ who were invited and offered positions in government by Duke Milo: ObrenoviC (1780–1860). Many foreign craftsmen, skilled in newly developed techniques, arrived from Western Europe in search of higher earnings, and likewise participated in this cultural awakening.     

Expansion of Liquid 4He Through the Lambda Transition

Zurek suggestedNature 317 , 505; 1985) that the Kibble mechanism, through which topological defects such as cosmic strings are believed to have been created in the early Universe, can also result in the formation of topological defects in liquid ⁴ He, i.e. quantised vortices, during rapid quenches through the superfluid transition. Preliminary experiments (Hendry et al., Nature 368 , 315; 1994) seemed to support this idea in that the quenches produced the predicted high vortex-densities. The present paper describes a new experiment incorporating a redesigned expansion cell that minimises vortex creation arising from conventional hydrodynamic flow. The post-quench line-densities of vorticity produced by the new cell are no more than 10 ¹⁰ m ^–2 , a value that is at least two orders of magnitude less than the theoretical prediction. We conclude that most of the vortices detected in the original experiment must have been created through conventional flow processes.     

A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always difficult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the fictitious nodal values & u circ; used as undetermined coefficients in the MLS approximation, u ^h (x) [u ^h (x)=Φ·& u circ;], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function u ^h (x) is presented here, to enforce the essential boundary conditions. This modified collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive definite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive definite system matrix.     

Cell attachment on diamond-like carbon coating

Preliminary results of diamond-like carbon (DLC) coating with its novel properties with no toxicity have caused a strong interest of commercial manufacturers of surgical implants. DLC coatings were prepared on polymethylmethacrylate (PMMA) at room temperature using ion beam assisted deposition (IBAD). It could be shown by X-ray photoelectron spectroscopy, Auger electron spectroscopy, and Raman spectroscopy that DLC coating prepared by 800 eV CHⁿ⁺ beam bombardment possessed a higher fraction ofsp ³ bonds in the structure of mixedsp ⁿ³ +sp ⁿ² bonding, resulting in a higher hydrophobicity. The results of the cell attachment tests indicated that DLC coatings exhibited low macrophage attachment and provided desirable surface for the normal cellular growth and morphology of the fibroblasts. At the same time, the number of both neutral granulocytes and platelets adhering to DLC coatings decreased significantly. These findings showed that DLC was a better coating with desirable tissue and blood compatibility.     

Application du Théorème D'échantillonnage à la Restitution D'objets Partiellement Connus

On cherche à restituer le mieux possible un signal b(x)dont un appareil nous fournit un enregistrement e(x)=b(x)*?(x); la fonction ?(x) est supposée couper toutes les fréquences de b(x) supérieures à une fréquence de coupure μ₁ = a. Il s'agit de rechercher une représentation b _c(x) du signal, identique à celle qu'aurait fourni un appareil de réponse impulsionnelle A _c(x) plus fine que ?(x) (i.e. dont la fréquence de coupure c est supérieure à a). On montre que l'on doit utiliser toutes les informations, même vagues, que l'on possède a priori sur b(x) afin d'imposer ‘des contraintes’ à b _c(x). On recherche alors, parmi les développements qui satisfont à ces contraintes, le développement b _c ⁿ qui rend minimum l'écart quadratique On montre que, grâce aux contraintes introduites, on peut restituer correctement une fraction importante des fréquences coupées : le développement b _c ⁿ ne diffère en effet pratiquement pas de l'approximation en moyenne quadratique sur -∞, +∞ de la représentation cherchée du signal, dans le système des bases